Stability Conditions and the Failure of Joint Function

In an open catalytic chain under constant organization, a stable positive stationary state is not evidence of cooperative self-organization. The stability of a stationary state is governed by two conditions: a rate-constant condition and a total-concentration condition.

Stability Conditions

Rate-constant condition: The first member’s rate constant must dominate all downstream constants: ≫ kⱼ for every .
Total-concentration condition: The total concentration must exceed the threshold ⱼ₌₂ⁿ (ⱼ)/kⱼ. Below this threshold, required downstream concentrations become nonpositive, so no positive stationary state exists in the concentration simplex.

Stronger dominance by the first member (larger ⱼ ratios) raises the required total concentration c₀.

Example

For a two-membered chain, the conditions reduce to and . If and , then . With , the stationary state gives and . Below , no positive stationary state exists.

Why Stability Does Not Imply Integration

When both conditions are satisfied, the system may settle to a stable stationary state. However, this state remains an open, first-member-dominated chain. The open chain lacks cyclic feedback: downstream species do not return to influence the first member. Consequently, perturbations do not engage the whole chain as a joint functional unit. Therefore, the chain cannot stabilize a joint function; stability of the stationary state is not equivalent to integration into a hypercyclic collective.

In this framework, selective advantage for a molecule corresponds to a larger rate constant. A molecule with a higher k outcompetes others, but this competitive advantage does not produce a joint function—it merely shifts the stationary state toward domination by that molecule.

Stability Threshold Diagram

Plot of the threshold c 0,min = Σⱼ₌ 2 ⁿ (k 1 − kⱼ)/kⱼ against the ratio k 1 /k 2 for

. The blue region above the line indicates where a positive stable stationary state is possible; the red region below indicates where no positive stationary state exists. Even in the blue region, the open chain does not achieve cooperative integration.

QUESTION 1

What are the two necessary conditions for a stable positive stationary state in an open catalytic chain?

≪ kⱼ and

(

ⱼ)/kⱼ

≫ kⱼ and

(

ⱼ)/kⱼ

ⱼ and

(

ⱼ)/kⱼ

ⱼ and

(

ⱼ)/kⱼ

QUESTION 2

For a two-membered chain with

and

, what is the threshold value c₀,min =

100

QUESTION 3

Why does a stable stationary state in an open catalytic chain fail to represent cooperative integration?

Because the first member is not dominant enough

Because the chain lacks cyclic feedback among its components

Because the total concentration is always too low

Because the rate constants are all equal

QUESTION 4

How does increasing the ratio

affect the required total concentration c₀?

It decreases c₀

It makes c₀ negative

It increases c₀

It does not affect c₀

QUESTION 5

In the context of an open catalytic chain, what does selective advantage mean for a molecule?

The ability to form cyclic feedback with other species

A higher rate constant that lets it dominate the stationary state

A lower rate constant that slows its own reaction

The capacity to reduce the total concentration of the system

Case Study: Three-Membered Open Chain

Analyze the stability conditions and the limits of integration.

Consider an open catalytic chain with three members

under constant organization. The rate constants are

. The total concentration is

1. Compute the threshold c₀,min = Σⱼ₌₂³ (

ⱼ)/kⱼ and determine whether a positive stationary state exists for

Answer:
c₀,min =

. Since

, the condition

,min is not satisfied, so no positive stationary state exists. The required downstream concentrations would become nonpositive.

2. Suppose the total concentration were increased to

. Explain why, even if a stable stationary state exists, this chain cannot stabilize a joint function.

Answer:
With

, a stable positive stationary state exists. However, the chain remains open and first-member-dominated:

≫

. There is no cyclic feedback from the downstream members back to the first member. Therefore, perturbations do not engage the whole chain as a cooperative unit. Stability of the stationary state is not equivalent to integration into a hypercyclic collective, so the chain cannot stabilize a joint function.