Stress sharing as cognitive glue for collective intelligences: A computational model of stress as a coordinator for morphogenesis
During evolution, we noticed significant differences between populations in higher fitness values than lower fitness values. To make this apparent, we zoomed into higher fitness values by exponentiating the l2distance as follows:d′=9d9
Given the presence of a developmental step, two separate configurations of the embryo existed during a single cycle of the GA: 1. The state of the embryo prior to development (as the genotype) and 2. The state of the embryo post development (as the phenotype). Consequently, two kinds of fitness existed: the genotypic fitness, which measured the fitness of the genotype; and the phenotypic fitness, which measured the fitness of the phenotype. An exception occurred in hardwired embryos: since they lacked competency, their phenotype was equivalent to their genotype and thus their phenotypic fitness was equivalent to its genotypic fitness.
The genotype of each embryo carried a “reorganization-marker” indicating the kind of developmental reorganization it could undergo during development. Each embryo was categorized into three different kinds based on this gene marker as follows: 1. With stress sharing embryo. 2. Without stress sharing embryo, and 3. Hardwired embryo. The first two categories had a general ability for reorganization, and were termed as “competent”, in the sense of problem-solving ability of active tissues to restore correct shapes despite induced barriers, perturbations, and other interventions [5,54]. Alternatively, the hardwired embryo was not-competent and possessed a fixed structure throughout development.
Our genetic algorithm evolved populations of embryos of three different kinds: 1. Stress sharing population, 2. Without stress-sharing population, and 3. Hardwired population. To create a population of size M; M 2D-matrices of size N were initialized by randomly scrambling the target pattern (also of size N). We initialized grids this way in order to maintain the same number of cell types (0’s and 1’s) across both the initialized embryo and the target. It ensured simplicity and allowed us to focus on reorganization rather than “growing” the required kind of cell types to satisfy the target pattern.
We evolved each population by passing it iteratively through a sequential process of: 1. Development, 2. Selection, and 3. Mutation. During the selection stage, phenotypic fitnesses were used to select the best genotypes for the subsequent generation (Darwinian selection). In order to repopulate the population with variant genotypes we mutated the selected genotypes by stochastically moving a few cells to random locations. At the end of each generation, we plot the phenotypic fitness and the genotypic fitness of the embryo with the highest phenotypic fitness in the population (see Supplement S2 for more details).
Embryos with competency (either with stress sharing or without stress sharing) were given a pre-determined “competency value”. This value provided an upper bound to the number of swaps the cells of the embryo could execute during the developmental step. We also monitored the total distance travelled by these cells by calculating the average Euclidean-distance moved by elements during a generation.
To observe the impact of stress sharing during evolution, we plot the fitness curves of each population (with stress sharing, without stress sharing, and hardwired), over 1000 generations. 95 % confidence intervals (CI) over 10 runs of each experiment were plotted to verify if any of the CI bands of any population overlapped with each other across 1000 generations. A lack of overlap suggested a significant difference in evolutionary dynamics. Further, we measured the statistical difference between different populations at arbitrary points during evolution and carried out a t-test between samples of any two populations to quantify their similarity.
To determine whether a cellular rearrangement process involving stress sharing helps in discovering the optimal morphogenetic process faster, we compared the evolution of three different populations in-silico, each with a different kind of cellular rearrangement process over 1000 generations.
The experiment consisted of running a population of embryos through a genetic algorithm. The genetic algorithm (GA) involved an iterative process of development, selection, and mutation. The developmental step was introduced to allow a genotype to change into a phenotype before selection (Fig. 2). Three different populations were evolved through our GA: 1. a population which relied on stress sharing to rearrange itself; 2. a population which relied on stress but had no sharing capacity, 3. a population which had no rearrangement capability (hardwired). We plot the phenotypic fitness (Fig. 3A.I) and the genotypic fitness (Fig. 3A.II) of the embryo with the highest phenotypic fitness in each of these populations over 1000 generations averaged over 10 runs to observe their morphogenetic ability.Fig. 3Stress Sharing results in solving the problem of morphogenesis faster than without stress sharing or without reorganization competency (hardwired)(A).I: Phenotypic fitness curves of populations of embryos (size 30x30) post competent cellular rearrangement are shown. Populations with stress sharing based competency (green) reached maximum fitness by generation 400, which was faster than populations without stress sharing (orange) (p≪0.01) or those without any competency at all (black curve; hardwired population) (p≪0.01) at generation 400. In the absence of stress sharing (orange), a competent population performed worse than a hardwired population (black) post generation 400.(A.II): Genotypic fitness curves of populations of embryos (size 30x30) prior to competent cellular rearrangement. The genomes of a competent population w/stress sharing improved in a manner similar to that of a hardwired population until ∼ generation 400 (p=0.9), post which it plateaued.(B): Embryos post competent cellular rearrangement (phenotypes) at different stages of evolution are shown.
Grids correspond to those embryos with the best phenotypic fitness in any generation.(B.I) (w/sharing): Starting from a randomly initialized grid of binary values (generation 0, row1, column 1), the target pattern (a smiling face) was achieved by ∼ generation 400 (Fig: A.I)(B.II) (w/o sharing): Similar to row 1, embryos were initialized randomly with a different seed value. By generation 1000 (row 2, column 4), the final pattern was partially achieved.(B.III) (hardwired): In case of a hardwired embryo, its cells were not allowed to move. Thus, improvements were possible only through mutations to the grid structure. We noticed that the target pattern was almost achieved in this case, outperforming a population w/o sharing (row3) (fitness of 0.975 (hardwired) vs 0.91 (w/o sharing) at generation 1000). (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)Fig. 3
We observed that populations with stress-sharing were able to discover the correct genotype and the competency necessary to form the target pattern faster than the other two populations (Fig. 3A). Members of this population discovered how to arrive at the target pattern by generation 500, at which a population with sharing had a significantly different phenotypic fitness from either the hardwired (p≪0.01), or that of the population without sharing (p≪0.01). Further, the confidence intervals of these curves did not overlap, indicating their distinct separation from one another (Fig. 3A.I).
The way in which stress sharing populations derived their advantage was by progressively improving their genotype during the early stages of evolution (Fig. 3A.II), behaving as if they were a hardwired population (we found no statistical difference between the two until generation ∼400 (p=0.9). In contrast, populations without stress sharing improved their genomes only fractionally above their initialized states (Fig. 3A.II) and relied exclusively on their competency to achieve high phenotypic fitness towards the target (Fig. 3A.I). Such a strategy paid off early during evolution (until generation ∼400) but resulted in diminishing gains post this point (Fig. 3A.I).
We verified the phenotypes of the best embryo in each of these populations at generation 0, 250, 750, and 1000. We observed that at generation 250, all three populations had a similarly distorted pattern as confirmed by their approximately similar fitness values (Fig. 3B). Further, the stress-sharing and the without stress-sharing populations had a similar fitness value (=0.88) at this point (Fig. 3A.I, p=0.97), even though the population without-sharing engaged in a different dynamic compared to the other two populations (Fig. 3). At generation 750, despite their similar dynamics early on during evolution, the embryos with stress sharing were capable of forming the target pattern, whereas the hardwired population could not; however, we noted that it does perform better than the population without-stress sharing. At generation 1000, we observed the hardwired embryos being closer to maximum fitness with a positive slope of 0.19 compared to the population without sharing (slope = 0.05).
We conclude that populations with stress sharing discover the correct morphological solution faster than hardwired populations or populations without stress sharing. Hardwired populations, relying on mutations alone, are eventually capable of achieving high fitness, while populations without-sharing are worse than hardwired populations over long time scales.
In our first experiment, each embryo within a population was a matrix of size 30x30. To determine how sharing benefits populations of cells of different sizes, we initialized multiple instances of the previous experiment each with a different grid size and compared their fitness curves over 1000 generations. Specifically, we ran three instances of the previous experiment, with grid sizes 20x20, 30x30, and 50x50. Within each instance, we compared the performance of the stress-sharing population with the hardwired and without-sharing populations by observing their fitness curves (Fig. 4).Fig. 4As grid size increases, the morphogenetic task grows harder; a competent stress sharing mechanism helps in navigating a population to the target solution(A): Fitness graphs (phenotypic and genotypic) for populations with embryo sizes of 20x20. A competent population w/stress sharing solved the problem by generation 100 (A.I), whereas both the hardwired as well as a population w/o sharing were unable to achieve perfect fitness by generation 1000. Further, the hardwired population performed comparatively better than a population w/o sharing post ∼ generation 350 (p≪0.01). Genomes of the w/sharing population evolved identically to that of a hardwired population until ∼ generation 100 (p = 0.96) post which we observed a plateauing of its genome quality (A.II). (B): Fitness graphs for populations with a grid size of 30x30. A population w/sharing solved the problem in ∼400 generations, whereas the hardwired population as well as the population w/o stress sharing, failed to reach maximum fitness by generation 1000, obtaining a fitness of 0.975 and 0.91 respectively (p≪0.01) (B.I). Genomes of the stress-sharing and hardwired populations had similar early stage dynamics (∼generation 400, p = 0.96), post which the stress-sharing population's genome stagnated.
The without-sharing population's genome improved by less than 20 % above its initialized state (4B.II). Further, we observed that these dynamics were all right-shifted versions (with respect to time) of the experiment with grid size 20x20. Generally, populations of size 30x30 took a longer time to reach the same phenotypic fitness level as in 20x20, and genomes of the stress-sharing population stabilized after a longer time (∼300 generations longer). (C): Fitness graphs for populations with embryo sizes of 50x50. A competent population w/sharing (green) improved identically (in phenotypic fitness) to a hardwired population (black) until generation 400 (p=0.95), by generation 1000 we noticed that its performance was climbing back up to hardwired levels (C.I). Genomes of the w/sharing population improved identically to that of the hardwired population until generation 400 (p = 0.98). Post which a deviation (identical to that observed in the phenotypic fitness curves) were noticed. By generation 1000, genomes of the w/sharing population were improving to the level of hardwired individuals (C.II). We were unable to run this experiment for longer timescales owing to resource constraints. However, we note that both the genotypic and phenotypic dynamics at this scale are right shifted (in-time) versions of lower grid sizes (20x20, and 30x30). Consequently, we expect similar future trends in their phenotypic and genotypic fitness. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)Fig. 4
We observed that for a grid size of 20x20 (Fig. 4A), stress-sharing populations were able to reach maximum phenotypic fitness by ∼ generation 100; hardwired populations were able to reach maximum phenotypic fitness at generation 1000 and a population without sharing was unable to reach phenotypic maximum fitness even after 1000 generations of evolution (Fig. 4A.I). The genomes of the stress-sharing and hardwired populations evolved identically during the early stages of evolution (until ∼ generation 100 throughout which we noticed no statistical difference (p=0.96) post which the stress-sharing population stabilized. (Fig. 4A.II). Genomes of the without-stress-sharing population improved marginally above its initialized state.
At a grid size of 30x30, a similar pattern of evolutionary behavior emerged: the stress-sharing population reached maximum phenotypic fitness faster, followed by hardwired with a rising slope (=0.19) at generation 1000, and the without sharing population with a stable phenotypic fitness around ∼0.91 at generation 1000 (Fig. 4B.I). Genomes exhibited identical patterns as well: the stress-sharing and hardwired populations had similar early stage dynamics (until generation 400) (p=0.96), post which the stress-sharing population's genome stagnated. The without-sharing population's genome improved by less than 20 % above its initialized state (Fig. 4B.II). Further, we observed that these dynamics were all right-shifted versions (with respect to time) of the experiment with grid size 20x20. Generally, populations of size 30x30 took a longer time to reach the same phenotypic fitness level as in 20x20, and genomes of the stress-sharing population stabilized after a longer time (∼300 generations longer).
At a grid size of 50x50, 1000 generations were insufficient for any of the three populations to reach maximum phenotypic fitness (Fig. 4C.I). However, general patterns similar to those seen at lower grid sizes were observed: phenotypic fitness of the stress-sharing population evolved similarly to the hardwired population until generation 400 (p=0.96), followed by a short term decrease, before rising back up to hardwired levels. The without-sharing population exhibited early gains but stabilized closer to generation 1000 (Fig. 4C.I). Trends in genotypic fitness revealed a similar dynamic to those seen in lower grid sizes (20x20, and 30x30): the hardwired and stress-sharing population evolved similarly, however, unlike lower grid sizes, its dynamics dropped momentarily before reaching hardwired levels. The without-stress sharing population exhibited relatively modest improvement of its genome, but we note that it is higher than either of the previous two grid sizes.
We conclude that as grid size increases, the problem of morphogenesis grows harder, with populations requiring a longer time to reach peak fitness. Despite the increase in complexity, a stress-sharing population can utilize its competency to reach maximum fitness. The benefit of stress-sharing is especially noticed in the later stages of evolution when mutations cannot be relied upon to solve the problem efficiently, and more precise cell movements are required.
How does stress sharing help the collective achieve its morphogenetic objective? We next tested the hypothesis that the sharing of stress information facilitates the movement of cells over longer distances. To quantify its effect to fitness improvement, we plotted the competency value and the average Euclidean distance travelled by cells of the best embryo in each of the three populations throughout evolution.
Specifically, we initialized populations with members of size 30x30 and ran them through our GA for 1000 generations. In each generation, we picked the embryo whose phenotypic fitness was the highest and traced its genotypic fitness (Fig. 5A.II), competency value (Fig. 5A.III), and the average Euclidean distance travelled by its cells within that generation (Fig. 5A.IV). Each embryo was given a maximum possible competency value of 4725 which was determined as a function of its grid size, N (see Supplement S3)Fig. 5Stress sharing is beneficial because it encourages cell movement over long distances. (A): Evolution of a population of embryos each with a grid-size of 30x30 over 1000 generations. Genotypic fitness of an embryo with the best phenotypic fitness (A.I), its corresponding phenotypic fitness (A.II), its competency value (A.III), and the average distance travelled by its cells (A.IV) at each generation is shown. A population w/o sharing increases its phenotypic fitness rapidly in the first 200 generations during which it utilizes a competency of 100 units and its cells move an average of 250 units. Post generation 200, its phenotypic fitness stabilizes owing to its low genotypic fitness and constant utilized competency. Populations with stress sharing behave as if they were hardwired for the first 400 generations despite utilizing a high degree of competency (∼4725 units) (A.III), post which they reach maximum fitness and reduce their competency usage by ∼150 units. A high utilization of competency consequently allows cell movement across large distances in the matrix. Greater freedom causes cells of a stress sharing population to move as if they were random during the first 400 generations, post which their freedom enables them to achieve cell states with high fitness.
(B): Stress sharing increases the radius over which a cell influences another.(B.I):Two embryos of size 30x30, one with stress sharing and one without are developed independently. At each stage, the maximum distance over which a cell influences another cell by changing its future fate is recorded and plot at different intervals in time. As a cell moves towards a distant target, it dislodges other cells lying in its path, effectively changing their future fates. The radius of influence serves to measure the maximum distance of such influence. The embryo with sharing has a radius of influence of 30 at developmental stage 1, and decreases in a non-linear manner, eventually reducing to zero at stage 85. On the other hand, an embryo without sharing has a radius of influence of 5 at stage 1 which terminates to zero at stage 10.(B.II, B.III): An alternate view of the radius of influence over time for an embryo with sharing and an embryo without sharing respectively. Rectangular blocks of width proportional to the radius of influence are stacked on top of each other. Blocks are representative of the radius of concern at developmental stages marked on the x-axis. An embryo with sharing exerts influence over a larger radius for a longer duration of developmental time (5.B.II) than an embryo without stress sharing (5.B.III).Fig. 5
We observed that the stress-sharing population utilized maximum competency (=4725) until ∼ generation 400, beyond which its value dropped to ∼4500 and remained approximately stable thereafter. The population without stress-sharing utilized a competency of ∼100 throughout evolution and the hardwired population used 0 competency because it wasn't allowed to use competency by design (Fig. 5A.III).
Further, the cells of the best embryo (i.e., with the highest phenotypic fitness) in the stress-sharing population moved an average Euclidean distance of ∼2500 units until ∼ generation 400, post which it varied around a value of ∼2400 units. In case of the population without stress-sharing - owing to its lower usage of competency – cells moved an average of ∼200 units throughout evolution (Fig. 5A.IV).
We conclude from these observations that stress-sharing is beneficial precisely because it enables cells to move over longer distances through a stress-communication mechanism (Fig. 5A). Owing to the approximate nature of such movement, stress-sharing embryos are capable of utilizing a high degree of competency (Fig. 5A.III) to solve the morphogenetic puzzle. In contrast, a population without stress-sharing lacks such a communication mechanism and is therefore restricted to local cell-movements. Movement restriction enforces a limit on competency causing embryos to perform poorly. The Hardwired population leverages mutations to solve the morphogenetic task better than a competent population with no stress sharing.
We then sought to study these dynamics from the perspective of the cognitive light cone model [55], which focuses on the radius in space and time of events that any given agent can use as a homeostatic setpoint. In effect, taking the agent's perspective in terms of what it functionally “cares about” [[56], [57], [58], [59]].
As a cell moves towards its target position during development, it disturbs other cells, knocking them out of position and altering their future fates towards a target position. To quantify the range over which a given cell alters the fate of another cell, we plot the maximum radius over which a cell dislodges another cell at each stage of development. Specifically, we took a single embryo of each kind (with sharing and without sharing), passed it through the developmental stage, and plot the maximum radius over which cells dislodged other cells in each developmental step (Fig. 5 B).
We observed that in an embryo with stress-sharing, a cell influenced another cell located a radius of 30 units on average at developmental step-1. This distance progressively decreased over development, with the value dropping to zero at step 85 (Fig. 5B.I). In contrast, in an embryo without sharing, a cell altered the fate of another cell located at a radius of only 5 units on average at development step 1, with development ceasing completely by step 10. (Fig. 5B.I).
An alternative view of the radius of influence over developmental time is shown in Fig. 5 B.II, and 5.B.III, which better illustrates the impact of stress-sharing over developmental time. We plotted the radius of influence as a rectangular bar of proportionate size, and stacked multiple such blocks at developmental steps of 5, 20, 35, 65, 80, and 85 for the stress-sharing embryo (Fig. 5B.II), and at steps 1 and 10 for the without-stress sharing embryo. (Fig. 5B.III).