Encoding Networks[chaos]
A Boolean network diagram shows genes joined by regulatory arrows. But where do those arrows come from?
That question separates a classical Kauffman network from AND. In a Kauffman network, the modeller specifies each gene’s incoming edges and Boolean rule. The network is the inherited object only if we decide to treat it that way. Evolutionary simulations therefore commonly mutate the network directly: rewire an input edge, or flip an entry in a truth table.
In AND, the inherited object lies one level below the network. It is a string of gene-like expressions whose binding regions name the factors to which each gene responds. The regulatory network is derived from that string. An arrow is not independently stored and then attached to the gene; it appears because a heritable binding motif makes one gene product relevant to another gene’s expression.
This is the central connection to Buchler’s account of combinatorial transcription logic (Buchler et al., 2003). Logic and connectivity are properties of cis-regulatory organisation. Change the binding region, then ask what rule and what network the changed region now produces.
Kauffman: edges and rules are specified
For each target gene in a classical NK network, two choices are made:
- choose the K source genes that point to it;
- choose the Boolean function of those inputs.
These choices directly define the network. For example, saying that C reads A and B installs the edges A \rightarrow C and B \rightarrow C. Saying that its rule is AND separately installs the truth table C' = A \land B.
If we later replace B with D, we have performed an explicit rewiring mutation:
That is a perfectly useful evolutionary convention, but the causal order is simply edit the edge, then obtain the rewired network. Nothing in the model has to encode why B was an input before the mutation or why D is an input afterwards.
AND: inherit the binding string
Here is a small AND genotype:
BNBab->C BNXc_->D The first expression represents a gene producing C.
Its binding region, BNBab, requires A and B.
The second produces D when C is present.
The rules and edges below are not separately supplied.
They are calculated from the parsed genotype: for every expression, the target comes from the
uppercase letter after ->; effective inputs come from the motifs on which the expression’s
Boolean equation depends.
| Inherited expression | Produced factor | Factors that can regulate it | Derived Boolean condition |
|---|---|---|---|
| BNBab->C | C | A, B | ab |
| BNXc_->D | D | C | c |
The resulting network is a summary of the string:
The string comes first; the network comes second. Different strings can derive the same Boolean rule or the same collapsed network, just as different DNA sequences can support similar regulatory behaviour. The diagram therefore discards information about the heritable organisation that generated it.
Mutate the binding region, not the arrow
Now mutate one character in the first binding region.
The motif b becomes d:
BNBab->C BNXc_->D
BNBad->C BNXc_->D
No instruction says “delete B \rightarrow C” or “add D \rightarrow C.” The mutation changes which factor can bind in the regulatory region of the C-producing gene. When the network is derived again, the old edge is absent and a new one has appeared.
| What is being described | Before b→d | After b→d |
|---|---|---|
| Inherited string | BNBab→C | BNBad→C |
| Binding consequence | A and B can satisfy C's region | A and D can satisfy C's region |
| Derived rule | C′ = AB | C′ = AD |
| Derived network | A→C, B→C | A→C, D→C |
The mutation also creates a feedback loop. The parent already had C \rightarrow D; the new D \rightarrow C edge closes the loop. This network-level structure was not inserted as a design choice. It arose from the local compatibility between a changed motif and a factor already produced by the genotype.
The comparison
| Question | Classical Kauffman network | AND |
|---|---|---|
| What is inherited? | Usually an input list plus a truth table for every node | A string of expressions containing binding regions |
| Where is an edge represented? | Directly in the target node's input list | Implicitly in a motif that responds to a factor |
| Where is gene logic represented? | Directly in its externally assigned truth table | In binding operators and composition within that region |
| How is rewiring produced? | Replace, add, or remove an input edge | Mutate a motif or operator, then derive the resulting edges |
| What is the network diagram? | The model's primary specification | A compressed phenotype of the inherited string |
| What does selection act on? | A directly mutated network representation | The string; network changes are consequences |
The distinction is not that Kauffman networks cannot evolve. They can, once a mutation scheme is added. The distinction is what the mutation scheme treats as primitive.
In a conventional evolving NK model, an edge mutation is a legitimate elementary event because the input list is part of the encoded network. In AND, an edge is an interpreted consequence. The elementary event occurs in the binding string; changes to connectivity, Boolean logic, feedback, trajectories, and phenotype follow from interpreting and running the mutated string.
What Buchler contributes
Buchler’s work makes this more than a change of file format. It supplies a biological reason for putting the rule in the inherited regulatory sequence. Binding-site identity determines which factors can participate; affinity tunes when they bind; spacing and interaction determine whether occupied sites cooperate, compete, activate, or repress. The links and logic visible in a gene-network diagram are therefore higher-level consequences of local, heritable cis-regulatory organisation.
AND strips away affinity, geometry, concentrations, and noise, but preserves that causal direction:
This is why AND does not evolve by reaching into a network diagram and moving arrows. It mutates a simplified regulatory genome and lets the arrows—and the behaviour associated with them—arise again.