Framework for modeling siderophore-mediated interaction and benefit transfer in microbial communities.

(A) Overview of the siderophore-mediated iron uptake. Microbes allocate internal resources between growth (αi0) and the production of siderophores (αij for j > 0). Secreted siderophores form siderophore-iron complexes, which are taken up via type-specific receptors with allocation fractions vij. Different types of siderophores with their matching receptors are distinguished by colors. (B) Microbial iron-scavenging strategies. Microbes are categorized into three major classes by siderophore production and uptake patterns: (i) “Pure-producers,” which produce and exclusively utilize their own siderophores; (ii) “Partial-producers,” which produce/utilize their own siderophores and also exploit foreign siderophores; (iii) “Pure-cheaters,” which rely entirely on siderophores produced by others. (C) Benefit Transfer Graph (BTG). Left panels illustrate example siderophore-mediated interactions; right panels show their BTG representations: Nodes denote species, and directed edges represent benefit transfer from siderophore producers to beneficiaries. Edge colors correspond to siderophore types.

Rules governing siderophore-mediated iron interactions in single- and two-species models.

A. Schematic of the single-species chemostat model with iron supply Rsupply and dilution rate d. Variables [Riron], [M1], [R1] denote the concentration of free iron, microbial biomass, and siderophore, respectively. B. Time courses of biomass (M1) starting from different initial inoculations, illustrating the threshold-dependent survival. C. Schematic of a two-species system where two pure-producers compete. Each species secretes and exclusively utilizes its own siderophore type (blue for species 1, green for species 2). D. The Benefit Transfer Graph (BTG) corresponding to (C). Nodes represent species; edges represent the benefit flow. This graph features self-loops (b1,1, b2,2) and cross-species benefit transfers (b1,2, b2,1) E-F. Phase diagram spanning the receptor profile (v11-v22 for E) and growth allocation (α10 - α20 for F). Distinct ecological outcomes are color-coded and numbered. Color intensity is proportional to the total steady-state biomass. G. Representative state-space simulation dynamics projected onto the M1-M2 plane, parameters are from four different regimes in (E)-(F). Arrows denote the directionality of trajectories. Solid circles indicate stable fixed points. H. BTG of a three-species system. Blue, orange, and yellow arrows represent benefit transfers mediated by different siderophores produced by species 1, 2, and 3, respectively. Two rock-paper-scissors loops emerge: clockwise (characterized by v321) and counterclockwise (characterized by v123). Self-loops indicate self-utilization (vself) I. Representative time courses showing sustained oscillations in the system of (H). J. BTG of a five-species system. K. State-space projection onto the Riron-M1 plane showing a chaotic trajectory.

Cheating breadth elevates both extinction risk and community diversity

A. Definitions of cheating breadth (number of exploitable foreign siderophore types) and pure-cheater ratio. B. Probability of community-level extinction increases with both cheating breadth and pure-cheater ratio (ratio is color-coded; consistent across panels B–C). C. In non-extinct communities, species richness increases with cheating breadth but decreases with pure-cheater ratio. D. Violin plots showing non-extinct community size distributions for steady (yellow) versus dynamic (green) outcomes (pure-cheater ratio = 2%), under different cheating breadth. Dynamic communities consistently support higher biodiversity. E. Schematic illustrating edge ranking in BTGs, where edges from producers i are ranked by their weights bij. Two subgraphs formed by low-biomass species (left) and high-biomass species (right) are bracketed by dashed lines, with the relative frequency of Rank-1 incoming edges shown below. F. Rank frequency distribution of benefit transfer edges in BTGs. Top-ranked edges were enriched in subgraphs formed by high-abundance species (biomass > 10−3)

Core loops of the maximal Benefit Transfer Graph (mBTG) predicts community fate and resolves the cheating paradox

A. Construction of the mBTG. For each producer, the single strongest outgoing benefit flow defines the “maximal beneficiary,” forming a rank-1 directed edge. B. Topology of the mBTG decomposes into Weakly Connected Components (WCCs), each containing exactly one “Core” (colored nodes and edges). Four WCCs are separated by dashed circles. Core classes (Sink, Self-loop, Cyclic) are marked on each top. C. Scatter plots showing the probability of entering dynamic attractors leaps at core loop length of three. D. Scatter plot showing how community size increases with core loop length, for steady-state (yellow) and dynamic (blue) communities. Filled and open circles indicate high-abundance and surviving species (biomass threshold 10−3 and 10−6), respectively. E. Node classifications in mBTG (left) and how their counts change with cheating breadth (right). Self-loop maximal beneficiaries (MBs) have edges directed to itself; Connector MBs possess both incoming and outgoing edges; Terminator MBs have only incoming edges and no outgoing edges; Leaf nodes have no incoming edges. F. Probabilistic explanation that increasing cheating breadth expands the pool of potential recipients, diluting the producer’s chance of retaining its own siderophore (Self-loop MBs decline). G. Broader cheating amplifies MB selection bias toward high-α0 species, creating heavy-tailed in-degree distributions and promoting Terminator MBs (CB is abbreviation for cheating breadth). H. The percolation transition. The curve shows the fraction of nodes occupied by the largest WCC, which grows with cheating breadth. Colors under the curve indicate the proportion of WCCs governed by different core lengths. The system transitions from fragmented Self-loop Cores (exclusion) to giant components dominated by either Terminators (extinction) or Cyclic Cores (coexistence), with examples shown in bottom insets.