Fixed Points of Open Catalytic Chains

Learning Objective: Analyze catalytic chains under constant organization and derive the six (and generally 2n) fixed points of the three-membered and n-membered systems.

Constant Organization and the Concentration Simplex

Under constant organization the total concentration c₀ is held fixed. It is therefore natural to use normalized concentrations for the n species of the chain. These coordinates satisfy and . The allowed state space is thus the concentration simplex : for it is a triangle; edges and corners represent states with one or more species absent.

Why Open Chains Have No Interior Fixed Point

Fixed points are found by setting the normalized rate equations to zero while enforcing . In an open catalytic chain the last member is not converted back into the first: there is no closing link such as C → A. As a result, the chain cannot balance all species simultaneously with every . The all-positive interior state is not a fixed point. Fixed points occur only when at least one species is absent, that is, on the boundary of . This is the fixed-point signature of an open chain that cannot integrate its members into a joint, closed function.

Counting the Fixed Points: and General n

For , there are six fixed points: three pure corner states (A, B, C) and three two-species edge equilibria (AB, BC, CA).
For general n, the structural count is 2n fixed points, corresponding to corner and mixed boundary states.
The 2n count is a fixed-point enumeration result. Stability and dynamical character are treated separately; a boundary fixed point need not be stable.

Worked Example: A → B → C

Consider a three-membered open chain A → B → C with no C → A closure. Solving the normalized fixed-point conditions under constant organization gives the six boundary fixed points: the three corners A, B, and C, plus the three two-species edge equilibria on AB, BC, and CA. The central interior state with A, B, and C all present is not a fixed point for the open chain. Under the chain dynamics, trajectories migrate toward the boundary, and the last member of the chain can eventually dominate in a once-for-ever selection.

Scope Check

The result 2n counts boundary fixed points only. It does not assert that every boundary fixed point is stable, nor that the system cannot fluctuate before reaching the boundary.

Fixed Points of Open Catalytic Chains

For a three-membered open chain A → B → C, the six fixed points lie on the boundary of the concentration simplex S 3: three corners (pure A, B, C) and three edge equilibria on AB, BC, and CA. The central interior point with all three species present is not a fixed point because the chain lacks the closing C → A link.

QUESTION 1

Under constant organization, which quantity is held fixed for an n-membered catalytic chain?

The rate constant of the first reaction only

The individual concentration of the final species only

The sum of all rate constants

The total concentration c₀

QUESTION 2

Which set of conditions defines the allowed state space for normalized concentrations ξᵢ?

ξᵢ ≥ 0 and Σ ξᵢ = 1

ξᵢ ≥ 0 and Σ ξᵢ = c₀

ξᵢ ≤ 0 and Σ ξᵢ = 0

ξᵢ arbitrary and Σ ξᵢ = 1

QUESTION 3

Why does an open catalytic chain generally lack an all-positive interior fixed point?

Because the chain has no closing catalytic link returning the last member to the first

Because every species is converted into every other species

Because the total concentration c₀ is allowed to vary without bound

Because the rate equations are linear and always have an interior solution

QUESTION 4

For a three-membered open chain, what are the six fixed points?

One interior state and five boundary states

Two corner states and four edge states

Six interior states with all three species present

Three pure corner states and three two-species edge equilibria

QUESTION 5

For a general n-membered open catalytic chain, how many fixed points are expected by the structural count?

n² fixed points

2ⁿ fixed points

2n fixed points

n fixed points

Case Study: Boundary Selection in an Open Three-Member Chain

Read the scenario and answer both questions.

A laboratory protocell maintains a constant total concentration c₀ of three catalytic species A, B, and C. The reactions form an open chain A → B → C, with no reaction returning C to A. The system is modeled under constant organization, so normalized concentrations

, and

satisfy

and each ξᵢ ≥ 0. The experimenter wants to know whether a stable coexistence of all three species can appear as a fixed point of the normalized rate equations.

Write the general fixed-point conditions for this normalized three-species system and explain why no solution with

, and

exists.

Answer:
Set the normalized rate equations to zero: dξ_A/dt = 0, dξ_B/dt = 0, dξ_C/dt = 0, together with the constraint

. Because the chain is open, there is no C → A conversion to close the cycle. A fully interior fixed point would require a balanced cyclic flow with all three species present and all rates zero simultaneously; the missing closing link prevents such a balance. Therefore any fixed point must have at least one ξᵢ = 0, so it lies on the boundary of the simplex, not in the all-positive interior.

Enumerate the six expected fixed points for this

open chain and classify each as a corner or an edge state.

Answer:
The six boundary fixed points are: (1) corner A: (

) = (1, 0, 0); (2) corner B: (0, 1, 0); (3) corner C: (0, 0, 1); (4) edge AB equilibrium:

with

and

; (5) edge BC equilibrium:

with

and

; (6) edge CA equilibrium:

with

and

. The corners are pure single-species states, while the edge states are two-species equilibria with the remaining species absent. The central interior state with all three species positive is not a fixed point for this open chain.