Bibliographic Reference
Li, X., & Thirumalai, D. (2019). Share, but unequally: a plausible mechanism for emergence and maintenance of intratumour heterogeneity. Journal of the Royal Society Interface, 16(150), 20180820. https://doi.org/10.1098/rsif.2018.0820
Originally posted as a bioRxiv preprint (2018, DOI: 10.1101/288670). The peer-reviewed version was published January 9, 2019 in J. R. Soc. Interface. The preprint and published versions are substantially identical in scientific content; the published title adds “plausible” and uses British spelling (“tumour”).
Core Argument
Prevailing explanations for intratumor-heterogeneity (ITH) invoke mutation-driven diversification and clonal selection — distinct subclones arise from genetic or epigenetic variation and coexist due to spatial constraints, neutral drift, or balancing selection. Li & Thirumalai propose a fundamentally different mechanism: ITH can emerge and persist without requiring genetic diversification, through unequal allocation of diffusible paracrine growth factors (“public goods”) between producer and non-producer cell subpopulations. Using replicator dynamics with evolving population size, the authors show that stable coexistence of producers and non-producers requires (a) nonlinear fitness functions of the fraction of producers, and (b) unequal sharing in which the producer retains more of the public good than the non-producer (an allocation rule they call “distribution according to work”). The model predicts that ITH is maintained only in a narrow range of exogenous resource concentrations — harsh conditions favor cooperation and heterogeneity, whereas resource abundance promotes competition and clonal homogenization. The theory is validated by quantitative fits to in vitro experiments on pancreatic cancer (IGF-II producer/non-producer cells; Archetti et al. 2015) and in vivo experiments on glioblastoma multiforme (wtEGFR/ΔEGFR cells; Inda et al. 2010).
Methods
Model framework: Replicator dynamics with evolving (non-constant) population size, tracking the fractions of producer (f_+) and non-producer (f_-) cells over time. The authors explicitly note that the constant-population-size assumption of standard replicator dynamics requires scrutiny (citing Gerlee & Altrock, 2015).
Fitness functions: Growth rates of producer and non-producer cells are modeled as Hill-like functions of total available IGF-II concentration. For non-producer cells, the growth rate w_- is a Hill function of exogenous IGF-II concentration c = c_0 + bf_+, where c_0 is the exogenous supply, bf_+ is IGF-II produced by the +/+ cells, and b is the allocation coefficient for non-producers. For producer cells, the growth rate w_+ is a Hill function of c_+ = c_0 + af_+, where a is the allocation coefficient for producers. The ratio b/a determines how public goods are shared. The producer also incurs a production cost p_0 subtracted from its growth rate.
Parameter estimation: Hill function parameters (a_1, lambda_1, alpha, a_2) were fit to experimentally measured growth rates of IGF-II null (-/-) cells at varying IGF-II concentrations from Archetti et al. (2015). Allocation coefficients a and b, plus the cost p_0, were determined by fitting to equilibrium fractions of +/+ cells at different serum levels. For the GBM application, parameters were fit using three tumor growth curves (0%, 50%, and 100% deltaEGFR cells) from Inda et al. (2010), and then predictions were made for 10% and 90% deltaEGFR mixtures.
Phase diagram analysis: The model produces two-dimensional phase diagrams in the space of (a) exogenous resource concentration vs. initial fraction of producers, delineating regions of three stable phases: homogeneous producer-only, homogeneous non-producer-only, and heterogeneous coexistence.
Mathematical accessibility: The paper is analytical (derivation of equilibrium conditions and stability criteria) with supporting numerical solutions of the replicator equations. Key equations involve Hill functions, logistic functions, and linear stability analysis of fixed points. The extraction could not capture all equations; the original PDF should be consulted for the exact mathematical derivations.
Key Findings
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Unequal allocation of public goods is necessary for stable coexistence. “A stable heterogeneous state arises if the producer can obtain the public goods more efficiently than the non-producer, ie, the public goods are allocated according to the rule of ‘distribution according to work’ (Sen 1966).” When public goods are shared equally (b/a = 1), the non-producer always grows faster and sweeps the population. When producers get a larger share (0 < b/a < 1), two internal equilibrium points appear — one unstable and one stable — enabling coexistence.
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Nonlinear fitness functions are required for heterogeneity. A heterogeneous state “cannot be realized if w_+ and w_- are linear fitness functions of f_+.” With linear fitness, even if unequal allocation produces an internal equilibrium (w_+ = w_-), it is unstable: a slight increase in producers tips the system to producer-only fixation, and a slight decrease tips it to non-producer-only fixation. Nonlinear fitness — which is “frequently observed in biological systems at all length scales due to cooperation or competition” — enables a stable internal equilibrium.
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ITH is maintained only in a narrow range of exogenous resources, and harsh conditions favor cooperation. “Maintenance of ITH requires cooperation among tumor cell subpopulations in harsh conditions, specified by lack of exogenous IGF-II, whereas surplus exogenous IGF-II elicits competition.” Below a critical resource concentration, a bistable system supports both heterogeneous coexistence and non-producer-only fixation. Above this threshold, non-producers always outcompete producers regardless of initial fractions. “The establishment of cooperation between different players shows strong dependence on environmental conditions.”
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The producer’s production cost (p_0) modulates the extent of ITH. “Higher price can decrease the demand for exogenous resources in order to establish cooperation and might also expand cooperation to a wide parameter range.” Lower p_0 (cheaper production) shrinks the parameter space for non-producer domination but also limits the heterogeneous region; higher p_0 expands the heterogeneous phase. This is because a higher cost makes producers less competitive, paradoxically requiring more cooperation for survival.
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The model quantitatively explains in vitro and in vivo experimental data without free parameters. For the pancreatic cancer experiments (Archetti et al., 2015), “our theoretical model with two free parameters a and p_0 fits the experimental results very well” across multiple serum concentrations. For the GBM experiments (Inda et al., 2010), “the theoretical predictions for the tumor growth at 10%, and 90% of delta-cells agree quantitatively with experimental observations,” using parameters determined solely from the 0%, 50%, and 100% delta-EGFR growth curves.
Concepts Introduced or Used
- Public goods game — An evolutionary game theory framework in which one cell type (producer) incurs a cost to produce a diffusible growth factor that benefits both producers and non-producers. Without unequal allocation, the system exhibits the “tragedy of the commons” and collapses to non-producer fixation. This paper extends the public goods game by incorporating unequal allocation and nonlinear fitness.
- Producer-nonproducer dynamics — A two-population system in which one subpopulation synthesizes and secretes a diffusible public good while the other consumes it without producing it. The authors show that stable coexistence requires the producer to capture a disproportionate share of the public good.
- Unequal allocation rule (“distribution according to work”) — The principle that public goods must be allocated preferentially to their producers for a heterogeneous state to be stable. The parameter b/a (ratio of allocation coefficients to non-producers vs. producers) must be less than 1.
- Nonlinear fitness — Fitness functions that depend nonlinearly on the fraction of producer cells. Required for stable heterogeneity; linear fitness yields only unstable equilibria.
- Replicator dynamics with evolving population size — A generalization of classical replicator dynamics that relaxes the constant-population-size assumption, allowing population size to evolve alongside strategy frequencies.
- intratumor-heterogeneity — The coexistence of genetically or phenotypically distinct cell subpopulations within a single tumor. This paper proposes a non-genetic cooperation-based mechanism for ITH.
- clonal-sweep — The fixation of a single clone due to selective advantage. In this model, equal sharing of public goods leads to non-producer sweeps, while resource abundance elicits competitive sweeps.
- driver-mutation — Mutations that confer a selective growth advantage. This paper’s mechanism operates orthogonally to driver-based evolution: ITH arises from ecological cooperation, not mutation accumulation.
Entities Referenced
- IGF-II (insulin-like growth factor II) — Upregulated in many cancers; functions as the “public good” in the pancreatic cancer model. Produced by +/+ cells, consumed by both +/+ and -/- cells.
- EGFR (epidermal growth factor receptor) — Amplified in GBM. The two GBM cell types are wtEGFR (cells with amplified EGFR) and deltaEGFR (cells with intragenic rearrangement/deletion of exons 2-7 of EGFR).
- IL-6 (Interleukin-6) — Secreted by deltaEGFR cells in GBM as a paracrine factor promoting proliferation and inhibiting apoptosis; acts as the public good in the GBM model.
- LIF (Leukemia inhibitory factor) — Another paracrine factor secreted by deltaEGFR cells, with similar function to IL-6 in the GBM model.
- Cancer types: Pancreatic neuroendocrine cancer (insulinoma), glioblastoma multiforme (GBM)
- Methods: Replicator dynamics, Hill function fitting, phase diagram analysis, public goods game model
- Genes: IGF-II (Igf2 gene), EGFR (wtEGFR and deltaEGFR variant)
Limitations (as stated by authors)
- The model is restricted to two subpopulations (producers and non-producers). Extending “to the case beyond two species which is more common in nature” is noted as a future direction.
- Fitness functions are assumed to follow a specific Hill-like functional form. The authors test a logistic function alternative and find “qualitatively similar results,” suggesting the exact form is not critical, but this is not a comprehensive robustness check.
- Parameters for the pancreatic cancer model (a, p_0, and b) were determined by fitting to the experimental data from Archetti et al. (2015), meaning some findings are model-dependent.
- The model assumes a single public good; real tumors likely involve multiple diffusible factors with complex interactions.
- Preprint-to-publication track: This paper was originally posted as a bioRxiv preprint (March 2018) but was subsequently published in the peer-reviewed Journal of the Royal Society Interface (January 2019, DOI: 10.1098/rsif.2018.0820). The preprint and published versions are substantially identical in scientific content. The paper has been through peer review.
Relevance to Clonal Evolution
This paper introduces a fundamentally different category of mechanism for intratumor-heterogeneity — one that does not rely on mutation-driven diversification, selection-driven clonal sweeps, or neutral drift. Instead, ITH emerges from ecological interactions (cooperation through unequal public goods sharing) between phenotypically distinct but genetically stable subpopulations. This has several implications for the wiki’s domain:
Contrast with mutation-driven ITH: The canonical view (Nowell, 1976; Gerlinger et al., 2012) explains ITH as the result of sequential mutation accumulation producing genetically distinct subclones. Li & Thirumalai show that ITH can arise and persist without genetic diversification — the producer/non-producer distinction is a phenotypic or epigenetic state, maintained by the allocation rule and resource conditions. This suggests that sequencing-based measures of ITH (which detect genetic heterogeneity) may systematically underestimate the functional heterogeneity present in tumors.
Contrast with selection-based ITH: In the selection paradigm, ITH persists because spatial constraints or balancing selection prevent any single clone from sweeping. Here, ITH persists because cooperation under harsh conditions creates a stable equilibrium where both cell types are equally fit (w_+ = w_-). This is an equal-fitness stable coexistence, not a transient state before a sweep.
Contrast with neutral drift: The neutral model (Sottoriva et al., 2015) posits that most ITH reflects passenger mutations accumulating at equal rates across all lineages. Li & Thirumalai propose a non-neutral, actively maintained heterogeneity whose stability depends on the allocation ratio b/a and resource availability. The model makes a sharp testable prediction: ITH should be lost when exogenous resources exceed a critical threshold — a prediction that neutral drift does not make.
Therapeutic implications: The model suggests counterintuitive treatment strategies. Rather than starving tumors (which favors cooperation and ITH), “it might be prudent to feed these cells instead of depriving them of nutrients so that competition between different subclones is promoted.” This is illustrated for GBM: adding exogenous cytokines could eliminate the producer (deltaEGFR) population, after which the non-producer (wtEGFR) cells alone cannot sustain tumor growth. This resource-modulation approach is orthogonal to mutation-targeted therapies and may be relevant for tumors where subclonal cooperation is prevalent.
Relationship to branching-evolution and clonal-evolution: The model operates at a different level than phylogenetic branching. It explains how two coexisting populations can be stably maintained — it does not address how new branches arise through mutation. As such, it is complementary to branching models: branching generates diversity; unequal sharing maintains it.
Relationship to clonal-expansion: The model shows that total tumor growth rate is maximized at an intermediate fraction of producers (e.g., 77% deltaEGFR cells in GBM), meaning heterogeneous tumors can grow faster than any homogeneous population. This provides a mechanism for the observation that mixed subpopulations often produce larger tumors than pure populations (Marusyk et al., 2014).