Branched Catalytic Networks and Once-for-Ever Selection

In a branched catalytic network, one catalyst or catalytic unit supports multiple competing replicators. Unlike a cyclic hypercycle, the coupling is open: there is no ring closure and no mutual dependence among all members. Under constant organization, the total concentration is fixed, and the dynamics of branch concentrations follow the standard selection equation:

Here is the concentration of branch i, is its selective value or productivity, and Φ is the average productivity in the population. Branching alters both and Φ, so motion on the concentration simplex depends on the relative productivity of the competing branches.

Fixed Points and Corner Selection

The concentration simplex has pure states at its corners. For a branched, non-cyclic network, interior coexistence fixed points are generally unstable or saddle-like when branches compete. Linearization near a corner gives a Jacobian whose eigenvalues determine stability: a corner with a higher selective value attracts the system, while the competing corner becomes unstable.

If one branch has a larger , it drives toward 1 while the other branches decline toward 0.
Fluctuations or small rate differences can tip the system, but once a branch gains an advantage, the fixed point migrates to its corner.
Losing competitors go extinct; the system does not settle at an interior cooperative state.

Once-for-Ever Selection

This outcome is called once-for-ever selection: a winner-take-all commitment rather than cooperative integration. The stable fixed point migrates to one of the competing corners, and the system remains there. Branched couplings lack the closure and mutual dependence needed for a hypercycle, so they fail to integrate multiple functions into a joint catalytic organization.

Source example: two-member branched/parasite case

The source gives

, with

, and

. This leads to

for

and

for

. The system has two fixed points at the corners of S₂, illustrating corner selection and the absence of stable interior coexistence.

Branched Catalytic Networks and Once-for-Ever Selection

One catalyst feeds two competing branches, while the concentration simplex S 2 shows the stable fixed point migrating from unstable interior coexistence to a pure-state corner. The winning branch attracts the system, and the losing branch goes extinct.

QUESTION 1

Under constant organization, where does the stable fixed point of a branched catalytic network migrate as one branch gains a selective advantage?

It settles onto a stable limit cycle.

It remains at the interior coexistence point.

It remains at the center of the simplex.

It migrates to a corner of the simplex.

QUESTION 2

In the selection equation ẋᵢ = xᵢ(Aᵢ − Φ), what does Φ represent?

The average productivity Σⱼ xⱼ Aⱼ.

The rate constant of the winning branch.

The total concentration c₀.

The largest eigenvalue of the Jacobian.

QUESTION 3

For the two-member branched/parasite case, which condition gives

, so the parasite grows?

QUESTION 4

In linearization near a corner of the simplex, which Jacobian eigenvalue pattern makes that corner stable?

Every eigenvalue equals zero.

At least one eigenvalue has a positive real part.

Every eigenvalue has a negative real part.

The eigenvalues lie on the imaginary axis.

QUESTION 5

Why do branched catalytic couplings fail to integrate multiple functions into a cooperative hypercycle?

They lack closure and mutual dependence among all members.

They produce a central fixed point for every dimension.

They have cyclic closure and mutual dependence.

They always stabilize an interior fixed point.

Case Study: Branched Parasite Competition in a Constant-Organization Reactor

Read the scenario and answer the two questions using the branched-network dynamics.

A constant-organization reactor contains a catalyst A that can support two competing replicators: a cooperative branch B and a self-reproductive parasite P. The dynamics follow ẋᵢ = xᵢ(Aᵢ − Φ), with

ⱼ xⱼ Aⱼ. For the parasite case, the source equations are

, and

. The system has two fixed points at the corners of S₂.

1. Determine the sign of ẋ for

and explain what it implies for parasite invasion.

Answer:
When

, so the parasite concentration x increases and the parasite invades. When

and the parasite declines. This shows that the outcome depends on total concentration and rate constants, and the system can move toward a pure-state corner.

2. If the parasite branch has a higher selective value than the cooperative branch, use corner stability and Jacobian eigenvalue reasoning to predict the long-term composition. Explain why this is once-for-ever selection rather than hypercyclic cooperation.

Answer:
The higher-selective-value branch has a locally stable corner in the concentration simplex. Linearization gives Jacobian eigenvalues with negative real parts at that corner, while the competing corner is unstable or saddle-like. The system commits to the dominant branch and the loser goes extinct. This is once-for-ever selection because the fixed point migrates to one corner and remains there; there is no closure or mutual dependence to sustain joint function, unlike a hypercycle.