TABLE 2.

Microbial integration into mathematical models of cancer evolution. Overview of each model’s characteristics and references provided with modeling examples, as well as suggested ways that both microbes and cancer cells could be incorporated into models if not yet commonly applied in this context. Hybrid models that include aspects of more than one model type are also utilized in practice. CRC: colorectal cancer.

Model Type	Overview of Model	Examples of Potential Incorporation of Host-Microbe Cancer Interactions	References
Genome-scale metabolic model (GEM)	• Models metabolic interactions between microbiome and host • Analyzes genotype-phenotype relationships by connecting metabolic genes with their corresponding metabolic pathways • Can be used to analyze the Warburg effect in cancer cells • GEM algorithms (e.g., CASINO) allow several microbial species (≥5 species) to be modeled	• Current uses include modeling host-microbe interactions (e.g., quantifying how microbiota interactions impact host physiology, including applications in CRC) • Potential to explicitly incorporate metabolite release by microbes and consumption (e.g., glucose) by cancer cells in an evolutionary model	[169–173,198–202]
Generalized Lotka-Volterra (gLV) model	• Predicts the population dynamics of cancer cells or microbes and incorporates growth rates, interaction strength or competition between groups, and environmental changes • Classical predator-prey LV model defined as two competing populations that affect one another’s growth, but can be generalized to an arbitrary number of coexisting populations	• Quantify competition between cancer cells using relative abundance of microbes • Microbes and cancer cells could both be considered as distinct populations competing amongst each other over shared resources or sustaining a mutualistic relationship in the model with defined interaction strengths	[170,203,204]
Agent-based model	• Define ‘agents’ as individuals or members of the microenvironment with specific properties and actions on a structured grid or 3D space • Can have stochastic and deterministic components with spatial constraints • Define environmental rules such as chemotaxis, and the presence of factors in space, such as signalling proteins like VEGF • Define agent-agent interaction rules	• Create microbe as one agent type in the microenvironment and cancer cell as another agent type • Allow clonal evolution of cancer cells and separate evolution of microbes in equations • Create biophysical rules accounting for spatial movement of microbes (e.g., quorum sensing) and effect of microbes on evolutionary rates, such as proliferation or death of cancer cells during drug delivery	[205–208]
Wright-Fisher type model	• Population size remains constant over time (can be extended to growing populations) • Models consider finite number of population species/k-alleles among either microbes (e.g., bacteria, viruses) or cancer cells • To create the next non-overlapping generation, alleles are randomly sampled with replacement • Allele frequency in the new generation is the combination of random sampling of population and the fitness of alleles • Captures genetic drift and natural selection if included	• Microbial species could undergo distinct Wright-Fisher evolutionary dynamics that are independent of, or, in turn, affect cancer cell evolution • Effects of microbes present could also be interwoven into cancer cell fitness evolving under Wright-Fisher dynamics • Fitness parameter of certain cancer cell genotypes may depend on metabolites, proteins, and antigens from intracellular bacteria, which in certain cases may drive differential immunoediting between cancer cell-bearing bacteria	[162,209–211]
Moran-type model	• Two or more species considered in a population of either microbes or cancer cells • Asexual reproduction, overlapping generations • Simultaneous birth and death events occur • As in the Wright-Fisher model, can be formulated as a diffusion approximation	• Similar to the Wright-Fisher type model, microbial species could be considered distinct population genotypes undergoing evolutionary dynamics, or the effects of microbes could be interwoven into the fitness of cancer cells • Fitness parameter of certain cancer cell genotypes may depend on metabolites, proteins, and antigens from intracellular bacteria, which in certain cases may drive differential immunoediting between cancer cell-bearing bacteria	[163,212–214]
Birth-death stochastic process	• Continuous time Markov model (branching process) where ‘birth’ or ‘death’ events can change the state/population size • A ‘birth’ increases the state by one, a ‘death’ decreases the state by one • Allows for multiple cell types (e.g., with/without driver mutations), fluctuations in total population size, stochastic extinction of cells, and mutation to other types • Can also be used to model eco-evolutionary dynamics of microbial communities	• Define properties of stochastic events such as survival for human cancer cells with:   ◦ Probability of birth, death, and/or mutation affected by products of microbes in the microenvironment   ◦ Probability of birth, death, and/or mutation dependent on a function of the fluctuating populations of intracellular microbes present • Define stochastic events in terms of both human cancer cells and microbial populations	[167,215,216]
Evolutionary game theory model	• Includes density-dependent fitness with cell-cell interactions • Models cooperation (e.g., between tumor and stromal cells, or between bacteria) •  Fitness landscapes in non-cancer models have been central to understanding microbial evolution such as E. coli	• Include microbes as a type of “player” in the modeled ecosystem alongside tumor cells for limited chemicals and nutrients (e.g., oxygen, sugars) • “Public good” produced by tumor cells, such as lactate, included in a game as competing resources with microbial populations	[217–222]