Linear scaling reveals low-dimensional structure in observable microbial dynamics
Microbial communities often exhibit apparently complex dynamics driven by myriad interactions among community members and with their environments. Yet, practical modeling and control are often based on limited number of observables, raising a fundamental question: to what extent are these observed dynamics predictable given unobserved background complexity? Here, we report an emergent simplicity that the temporal dynamics of observable microbial populations can be captured by low- dimensional representations. Using variational autoencoders (VAEs), we define a critical latent dimension (Ec) that quantifies the minimal number of variables required to represent observable microbial dynamics. We find that Ec scales linearly with the number of observables, despite the complexity of unobserved background dynamics. This principle holds across simulations of ecological, spatial, and gene-transfer models, experiments with engineered and environment-derived communities, and human microbiomes. Our findings establish a scaling law for microbial community dynamics and demonstrate observable dynamics alone contain sufficient information for prediction and control, even without full knowledge of the community.