Modeling endosymbioses: Insights and hypotheses from theoretical approaches

Abstract

Endosymbiotic relationships are pervasive across diverse taxa of life, offering key avenues for eco-evolutionary dynamics. Although a variety of experimental and empirical frameworks have shed light on critical aspects of endosymbiosis, theoretical frameworks (mathematical models) are especially well-suited for certain tasks. Mathematical models can integrate multiple factors to determine the net outcome of endosymbiotic relationships, identify broad patterns that connect endosymbioses with other systems, simplify biological complexity, generate hypotheses for underlying mechanisms, evaluate different hypotheses, identify constraints that limit certain biological interactions, and open new lines of inquiry. This Essay highlights the utility of mathematical models in endosymbiosis research, particularly in generating relevant hypotheses. Despite their limitations, mathematical models can be used to address known unknowns and discover unknown unknowns.

Fig 1. Integrating contrasting effects.

(A) Sketch illustrating how the fitness of a host with or without endosymbionts changes in different contexts. These contexts affect whether selection favors mutualistic or exploitative relationships. Integrating over all contexts (e.g., the whole life cycle) gives a perspective on the overall behavior. (B) Schematic showing how 2 contrasting behaviors might be observed under different contexts. Integrating over the whole time period identifies the net outcome. (C) An example hypothesis generated for an endosymbiosis that appears to exhibit 2 contrasting behaviors—exploitation and mutualism—in different contexts.

Fig 2. Broad generalizations.

(A) Sketch illustrating how theoretical abstraction can be used to compare between different biological systems. For example, this can help to identify similarities in division of labor between some endosymbioses and microbial communities. Theoretical abstractions also highlight distinctions between systems (e.g., lichens have a lower frequency of horizontal gene transfer [43] than microbial communities). (B) Schematic showing how different axes can be used to separate biological systems. Here, systems A, B, and C could display similar dynamics along axis 2, but system C might be incomparable with A and B along axis 1. Moving along the axes could be interpreted in 2 ways depending on the scenario: (i) changing a parameter value, such as rate of horizontal gene transfer, shifts the system into a different regime where categorically different behaviors are observed; and (ii) entirely different models, such as mode of transfer, are required. (C) An example hypothesis generated for endosymbioses that broadly share similar rates of horizontal gene transfer.

Fig 3. Identifying potential mechanisms.

(A) Sketch illustrating 2 models for the growth of an endosymbiont under different nutrient concentrations with experimental results (inspired by [15]); model B is a better fit for the data. (B) Schematic showing how models can propose mechanisms for an observed phenomenon when the full details of the underlying mechanism are unknown. More than 1 model system can be explored to see when the outputs match the observations. (C) An example hypothesis generated for the regulation of endosymbionts where the primary mechanism is currently unknown.

Fig 4. Exploring the unknown.

(A) Several examples of endosymbioses with a eukaryote are known and well studied, but prokaryote hosts are lacking. Prokaryote hosts with eukaryote endosymbionts have not been observed and might be unfeasible due to space limitations. Prokaryote–prokaryote endosymbioses are extremely rarely observed. These endosymbioses can be explored with theoretical models, for example, to estimate the proportion of prokaryote–prokaryote pairs that could form viable endosymbioses based on metabolism (as in [51]). (B) Schematic showing how modeling can be used to give insights for empirical systems where lab experiments have yet to be devised. The model can suggest potential experimental systems, for example, by identifying species that have a high propensity to become hosts and endosymbionts. (C) An example hypothesis generated for prokaryote–prokaryote endosymbioses where experimental systems are currently unknown.

Fig 5. New lines of inquiry.

(A) A representative output from agent-based model simulations of a coral ecosystem, adapted from [53]. Each grid square’s color represents the relative proportions of sand, various coral species, and algal types. To enhance this model by incorporating the impact of endosymbionts on coral growth rates and mortality during stressful events, such as heatwaves [54], assumptions are required regarding the interactions between corals and their endosymbiotic algae (for example, how corals may alter their internal symbiont composition in response to such events). (B) Schematic illustrating the assumptions necessary for constructing a model. Two potential assumptions for component C are presented. By running simulations with both assumptions, we can test the robustness of the results; identical outcomes suggest that the model’s results are not sensitive to the assumptions about C. Conversely, divergent outcomes indicate that further investigation into assumption C could lead to new research avenues. (C) An example hypothesis generated for a coral endosymbiosis where different assumptions can be made.