THE EQUATION — Hachimoji Extension

From 4 Bases to 8: The Generalized Form

The Generalized Equation

For a genetic alphabet of size N = 2^m bases:

encode?(n) = κ_A(n) ∧ κ_C(n) ∧ [J_m(n) > 0]

where:

n = k^m + a,    b = (k+1)^m - n,    k = ⌊n^(1/m)⌋

J_m(n) = Σᵢ₌₁^m aᵢ·bᵢ·F_{m,i}  +  Σᵢ₌₁^m (aᵢ-bᵢ)·F_{p,i}  +  ⟨χ, F_c⟩

The sum runs over the m base-pair types.

DNA (m = 2, N = 4)

The original equation:

J₂(n) = a₁·b₁·F_{m,1} + a₂·b₂·F_{m,2}
      + (a₁-b₁)·F_{p,1} + (a₂-b₂)·F_{p,2}
      + ⟨χ, F_c⟩

a₁,b₁ = GC content     F_{m,1} = 3-H-bond field (~41 kJ/mol)
a₂,b₂ = AT content     F_{m,2} = 2-H-bond field (~27 kJ/mol)

Hachimoji (m = 3, N = 8)

The extended equation:

J₃(n) = a₁·b₁·F_{m,1} + a₂·b₂·F_{m,2} + a₃·b₃·F_{m,3}
      + (a₁-b₁)·F_{p,1} + (a₂-b₂)·F_{p,2} + (a₃-b₃)·F_{p,3}
      + ⟨χ, F_c⟩

a₁,b₁ = GC content    F_{m,1} ~ 41 kJ/mol  (3 H-bonds)
a₂,b₂ = SB content    F_{m,2} ~ 43 kJ/mol  (3 H-bonds)
a₃,b₃ = AT+PZ content F_{m,3} ~ 28 kJ/mol  (2 H-bond average)

F_{p,1} = GC skew     (leading vs lagging strand)
F_{p,2} = SB skew     (replication-induced asymmetry)
F_{p,3} = (AT+PZ) skew

What Changes

Component	DNA (4 bases)	Hachimoji (8 bases)
Shell	`n = k² + a`	`n = k³ + a`
Mass field	2 parameters (GC, AT)	3 parameters (GC, SB, AT+PZ)
H-bond energies	2 types (2, 3)	3 types (2, 3-strong, 3-weak)
Genetic code	4³ = 64 codons	8³ = 512 codons
codonLUT entries	64	512
Top handle count	2 (GC, AT)	3 (GC, SB, AT+PZ)

What Stays the Same

Invariant	Reason
3-point contact	Physical B-DNA helix structure (3 recognition surfaces)
Gate logic κ_A ∧ κ_C ∧ J > 0	Multi-layer consensus required
1/n progressive binding	Topological entropy law
Score law ℓₜ structure	Thermodynamic cost structure

The Key Difference

In DNA, the interaction score J₂(n) = 0 defines a curve in the (a₁, a₂) plane — the set of positions where energy vanishes.

In hachimoji, J₃(n) = 0 defines a surface in the (a₁, a₂, a₃) volume — a 2D manifold embedded in 3D composition space.

This means:

DNA has 1 throat dimension (the GC-AT balance axis)
Hachimoji has 2 throat dimensions (a surface of stable configurations)

The hachimoji genome has MORE stable configurations — a larger "throat" — because the extra base pairs provide additional degrees of freedom for energy optimization.

Thermodynamic Data (Kumawat & Sherrill 2023)

Pair	H-bonds	Energy (kJ/mol)	S3C Classification
G:C	3	~41	F_{m,1} — strong 3-bond
S:B	3	~43	F_{m,2} — strong 3-bond (slightly stronger)
A:T	2	~27	F_{m,3} — weak 2-bond
P:Z	2	~29	F_{m,3} — weak 2-bond (slightly stronger)

The energy differences (~2 kJ/mol between GC and SB, ~2 kJ/mol between AT and PZ) are small but measurable, and they define the separate mass fields F_{m,1}, F_{m,2}, F_{m,3}.

References

Hoshika S. et al. (2019). "Hachimoji DNA and RNA: A genetic system with eight building blocks." Science, 363, 884-887. (554 citations)
Kumawat A., Sherrill C.D. (2023). "Evaluating the hydrogen bonding of hachimoji base pairs." Physical Chemistry Chemical Physics, 25, 22699-22711. (16 citations)
Negi S. et al. (2023). "Hachimoji DNA: Structure and its comparison with the four, six, and eight-lettered natural and unnatural genetic alphabets." International Journal of Molecular Sciences. (2 citations)
Eberlein C.K. et al. (2020). "Tautomeric equilibria in DNA: a study of hachimoji expanded alphabets." Chemical Science, 11, 12307. (45 citations)

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