Chapter 15: Elongation as ψ-Extension Path "In elongation, ψ walks its path—each step adding an amino acid, each cycle extending the chain, information becoming structure one residue at a time."
15.1 The Elongation Cycle
Translation elongation represents ψ's iterative process—a cyclic mechanism that reads codons and adds amino acids with remarkable speed and accuracy. Each cycle is identical yet unique, building diversity through repetition.
Definition 15.1 (Elongation Cycle):
Cycle = { Decoding , Peptidyl transfer , Translocation } \text{Cycle} = \{\text{Decoding}, \text{Peptidyl transfer}, \text{Translocation}\} Cycle = { Decoding , Peptidyl transfer , Translocation }
Three steps repeated until termination.
15.2 EF-Tu Delivery
Theorem 15.1 (Ternary Complex):
EF-Tu \cdotp GTP \cdotp aa-tRNA → A site \text{EF-Tu·GTP·aa-tRNA} \rightarrow \text{A site} EF-Tu \cdotp GTP \cdotp aa-tRNA → A site k on ≈ 10 7 M − 1 s − 1 k_{\text{on}} \approx 10^7 \text{ M}^{-1}\text{s}^{-1} k on ≈ 1 0 7 M − 1 s − 1
Near diffusion-limited delivery of substrates.
15.3 The Decoding Pause
Equation 15.1 (Selection Time):
τ selection = τ 0 + Δ τ induced fit \tau_{\text{selection}} = \tau_0 + \Delta\tau_{\text{induced fit}} τ selection = τ 0 + Δ τ induced fit
Time investment ensuring accuracy.
15.4 GTPase Activation
Definition 15.2 (Fidelity Switch):
Codon match → 30S closure → EF-Tu activation \text{Codon match} \rightarrow \text{30S closure} \rightarrow \text{EF-Tu activation} Codon match → 30S closure → EF-Tu activation k GTP cognate / k GTP near-cognate > 10 5 k_{\text{GTP}}^{\text{cognate}}/k_{\text{GTP}}^{\text{near-cognate}} > 10^5 k GTP cognate / k GTP near-cognate > 1 0 5
Correct matching triggers rapid GTP hydrolysis.
15.5 Accommodation
Theorem 15.2 (tRNA Movement):
A/T state → EF-Tu release A/A state \text{A/T state} \xrightarrow{\text{EF-Tu release}} \text{A/A state} A/T state EF-Tu release A/A state
Large conformational change positioning aminoacyl end.
15.6 Peptidyl Transfer
Equation 15.2 (Bond Formation):
P-site peptidyl + A-site aminoacyl → P-site deacylated + A-site peptidyl + 1 \text{P-site peptidyl} + \text{A-site aminoacyl} \rightarrow \text{P-site deacylated} + \text{A-site peptidyl}^{+1} P-site peptidyl + A-site aminoacyl → P-site deacylated + A-site peptidyl + 1
The chemical heart of protein synthesis.
15.7 The Catalytic Mechanism
Definition 15.3 (Substrate Positioning):
d attacking N − carbonyl C < 3 A ˚ d_{\text{attacking N}-\text{carbonyl C}} < 3 \text{ Å} d attacking N − carbonyl C < 3 A ˚
RNA positions substrates for spontaneous reaction.
15.8 EF-G and Translocation
Theorem 15.3 (Ribosome Movement):
Pre → EF-G \cdotp GTP Post \text{Pre} \xrightarrow{\text{EF-G·GTP}} \text{Post} Pre EF-G \cdotp GTP Post Δ x = 3 nucleotides = 1 codon \Delta x = 3 \text{ nucleotides} = 1 \text{ codon} Δ x = 3 nucleotides = 1 codon
Precise stepping maintaining reading frame.
15.9 Hybrid States
Equation 15.3 (tRNA Positions):
Classical ⇌ Hybrid \text{Classical} \rightleftharpoons \text{Hybrid} Classical ⇌ Hybrid A/A, P/P ⇌ A/P, P/E \text{A/A, P/P} \rightleftharpoons \text{A/P, P/E} A/A, P/P ⇌ A/P, P/E
Intermediate states facilitating movement.
15.10 Elongation Rate
Definition 15.4 (Speed):
v elongation = 15 − 20 aa/s (prokaryotes) v_{\text{elongation}} = 15-20 \text{ aa/s} \text{ (prokaryotes)} v elongation = 15 − 20 aa/s (prokaryotes) v elongation = 3 − 8 aa/s (eukaryotes) v_{\text{elongation}} = 3-8 \text{ aa/s} \text{ (eukaryotes)} v elongation = 3 − 8 aa/s (eukaryotes)
Rapid yet accurate synthesis.
15.11 Energy Cost
Theorem 15.4 (GTP Consumption):
Cost per aa = 2 GTP + 2 ATP \text{Cost per aa} = 2 \text{ GTP} + 2 \text{ ATP} Cost per aa = 2 GTP + 2 ATP
Significant energy investment ensuring fidelity.
15.12 The Extension Principle
Elongation embodies ψ's method of incremental creation—building complexity through repeated simple operations, each cycle identical in mechanism yet unique in outcome.
The Elongation Equation :
ψ protein ( n ) = ∏ i = 1 n E [ ψ codon i ] \psi_{\text{protein}}(n) = \prod_{i=1}^{n} \mathcal{E}[\psi_{\text{codon}_i}] ψ protein ( n ) = ∏ i = 1 n E [ ψ codon i ]
Where E \mathcal{E} E
Thus: Elongation = Extension = Growth = Creation = ψ
"In elongation, ψ demonstrates the power of iteration—that complexity emerges from simplicity repeated, that proteins are written like sentences, one letter at a time. Each cycle is a step in ψ's walk from information to form."