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Chapter 40: Homomeric vs Heteromeric ψ-Coding

"In oligomerization, ψ explores symmetry and asymmetry—proteins assembling with identical copies or different partners, each choice encoding different functional possibilities."

40.1 The Oligomerization Dichotomy​

Protein oligomerization represents ψ's exploration of combinatorial assembly—homomeric complexes achieving symmetry through self-association, heteromeric complexes creating diversity through different subunit combinations.

Definition 40.1 (Oligomer Types): Homomeric=An\text{Homomeric} = \text{A}_n Heteromeric=AαBβ...Zζ\text{Heteromeric} = \text{A}_\alpha\text{B}_\beta...\text{Z}_\zeta

Identity versus diversity in assembly.

40.2 Symmetry Advantages​

Theorem 40.1 (Homomeric Benefits):

  • Genetic economy (one gene → multiple subunits)
  • Error correction through averaging
  • Cooperativity through symmetry

Efficiency through repetition.

40.3 The Dihedral Symmetry​

Equation 40.1 (D_n Symmetry): Dn=Cn+nC2\text{D}_n = \text{C}_n + n\text{C}_2

N-fold rotation plus perpendicular 2-folds.

40.4 Isologous vs Heterologous​

Definition 40.2 (Interface Types): Isologous:Same surface on both subunits\text{Isologous}: \text{Same surface on both subunits} Heterologous:Different surfaces\text{Heterologous}: \text{Different surfaces}

Symmetry of interaction surfaces.

40.5 Domain Swapping​

Theorem 40.2 (3D Domain Swapping): Monomerclosed⇌Dimerswapped\text{Monomer}_{\text{closed}} \rightleftharpoons \text{Dimer}_{\text{swapped}}

Exchange of identical structural elements.

40.6 Heteromeric Complexity​

Equation 40.2 (Subunit Diversity): Functions=f(∏iSubunitiαi)\text{Functions} = f(\prod_i \text{Subunit}_i^{\alpha_i})

Emergent properties from subunit combination.

40.7 The Hemoglobin Model​

Definition 40.3 (α₂β₂ Tetramer): Hb=α2β2\text{Hb} = \alpha_2\beta_2 Interfaces:α1β1,α1β2,α1α2\text{Interfaces}: \alpha_1\beta_1, \alpha_1\beta_2, \alpha_1\alpha_2

Multiple interface types in one complex.

40.8 Evolution of Quaternary Structure​

Theorem 40.3 (Gene Duplication): Homomeric→Duplication + DivergenceHeteromeric\text{Homomeric} \xrightarrow{\text{Duplication + Divergence}} \text{Heteromeric}

Asymmetry evolving from symmetry.

40.9 Allosteric Regulation​

Equation 40.3 (Symmetry and Cooperativity): Hill coefficient≤nsubunits\text{Hill coefficient} \leq n_{\text{subunits}}

Maximum cooperativity limited by oligomeric state.

40.10 Assembly Specificity​

Definition 40.4 (Partner Selection): Kdcorrect<<KdincorrectK_d^{\text{correct}} << K_d^{\text{incorrect}}

Discrimination ensuring proper assembly.

40.11 Stoichiometry Determination​

Theorem 40.4 (Balanced Expression): Stoichiometry=f(Expression levels,Kd values)\text{Stoichiometry} = f(\text{Expression levels}, K_d\text{ values})

Cellular mechanisms ensuring correct ratios.

40.12 The Coding Principle​

Homo- and heteromeric assemblies embody ψ's different strategies for creating functional complexity—symmetry providing robustness and cooperativity, asymmetry enabling specialization and regulation.

The Oligomerization Equation:

\text{Symmetry}(\psi^n) \quad \text{Homomeric} \\ \sum_i f_i(\psi_i) \quad \text{Heteromeric} \end{cases}$$ Different assembly logics for different needs. Thus: Oligomerization = Choice = Symmetry/Asymmetry = Function = ψ --- *"In choosing between homomeric and heteromeric assembly, ψ reveals that both unity and diversity have their place—that repetition creates strength while variation enables sophistication. Each oligomer encodes its functional logic in its assembly pattern."*