Stress sharing as cognitive glue for collective intelligences: A computational model of stress as a coordinator for morphogenesis

Individual cells have numerous competencies in physiological and metabolic spaces. However, multicellular collectives can reliably navigate anatomical morphospace towards much larger, reliable endpoints. Understanding the robustness and control properties of this process is critical for evolutionary developmental biology, bioengineering, and regenerative medicine. One mechanism that has been proposed for enabling individual cells to coordinate toward specific morphological outcomes is the sharing of stress (where stress is a physiological parameter that reflects the current amount of error in the context of a homeostatic loop). Here, we construct and analyze a multiscale agent-based model of morphogenesis in which we quantitatively examine the impact of stress sharing on the ability to reach target morphology. We found that stress sharing improves the morphogenetic efficiency of multicellular collectives; populations with stress sharing reached anatomical targets faster. Moreover, stress sharing influenced the future fate of distant cells in the multi-cellular collective, enhancing cells’ movement and their radius of influence, consistent with the hypothesis that stress sharing works to increase cohesiveness of collectives. During development, anatomical goal states could not be inferred from observation of stress states, revealing the limitations of knowledge of goals by an extern observer outside the system itself. Taken together, our analyses support an important role for stress sharing in natural and engineered systems that seek robust large-scale behaviors to emerge from the activity of their competent components.

Morphogenesis, broadly considered, is the ability of groups of cells to build complex, functional anatomical structures. Large-scale manifestations of this process occur during embryonic development, regeneration, and metamorphosis; smaller-scale, but equally important, is the on-going maintenance of order that can resist aging, degradation, and cancer over decades [[1], [2], [3], [4]]. It's a remarkable process in which large numbers of cells cooperate to reliably build, and remodel toward an invariant target morphology.

A key aspect of morphogenesis is that it is not an open-loop (purely feed-forward) process [5]. It is often assumed that the conceptual tools of emergence and complexity science can capture this process: local rules executed recursively in parallel across all cells, will result in the generation of complex growth and form. This is certainly true; indeed there are now many computational models of complex patterns emerging from simple processes [[6], [7], [8], [9], [10]]. However, there is an additional and crucial component of biological morphogenesis which is missed by purely emergent models: anatomical homeostasis (or more accurately, homeodynamics), in which the system can pursue multiple different paths in order to achieve the same anatomical outcome despite novel circumstances. In effect, biological morphogenesis is not well-described by bottom-up, open-loop, mechanical models [[11], [12], [13]], but instead exhibits more complex policies as it navigates the space of possible anatomical shapes despite various perturbations [14]. This includes the ability to reach the correct target morphology despite damage (e.g., bisection of early embryos), scrambling of tissue positions, changes of cell size and number, as well as numerous alterations of the genome (reviewed in Refs. [5,[14], [15], [16], [17]]). Numerous kinds of biochemical, biomechanical, and bioelectric prepatterns have been described which can encode setpoints for anatomical homeostasis [[18], [19], [20], [21], [22], [23]].

Understanding how bodies navigate anatomical morphospace, including the nature and limits of reliability, the possible failure modes, and the optimal intervention strategies is crucial for advances in evolutionary developmental biology, the bioengineering of synthetic life forms, the biomedicine of birth defects and regenerative therapeutics, and adaptive robotics [24,25]. This specifically requires models of how the behaviors of individual cells scale up to adaptive responses at the level of the entire body. What mechanisms allow the coordination of cell activity toward one consistent morphology? In particular, any theory has to provide a hypothesis for how the homeostatic capabilities of individual cells relate to the much more complex homeostatic setpoint of the collective.

One candidate mechanism that has been proposed is stress – a concept with a rich history in physics and materials engineering and a good candidate for a generic principle of biological organization at the system level [[26], [27], [28], [29]]. Stress is a popular topic in biology, studied at the level of DNA damage and protein misfolding, tissue morphomechanics, immunomodulation, behavior, and even whole ecosystems [27,[30], [31], [32], [33], [34], [35], [36], [37]]. We have previously proposed [38] that the generic concept of stress – as a biophysical marker of the computed error (distance between the current state and a homeostatic setpoint) – can be extended to the morphogenetic process. For example, the limb of the salamander will re-grow when amputated; the most remarkable part of this process is that it knows when to stop: growth and remodeling ceases when a correct salamander limb is completed [39,40]. Similarly, when a tail is transplanted to a limb-specific location in amphibia, it slowly remodels into a limb – the structure more appropriate to the bodyplan [41,42]. This, and other examples such as craniofacial corrections of scrambled amphibian faces [[43], [44], [45]], can be modeled as a progressive error minimization: cells continue to move and adjust until the error is within tolerance. The impetus that maintains activity could be seen as a stress reduction drive, as occurs at every other level of biological organization. Thus, in our models, stress reflects the degree of error – the distance (in this case, in anatomical morphospace) that the system can detect with respect to its prepattern. It is clear that many biological systems can detect and correct anatomical error (see examples in Refs. [5,17]); here, we focus not on these mechanisms but on the dynamics of the mediator of error information: stress.

While the molecular mechanisms of stress sensing, and the degree of conservation of specific molecules as bearers of the systemic stress signal from subcellular to multicellular scales, are currently under study and largely unknown, one interesting hypothesis concerns the role of stress sharing as a mechanism for the functional cooperation of cells in vivo. Consider a cell in the wrong position of a developing tissue and its positional information gradient (cell “I”, in Fig. 1A). The cell is motivated to move, to relieve its stress; however, its neighbors are at their correct positions and their low stress causes strong functional inertia not to move away from their current low-energy configuration. The individual cell-scale homeostatic loops prevent cooperation in this case and the optimal anatomical configuration is not reached (Fig. 1.A1).Fig. 1Schematic of experimental setup(A): A simplified diagram of one typical scenario during morphogenesis in which stress sharing is hypothesized to help solve the problem. Cell I (red) is motivated to move upward until it reaches a specific position in a morphogen gradient. (A1) Before and after, in the absence of stress sharing: cell I will not get to its intended spot because all the other cells' target states are met and they have no incentive to move enough to let cell I pass, no matter how much stress cell I is feeling due to the error state of its current position. Thus, a permanent defect will remain. (A2) Before and after, in the presence of stress sharing: if the stress molecule(s) representing the delta between its gradient setpoint and its current position can leak (or be exported) our of cell I, they will affect nearby cells.

Since stress molecules are conserved among all cells within a given body, the neighboring cells will be stressed by their microenvironment, which raises their plasticity and enables them to let cell I pass through. When it gets to the correct position, its stress levels reduce and the whole cell sheet's stress levels reach low levels because the entire collective is in a low-energy state with respect to all of their setpoints. In this way, a systemic stress response mechanism, implemented by conserved stress molecules that leak from specific cells, could facilitate large-scale problem-solving by de-localizing incentive to work toward group goals (cooperation without built-in altruism, since all the cells are simply trying to reduce their own local stress levels). The leak mechanism enables the collective to bend the energy landscape for the individual cells to exploit their homeostatic optimization behavior towards a tissue-level patterning goal. (B): Process of cell movement with and without stress sharing in embryos: The target task was for cells within a randomly initialized 2D matrix/embryo to re-arrange themselves towards a pre-set target pattern (a smiling face binary image in our setup). Stressed cells (B.I) (i.e, cells which are not in their correct positions to form the pre-set morphogenetic pattern) were allowed to move by picking other stressed cells as target and moving towards them through local swaps. By default, a stressed cell was not allowed to disturb other fixed cells (cells which were already in their correct positions). However, when given the ability, stressed cells could share their stress with their fixed neighbors, urging them to provide a path for movement towards the target (B.II). Consequently two kinds of competencies were setup: One with stress sharing (B.II), and one without (B.IV).

In the case where sharing was not allowed, a stressed cell surrounded by other fixed cells would have no way of moving through, causing it to get blocked locally (B.IV). In the case where sharing was allowed, blocked stressed cells would spread their stress to an area of size 3x3 around them (shown in orange), thereby communicating an intent of movement; in response, a channel would be created by the fixed cells for the stressed cell to move towards the target location. (Stages I, II, and III). With each resolved cell movement, the stress map was updated (B.III). Such a process occurred iteratively until either the competency limit was reached or the target pattern was formed. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)Fig. 1

However, one simple way to resolve this problem would be to allow stress sharing molecules to leak out of the source cell and diffuse outward to neighboring cells. If all cells interpret these conserved, shared molecular signals as stress, a given cell cannot tell if its high stress sensation is due to a problem it has or a problem its neighbors have. A cell that is leaking stress raises the exploratory activity level temperature of the nearby cells [46] (a quantity known as temperature, in the physics of annealing systems), making them more plastic and willing to perform active behaviors because they do not rate their current state as a satisfactory, low-energy stable state. This lowers the barrier for them to undergo exploratory motion in the space, allowing the other cell to move through to lower its stress level, at which point the whole tissue is at the optimal lowest-energy configuration and every cells' stress is reduced (Fig. 1A2). This kind of scheme is common in computational models of biological networks [[47], [48], [49]], and produces cooperation without explicit mechanisms of altruism. By sharing stress with neighbors, one cell's (or region's) problem becomes everyone's problem, leading to a higher willingness to adopt new configurations in which everyone's free energy is lower (and thus, more optimal morphogenetic problem-solving).

This hypothesis makes some strong predictions about the robustness of morphogenesis with and without such a stress sharing mechanism. However, in multiscale systems with emergent behaviors it is often not obvious how well such a mechanism would work, or what the resulting dynamics would be, without quantitative simulation. Here, we test those predictions in an in silico model of morphogenesis. In our model, migrating cells working to implement homeostatic target morphologies were either allowed or disallowed from sharing their stress during the process of rearranging in space.

We analyzed this model in three conditions, with the first being designed to test the impact of stress sharing and the last two being null hypothesis controls. First, our experimental mode with stress sharing, in which the individual cellular homeostats, in which stress is proportional to distance from their homeostatic setpoint, do propagate their stress to other cells. By analyzing these results, we find interesting phenomena that suggest that stress sharing is an easy way to achieve robustness in collectives made up of homeostatic subunits.

Second, a mode in which cells are agents and possess certain competencies that modify the resulting phenotypic fitness – they still have homeostatic loops governed by stress but they do not export it to their neighbors [50,51].

Third, a “direct development” mode in which phenotypes are hardwired from the genotype, mirroring many uses of genetic algorithms and numerous theoretical biology or artificial life studies which omit the developmental layer between them (no homeostatic competency or regulative development). By comparing the data, we find interesting and important roles of stress sharing as a dynamic that facilitates collective problem-solving.

We modeled (see Supplements S1 and S2 for technical approach and code execution details) embryos achieving a target morphology in-silico; the target morphology we chose was similar to the “electric face” bioelectric prepattern known to exist during frog embryogenesis [52]. We assessed the impact of embryogenesis on the morphogenetic process by encoding embryos with different developmental routines and comparing their reorganization potential. All developmental routines involved varying degrees of stress sharing between cellular components as encoded by their genome. Apart from comparing developmental routines on an individual scale, we also sought to compare its effects on an evolutionary scale by wrapping the developmental process within an evolutionary algorithm.

In our framework, an embryo was treated as a collection of cells within a two-dimensional (2D) grid. We chose a two-dimensional (2D) matrix data structure to be its computational equivalent for modelling, referring to its elements as cells (analogous to biological cells). Thus, our biological tissue is simplified as a discretized lattice, conceptually related to Chua's cellular neural networks [53]. Embryos were grouped into three different kinds based on their cells' ability to reorganize themselves during development: (1) stress-sharing embryo: wherein cells moved by sharing their stress, (2) without-stress-sharing embryo: where cells could move, but could not share their stress levels with others, and (3) hardwired: where cells could not move at all. The first two types thus included a developmental layer between genotype and phenotype, while the third simulated an organism with no regulative development, where the connections between genotype and phenotype was much more direct [51].

Our primary goal was to observe how the presence of a developmental process, -- which acted as a layer separating the initial substrate of the embryo from its reorganized state -- altered the evolutionary dynamics of a population. To quantify this, we evolved populations of embryos using a genetic algorithm (GA). Since our genetic algorithm had to account for a developmental stage during the life cycle of an embryo, it was an iterative sequence of the following stages: 1. Development, 2. Selection, and 3. Mutation (Fig. 2).Fig. 2Genetic algorithm Schematic(A): Initialization: Prior to evolution, a population of genomes which served as the substrate for evolution were initialized. A genome in our framework was a randomly initialized two-dimensional matrix containing two cell-types (0, indicated in black, and 1, indicated as white) (A.I). The genome also carried a marker indicating the type of development an embryo could undergo during the first stage of evolution i.e: development. Three kinds of reorganizaton processs were defined: 1. Stress sharing based reorganization, 2. Reorganization without stress sharing, and 3. Hardwired (no reorganization). A target for evolution was also set. To this end, we chose a downsampled binary image of a smiling face as the target pattern which evolution had to optimize over (A.II). (B): Development: The first stage of evolution involved a process of cellular reorganization of the initialized matrices. Reorganization would be based on the gene marker encoded in the genomes of each embryo. This stage served to mimic the developmental stage observed during an organism's lifetime. Post development, the reorganized cell-structure was termed the embryo's phenotype. Thus, the developmental process served as an independent layer separating the genotype of an embryo from its phenotype.

(C): Selection: Prior to development, the fitness of each genome was recorded by comparing it with the target (l2−distance). And Post development, the fitness of the corresponding phenotype was also recorded using its l2−distance with the target. During selection, phenotypic fitnesses of the top 10 % of the population were chosen and, their corresponding genomes were selected (Darwinian selection) for evolution in the next generation and the rest were discarded. (D): Mutation: The selected for genomes constituted 10 % of the original population. To repopulate the population back to its original strength, selected genomes were mutated by randomly swapping cell-pairs within the genome. The resulting mutated genome served as a new embryo. Post re-population, a new cycle of evolution began marked by the onset of development.Fig. 2

During initialization, each embryo (i.e., each 2D matrix) was initialized with one of two cell types (0 or 1) and were patterned in a stochastic manner to give a scrambled appearance to their structure (Fig. 2A.I). Each embryo within a population was initialized similarly with a unique scrambled pattern and the specific pattern itself was treated as its genome. In addition, the genome also encoded a separate value (the reorganization gene marker) which indicated the type of developmental reorganization it could undergo during the upcoming developmental stage.

Post initialization, a population of embryos underwent development (the first stage of the genetic algorithm). During development, embryos reorganized their cells to resemble a target two-dimensional pattern which we pre-specified in our experiments to be binary image of a smiling face (Fig. 2A.II). Based on the kind of reorganization gene marker encoded in its genome, an embryo could either undergo reorganization by sharing its stress, by not sharing its stress, or by not using stress at all.

Stress was a binary signal which indicated whether a particular cell was in its correct x,y coordinate position (with respect to the target) or not. Since our initial embryo was scrambled, most of its cells were bound to be stressed. Stressed cells were inherently driven to move somewhere (they possessed a sense of restlessness), the question of where to move to was based on another signal which we called the distress signal.

A distress signal originated in those cells which were not stressed (we called these “fixed cells”). Each of these fixed cells had a tendency to observe their neighbors (a 3x3 neighborhood around them) and to check if there were any stressed cells present. On recognizing the presence of a stressed cell in a particular direction within their 3x3 neighborhood, they would send a radially diminishing signal (which had a strength of 1.0 at its origin, but radially diminished in strength) encoding the presence and the location of a stressed cell. This process was analogous to how cells at a wounded site send signals elsewhere requesting assistance. In our case, a fixed cell, requested assistance in a particular location by sending a distress signal.

It is important to note that not all cells received this distress signal. First, only stressed cells received it, and second, only those stressed cells carrying a cell-value of the opposite kind received it: As an example, suppose a fixed cell (say, F1) identifies a stressed cell (say, S1) of cell-type “0” located on top of it; the stressed cell, S1, is precisely stressed because it is of the wrong cell type at the wrong x,y location). Thus, the fixed cell, F1 recognizes this and sends a distress signal to the rest of the 2D grid, requesting some other distant stressed cell (say S2) with the opposite cell-type “1” to move to the place of S1, thereby relieving its stress.

The above example is a simplification; in practice, multiple distress signals were sent out simultaneously, and stressed cells could receive and accumulate these signals to move to a specific target location. Once distress signals were captured by stressed cells, they moved.

Movement occurred sequentially. At each instance, a single stressed cell in the grid was chosen at random, and its distress-signal-record from various sources was consulted. Its destination for movement was chosen as that location from which the distress signal was strongest in value. The chosen stressed cell moved to its target by repeatedly swapping with intermediate elements in the direction of shortest path.

During cell movement, we had to deal with a problem: what would happen to the fixed elements which were in the way of a stressed cell as it moved towards its destination? If the stressed element swapped with fixed cells, it would dislodge them causing them to become stressed (thereby increasing the overall stress rather than decreasing it). We solved this problem by disallowing stressed elements to swap with intermediate fixed elements i.e., by default, if a stressed element encountered a fixed element while making its way towards its chosen destination, it could not move any further and would get stuck.

A special case which over-rode this default, was when we pre-specified stressed cells to share their stress with its neighbors. Since stress was a binary signal (0 or 1), a stressed cell communicated its status as a stressed cell to a fixed neighbor during movement. This communication process was termed “stress-sharing” and the effect it had was a coercion of the fixed cell to create a passage (a de facto tunnel) for movement through it rather than by swapping with it (Fig. 1B). Thus, on receiving a shared signal of stress from a stressed cell moving towards its destination, the fixed cell created a temporary channel, which allowed the stressed cell to swap with other stressed cells through it. In case multiple fixed cells stood in the way of the moving stressed cell, stress sharing ensured a tunnel was created through all of them for movement.

The tunnel induced as a result of the shared stress was temporary. Once the stressed cell moved away from the tunneled fixed cell, it reverted back to its original “blocking state”. Such a stress sharing mechanism, allowed the possibility of free, unrestricted movement towards all parts of the discrete grid, preventing dislocation of fixed cells from their grid positions during cell-movement. An alternate form of the reorganization process, one which did not involve stress sharing was also considered. In such cases, a stressed cell would get stuck as soon as it encountered a fixed cell while moving towards its destination in the direction of shortest path.

To summarize, an embryo could undergo one of three different kinds of reorganization: 1. With stress sharing, 2. Without stress sharing, and 3. Hardwired. Since the first two methods involved local cell movement they were termed competent strategies. The hardwired case served as a control to the other two competent strategies. The cells of a hardwired embryo could not move and thus lacked a reorganization process; development during evolution only served as a placeholder, without any functional significance.

Post-development, embryos possessed a transformed structure and the resulting grid organization was termed its Phenotype. Thus, the developmental stage served as an independent layer separating the genotype from the phenotype. Definitions of the genotype and the phenotype, as they have been used here, have loose connections with biology and assume special meaning only here. Demarcating these two phases allowed us to monitor the impact of the reorganization process on fitness throughout evolution (Fig. 2).

Our genetic algorithm served as a search process over genotypes which best reorganized themselves according to the target pattern (a smiling face in our experiments). The fitness value of any embryo was determined by its l2distance, d with the target.d=1N2(∑i=0N∑j=0N(Ei,j−Ti,j)2)12Where Ei,j and Ti,j are the (i,j)th elements of the embryo and the target respectively. Intuitively, the fitness function provided a quantitative metric of how close the embryo was to the target morphogenetic pattern.