NOTE-1211: Athreya, Czuppon & Gokhale — The evolution of dependence and cohesion in incipient endosymbioses

Contribution

It treats the early evolution of an endosymbiosis as movement in a plane with two axes. One is mutual dependence, how well the partners grow alone against together. The other is reproductive cohesion, how often they reproduce as a unit. Each axis is set by a trait in each partner. Earlier models (Law & Dieckmann 1998; Nguyen & van Baalen 2020) let populations grow exponentially, or held the host’s traits fixed. This one gives all three populations logistic growth and lets both partners’ traits evolve. Three results follow:

• carrying capacity, not growth rate, decides whether dependence starts at all;
• which partner becomes obligate first is set by population growth parameters;
• early on, dependence outruns cohesion.

Key insight

Because the collective’s reproduction aligns the partners’ interests, the partner that evolves faster gets there first. The symbiont, with the larger population and faster growth, gives up free-living reproduction first. At that point the host’s own obligacy stops mattering in the base model: “R₀(Ω̃_H, Ω_H, 1) = 1”. What happens next depends on details of life inside the collective. If partners die there more than they reproduce, the host follows into full dependence. If they reproduce there more than they die, the host is selected back to independence. So full mutual dependence, the model’s mark of a transition, is a parameter-dependent outcome, not the default. In the authors’ words, a transition “is far from an inevitable outcome of the existence of a higher level of selection”.

Assumptions

• Three populations, logistic growth (Eq. 1). Free hosts H, free symbionts S and collectives C, with growth rates f and carrying capacities K. Hosts and symbionts associate at rate a and dissociate at rate d, and dissociation returns exactly one of each.

• Benefits only inside the collective. A free host’s growth does not depend on the symbiont’s traits, and vice versa. There are no Lotka–Volterra cross terms.

• A growth trade-off (Eq. 2, “default”).

  • f_H = r_H(1 − Ω_H) and f_S = r_S(1 − Ω_S): each partner’s free growth falls with its own obligacy only;
  • f_C = r_C Ω_H Ω_S α_H α_S;
  • d = d₀(1 − α_H α_S): dissociation falls only when both adhesions are high;
  • a = a₀, a constant.

  Note S1 states the general constraint ∂a/∂αᵢ ≥ 0, but every analysis holds a fixed.

• The symbiont is faster. K_S > K_H and r_S > r_H throughout, so more symbiont mutants arise and the symbiont’s traits evolve faster (Dieckmann & Law’s canonical equation).

• Collective carrying capacity is a constant. It is not a function of the traits, on the view that the partnership opens a niche with a fixed capacity.

• One symbiont unit per host, acquired once. There is no growth of symbionts inside a host, no mixed infections, and no competition between free types and collectives.

• Separation of timescales and small mutations, with invasion judged by R₀ from the next-generation theorem. To suppress floating-point artefacts, the numerics accept a mutant only when R₀ > 1 + 10⁻⁷ (Note S6).

• The definition of a transition. A collective is an obligate endosymbiosis if the symbiont is intracellular, at least one partner is obligate, and the collective reproduces as a unit. A transition in individuality is “complete mutual dependence and reproductive cohesion”, Ω = α = 1 for both partners. This follows van Gestel and Tarnita, for whom a life cycle marks an evolving entity.

Key results

• Obligacies alone (Fig. 2). The symbiont’s obligacy reaches 1 first. The host’s then stops evolving, because once f_S = 0 the host’s R₀ is exactly 1. The model therefore predicts neutral variation in host obligacy loci. At Ω_S = 1, the symbiont’s equilibrium population exceeds its own carrying capacity, sustained by “backflow” from dissociating collectives. The authors call this counterintuitive: the partner that gave up independent reproduction is the more abundant.

• Adhesions alone (Fig. 2). The symbiont’s adhesion reaches 1 first. The host’s then rises to an evolutionarily stable value, and the pair drifts along a ridge of evolutionarily stable strategies near (1,1). The authors attribute the drift in their figures to floating-point error, though drift along a neutral ridge is what they predict for real populations. The ridge exists because adhesion helps hosts and symbionts at first, by raising collective growth, and hurts later, by cutting the dissociations that supply free individuals.

• Births and deaths inside the collective (Eq. 6, Fig. 3, Table 2).

  • Death without birth (d_H = d_S > 0): obligacies rise only from initial values above a threshold, which grows with the death rate. Once Ω_S = 1, the host also goes to 1, giving full mutual dependence.
  • Birth without death (b_H = b_S > 0): obligacies rise from (0,0) with no threshold. Once Ω_S = 1, the host’s obligacy falls to 0, a one-sided dependence. There is then no fixed point: the symbiont goes extinct while host and collective populations grow without bound.
  • Adhesion results are unchanged, except that adhesions do not rise at high death rates (Fig. S8).
  • With Ω_S = 1, the equilibrium shows that the sign of b_S − d_S decides whether more host obligacy raises or lowers the collective’s abundance (Eq. 7).

• Carrying capacity, not growth rate (Fig. 4, Notes S3.B and S6).

  • With K_C = 10, the traits stay at (0,0). With K_C = 50 or 500 they rise, and r_C (1 or 40) makes no difference.
  • Suggested sufficient condition: K_C > aK_HK_S/d. That is the collective population that association and dissociation alone would maintain. At the Figure 4 parameters it is 0.1·100·200/50 = 40, as checked here.
  • Perturbing a (0.01, 5) moves the outcome as predicted. The authors have not shown the condition is necessary.
  • K_C sets where adhesion ends up, but not obligacy.
  • With small r_C, the 10⁻⁷ acceptance cutoff hides real increases (Note S6), so “r_C does not matter” partly depends on the numerics.

• Dependence before cohesion (exponential model, Fig. 5).

  • Letting K_C → ∞ makes the invasion criterion exact and “separable”: f̃_H(1 − d̃/(a f̃_C)) > f_H(1 − d/(a f_C)).
  • From this follow an optimization principle, no evolutionary branching, and monotone increase in all four traits.
  • Simulations then show Ωᵢ > αᵢ for both partners. Mutual dependence (Ω_HΩ_S) runs ahead of cohesion (α_Hα_S) until the trajectories hit the feasibility bound, where the collective grows without limit.
  • The authors’ reason is that free growth depends on one’s own obligacy alone, while dissociation depends on the product of both adhesions. A gain in adhesion is therefore “damped by the other adhesion”.

• The bias depends on that choice (Note S4.E, Table S1). The appendix varies the two structural choices.

  • Free growth depending on one’s own obligacy, with dissociation on both adhesions (the default): dependence-biased.
  • Free growth on both obligacies, with dissociation on one adhesion: biased towards cohesion.
  • Matched choices (own/one, or both/both): little or no bias. The slight bias in own/one is called an artefact.

  The robustness checks of Figure S14 vary other things, all within the default structure: additive instead of multiplicative effects, symmetric or tenfold-asymmetric growth parameters, and a higher d₀. The authors “arrived at case i-II from biological considerations, and we therefore make the associated prediction”.

• The empirical check (Table 3). Ten symbioses are classified by dependence and transmission mode, after Estrela, Kerr & Morris (2016). The authors say the data are qualitative and a snapshot, and “does not allow us to confirm/refute the bias”.

• An ephemeral limit (Note S1.B). When collectives are fleeting (ẋ_C ≈ 0), the model reduces to a symmetric Lotka–Volterra mutualism with cross-term coefficient a(d/(d − f_C) − 1). That is positive when d > f_C, a mutualism, and harmful to both when the collective grows faster than it dissociates. The sign logic was checked here.

Claims

id	claim	strength	support
C1	The partner with the faster population growth (more mutants) becomes fully obligate first, an instance of the Red King effect in which the collective, not a mutualism, aligns interests	moderate within the model	simulations (Figs 2, 4) and the canonical equation; the direction follows from the assumed K_S > K_H, r_S > r_H
C2	Once the symbiont is fully obligate, the host’s obligacy is neutral, or rises, or falls, depending on whether within-collective deaths or births dominate	moderate within the model	R₀ = 1 shown analytically for the base case; Fig. 3 and Eq. 7 for the others, with only b_H = b_S and d_H = d_S tested
C3	Whether dependence and cohesion start to evolve depends on the collective’s carrying capacity, not its growth rate; K_C > aK_HK_S/d suffices	moderate within the model	Fig. 4, Fig. S5; sufficiency supported numerically, necessity not shown; Note S6’s cutoff caveat
C4	Early in an endosymbiosis, mutual dependence evolves faster than reproductive cohesion, “even if investments in dependence and cohesion are uncorrelated”	weak as a claim about nature, moderate within the model	Fig. 5 under the default mapping; Table S1 shows the bias set by the mapping’s structure and reversed under the alternative
C5	Endosymbioses in nature are “more likely to be mutually dependent than cohesive”	weak	follows from C4 if the default mapping is right; Table 3 cannot test it, by the authors’ account; indirect support cited (metabolic exchange is common, Black Queen dynamics, rapid experimental evolution of dependence)
C6	A transition in individuality is not an inevitable outcome of a higher level of selection	moderate	C2 and C3: full mutual dependence and cohesion need particular parameters

Concepts

• Obligacy (Ωᵢ): partner i’s investment in collective over free-living growth, in [0, 1]. At 1 it cannot reproduce alone.
• Adhesion (αᵢ): partner i’s investment in synchronized reproduction. It raises the collective’s growth and lowers dissociation.
• Mutual dependence: Ω_HΩ_S. The product is motivated by Ω’s multiplicative effect on collective growth (footnote 1).
• Reproductive cohesion: α_Hα_S. It is “similar to, but stronger than, vertical symbiont transmission”, since it requires the partners’ replication to be coupled, as mitochondrial fission is to the cell cycle. The fig–wasp mutualism is the example of high obligacy with low adhesion.
• Egalitarian transition: one between unrelated partners (Queller), as against fraternal transitions such as multicellularity.
• Red King effect: in a mutualism, the slower-evolving partner gets the better share (Bergstrom & Lachmann 2003). Here the faster partner gives up independence first.
• Feasibility bound: d > f_C(1 + a√(K_HK_S/(f_Hf_S))). Beyond it the exponential model has no positive equilibrium and collectives grow without limit.

Connections

The authors build on Law & Dieckmann (1998), Nguyen & van Baalen (2020) and Estrela, Kerr & Morris (2016), from whom the two-dimensional framing comes. None of these is in the record. Their Red King argument is Bergstrom & Lachmann’s. They place reductive genome evolution under “other mechanisms such as Muller’s ratchet” (LIT-1573, Moran 1996) “or environment-induced redundancy”, the latter citing the Black Queen (LIT-1568, read in NOTE-1212), McCutcheon & Moran (2010) and Husnik et al. (2013, LIT-1569). These, they say, are “predicated on endosymbionts occurring within the hosts”. They cite symbiont replacement in aphids (Chong & Moran 2018) for the host side of the asymmetry. Frank’s models of host control (LIT-1605, LIT-1617) are not cited.

Bearing on the record

• QUESTION-277. The question rested its first two dimensions on this paper, and the reading partly bears that out. D and R are separate traits with separate dynamics, and the model gives conditions under which either stalls. But:
  • The ordering result does not stand as a finding about endosymbiosis. “Dependence evolves faster than cohesion” is fixed by the structural choice of how traits enter free growth and dissociation, and reverses under the alternative. The question’s summary repeated the exchange’s version, “even when investments in the two are uncorrelated”. That is true of the model’s traits, but the bias comes from how they act, not from correlation. A note goes on QUESTION-277.
  • The model has no gene loss, no transfer and no host control. Obligacy is a phenomenological trade-off between free and collective growth. The paper does not model the question’s routes (Black Queen, drift, transfer), and says reductive evolution may be driven by “other mechanisms”. So the paper speaks to D and R, not to G or C.
  • One-sided dependence is a natural outcome. Symbiont obligate and host free is what the birth-dominated case gives. It is consistent with the empirical asymmetry the paper notes: symbionts are more often obligate than hosts.
• THEORY-141. Here the collective’s shared fate “aligns interests” through the Red King effect without any policing. That is a mechanism of alignment that THEORY-141’s list (policing and similar) does not name. The model has no within-collective conflict, though, so it cannot test whether selection between collectives trumps selection within. The comparison is mine.
• THEORY-443 and the transition criterion. The paper’s operational definition of a transition, full dependence plus full cohesion, giving a life cycle, is one concrete criterion for Darwinian individuality of a partnership. Under it, aphid–Buchnera (Table 3: mutually obligate, vertical) is a candidate, and the base model predicts the host’s dependence will not follow unless collective mortality dominates. It offers a parameter to measure for CASE-173: the balance of births and deaths of partners inside the collective.
• CASE-177 (Hodgkinia). The model fixes one symbiont unit per host and so cannot represent lineage splitting or symbiont titre. The authors list symbiont growth inside a host as the extension most needed.
• Filed as THEORY-1050 (the ordering result, with its dependence on the mapping) and THEORY-1051 (the asymmetry, and the outcomes after the symbiont becomes obligate). No instruction for machine-learning practice.

Limitations

• The headline result is a modelling choice. Table S1 is the authors’ own evidence that the dependence-first bias follows from the i-II structure, and Figure S14’s “remarkably robust” checks never leave it.

• The co-evolution result uses the exponential model, trusted “only when the traits have low enough values”. The bias is shown only for early trajectories, before the feasibility bound. Whether it survives later is named as open.

• Within-collective rates are shared. Only b_H = b_S and d_H = d_S were tested, though these rates decide the host’s fate. Unequal and trait-dependent rates are named as the next study.

• No test against data. Table 3 is qualitative and cannot estimate the traits, as the authors say.

• Constant a, constant K_C, one symbiont unit per host, no competition between free and collective types. All are named by the authors as extensions.

• Slips in the preprint.

  • Figure 2’s caption assigns obligacy and adhesion to panels differently from the text.
  • The adhesion ridge is described as connecting states written in Ω rather than α.
  • Table 2 reads “b_H = b_S = d_H = d_H”.
  • Figure S14 labels two panels “(c)” and writes C_H, C_S for K_H, K_S.
  • The mutant-host equation of S14 has f̃_H x_H where y_H is meant.
  • The main text cites Eq. S14 for the within-collective model, which is S13.

  None of these changes a result. Whether the journal version corrects them is unknown.

Open questions

• Which trait mapping is right? Whether a partner’s free growth depends on its own obligacy or both, and whether dissociation needs both partners’ adhesion, is an empirical question. The model’s prediction turns on it. Measuring how symbiont loss affects host free-living growth, and how transmission fidelity depends on each partner, would bear on it.
• What are b and d, partner births and deaths inside the collective, in real symbioses? The authors call measuring them “imperative”.
• Does the dependence-first bias survive to later evolutionary times?
• How would a mechanistic route to obligacy, such as gene loss under the Black Queen or the ratchet, change the trade-off curve f_i(Ω_i)? The authors cite Estrela, Morris & Kerr (2015) for accelerating benefits of function loss giving “obligate exploitation” instead of mutual dependence.

Note of 2026-10-30: sufficient, not necessary

The summary’s “Obligacy rises only if the collective’s carrying capacity is high enough” overstates what the paper shows. The simulations show the traits leaving (0,0) at K_C = 50 and 500 and not at 10, and the authors’ condition K_C > aK_HK_S/d is shown sufficient, not necessary, as C3 in the claims table and the key results already say. The twenty-fourth status review narrowed THEORY-1051’s title to match.