Part 6: Translation and Protein Synthesis

Step 1: tRNA gene copy number as a proxy for abundance

In rapidly growing bacteria and yeast, tRNA abundance is approximately proportional to tRNA gene copy number. For each codon $c$, the available pool of cognate tRNAs includes exact-match (Watson-Crick) and wobble-pairing species. Define the absolute adaptiveness:

$$W_c = \sum_{j} (1 - s_{cj}) \cdot \text{tGCN}_j$$

where $\text{tGCN}_j$ is the gene copy number of tRNA isoacceptor $j$, and $s_{cj}$ is a penalty for wobble pairing (0 for Watson-Crick, 0.5 for G-U wobble, etc.).

Step 2: Normalize to get relative adaptiveness

For each amino acid, normalize by the maximum $W$ value among its synonymous codons:

$$w_c = \frac{W_c}{\max_{c' \in \text{syn}} W_{c'}}$$

This ensures $0 < w_c \leq 1$ for each codon. Codons with $w_c = 1$ are optimally adapted; lower values indicate slower decoding.

Step 3: Define the tAI for a gene

The tAI is the geometric mean of $w_c$ values over all $L$ codons in the coding sequence (analogous to CAI):

$$\text{tAI} = \left(\prod_{i=1}^{L} w_{c_i}\right)^{1/L} = \exp\left(\frac{1}{L}\sum_{i=1}^{L} \ln w_{c_i}\right)$$

Step 4: Relationship between tAI and translation speed

Since the elongation rate at each codon follows Michaelis-Menten kinetics with cognate tRNA as substrate, and tRNA abundance $\propto$ tGCN, the per-codon rate is approximately:

$$k_i \propto \frac{[\text{tRNA}]_{\text{cognate}}}{K_M + [\text{tRNA}]_{\text{cognate}}} \propto w_{c_i} \quad \text{(when not saturated)}$$

Step 5: tAI predicts protein abundance

The overall translation rate for a gene scales with the harmonic mean of per-codon rates (since slow codons bottleneck the ribosome). The tAI (geometric mean) approximates this and correlates strongly ($r \approx 0.6\text{--}0.7$) with measured protein abundance in E. coli and yeast.

Step 6: Comparison: tAI vs. CAI

CAI uses observed codon frequencies in highly expressed genes as a reference (empirical). tAI uses tRNA gene copy numbers and wobble rules (mechanistic). Both predict expression level, but tAI has a clearer biophysical basis. For E. coli: ribosomal protein genes have CAI $\approx 0.8\text{--}0.9$ and tAI $\approx 0.7\text{--}0.9$, while horizontally transferred genes have CAI $\approx 0.3\text{--}0.5$ and tAI $\approx 0.2\text{--}0.4$.

Step 1: Define the ternary complex delivery scheme

EF-Tu-GTP-aa-tRNA ternary complexes sample the A site. The cognate complex binds with association rate $k_1$ and either dissociates ($k_{-1}$) or triggers GTP hydrolysis ($k_2$):

$$\text{Ribosome} + \text{TC}_{\text{cog}} \underset{k_{-1}}{\overset{k_1}{\rightleftharpoons}} \text{Initial complex} \xrightarrow{k_2} \text{GTP hydrolysis} \xrightarrow{k_3} \text{Accommodation}$$

Step 2: Effective M-M rate for cognate tRNA selection

Applying the steady-state approximation to the initial recognition complex and combining the subsequent steps into an effective $k_{\text{cat}}$:

$$k_{\text{elong}} = \frac{k_{\text{cat}} \cdot [\text{TC}_{\text{cog}}]}{K_M + [\text{TC}_{\text{cog}}]}$$

where $K_M = (k_{-1} + k_2)/k_1$ and $k_{\text{cat}}$ combines GTP hydrolysis, proofreading, accommodation, peptide bond formation, and translocation.

Step 3: Include competitive inhibition by near-cognate tRNAs

Near-cognate ternary complexes compete for the A site but are mostly rejected during initial selection and proofreading. They act as competitive inhibitors with inhibition constant $K_I$:

$$k_{\text{elong}} = \frac{k_{\text{cat}} \cdot [\text{TC}_{\text{cog}}]}{K_M\left(1 + \frac{[\text{TC}_{\text{near}}]}{K_I}\right) + [\text{TC}_{\text{cog}}]}$$

Step 4: Kinetic proofreading (Hopfield, 1974)

After GTP hydrolysis, near-cognate tRNAs have a second chance to dissociate before accommodation (proofreading step). This adds an irreversible energy-consuming step that amplifies discrimination beyond thermodynamic equilibrium:

$$\text{Selectivity} = \underbrace{\frac{(k_2)_{\text{cog}}}{(k_2)_{\text{near}}}}_{\text{initial selection}} \times \underbrace{\frac{(k_3)_{\text{cog}}}{(k_3)_{\text{near}}}}_{\text{proofreading}} \approx 10^{2} \times 10^{1} = 10^{3}$$

Overall error rate per codon: ~$10^{-3}\text{--}10^{-4}$.

Step 5: Codon-specific rate variation

Since $[\text{TC}_{\text{cog}}] \propto$ tRNA abundance, which varies 10-fold between common and rare codons, the elongation rate varies accordingly. At a rare codon ($[\text{TC}] \ll K_M$):

$$k_{\text{elong}}^{\text{rare}} \approx \frac{k_{\text{cat}}}{K_M} \cdot [\text{TC}_{\text{rare}}] \quad \text{(first-order, slow)}$$

At an optimal codon ($[\text{TC}] \gg K_M$):

$$k_{\text{elong}}^{\text{optimal}} \approx k_{\text{cat}} \quad \text{(zero-order, maximal speed)}$$

Step 6: Overall translation rate for an mRNA

The total time to translate an mRNA of $L$ codons is the sum of per-codon dwell times. The overall rate is limited by the slowest codons (harmonic mean):

$$v_{\text{overall}} = \frac{L}{\sum_{i=1}^{L} 1/k_i} = L \cdot \left(\sum_{i=1}^{L} \frac{1}{k_i}\right)^{-1}$$

For E. coli: optimal codons give $k \approx 20$ aa/s, rare codons $k \approx 2$ aa/s. A cluster of rare codons can cause ribosome pausing, traffic jams, and co-translational folding pauses that may be biologically functional.

Step 1: Define the ribosome loading rate

Ribosomes initiate translation at the 5′ end of the mRNA at rate $k_{\text{init}}$ (ribosomes per second). Once initiated, each ribosome moves along the ORF at elongation speed $v_{\text{elong}}$ (codons per second).

Step 2: Ribosome transit time

For an ORF of length $L$ codons, the time for a ribosome to traverse the entire mRNA is:

$$\tau_{\text{transit}} = \frac{L}{v_{\text{elong}}}$$

For a 300-codon ORF at 6 aa/s (eukaryotic): $\tau = 50$ s.

Step 3: Steady-state number of ribosomes per mRNA

By Little's law (queueing theory), the average number of ribosomes on an mRNA equals the loading rate times the transit time:

$$\langle N_{\text{rib}} \rangle = k_{\text{init}} \times \tau_{\text{transit}} = \frac{k_{\text{init}} \cdot L}{v_{\text{elong}}}$$

Step 4: Linear density of ribosomes

The linear density (ribosomes per codon of mRNA) is:

$$\rho = \frac{k_{\text{init}}}{v_{\text{elong}}} \quad \text{(ribosomes per codon)}$$

A ribosome occupies ~30 nt (10 codons) of mRNA, so the maximum density is $\rho_{\max} = 1/10$ per codon. Initiation cannot exceed $k_{\text{init}}^{\max} = v_{\text{elong}}/10$ without causing queuing.

Step 5: Protein production rate per mRNA

The rate of completed proteins from one mRNA equals the initiation rate (in steady state, each initiated ribosome eventually produces one protein):

$$\frac{d[\text{protein}]}{dt}\bigg|_{\text{per mRNA}} = k_{\text{init}}$$

Total cellular protein production rate: $k_{\text{init}} \times [\text{mRNA}]$.

Step 6: Numerical examples

Highly expressed mRNA in E. coli: $k_{\text{init}} \approx 1$ s$^{-1}$, $v_{\text{elong}} = 15$ aa/s, $L = 300$ codons. Then $\langle N_{\text{rib}} \rangle = 300/15 = 20$ ribosomes per mRNA, $\rho = 1/15 \approx 0.07$ per codon. Electron micrographs of polysomes confirm 10-70 ribosomes on highly expressed mRNAs. Eukaryotic average: $k_{\text{init}} \approx 0.1$ s$^{-1}$, $v = 6$ aa/s, giving $\langle N \rangle \approx 5$ ribosomes per mRNA.

Step 1: Aminoacyl-tRNA synthesis (charging)

Aminoacyl-tRNA synthetase activates the amino acid using ATP:

$$\text{AA} + \text{ATP} \rightarrow \text{AA-AMP} + \text{PP}_i$$

The PP$_i$ is hydrolyzed by pyrophosphatase: $\text{PP}_i \rightarrow 2\text{P}_i$. This makes the reaction irreversible. Net cost: 2 high-energy phosphate bonds (ATP $\rightarrow$ AMP + 2P$_i$, equivalent to 2 ATP $\rightarrow$ 2 ADP).

Step 2: EF-Tu GTP hydrolysis (A-site delivery)

The EF-Tu-GTP-aa-tRNA ternary complex delivers the aminoacyl-tRNA to the ribosomal A site. Codon recognition triggers GTP hydrolysis:

$$\text{EF-Tu-GTP} \rightarrow \text{EF-Tu-GDP} + \text{P}_i$$

Cost: 1 high-energy phosphate bond. EF-Ts then recycles EF-Tu-GDP back to EF-Tu-GTP (no additional NTP cost).

Step 3: EF-G GTP hydrolysis (translocation)

After peptide bond formation, EF-G-GTP binds the ribosome and hydrolyzes GTP to drive translocation of the mRNA-tRNA complex by one codon:

$$\text{EF-G-GTP} \rightarrow \text{EF-G-GDP} + \text{P}_i$$

Cost: 1 high-energy phosphate bond.

Step 4: Total per amino acid

Summing all steps for one elongation cycle:

$$\text{Total} = \underbrace{2}_{\text{charging}} + \underbrace{1}_{\text{EF-Tu}} + \underbrace{1}_{\text{EF-G}} = 4 \text{ high-energy phosphate bonds per amino acid}$$

Step 5: Energy in thermodynamic terms

Each high-energy phosphate bond hydrolysis releases $\Delta G \approx -7.3$ kcal/mol under cellular conditions. The total energy invested per amino acid:

$$\Delta G_{\text{total}} = 4 \times 7.3 = 29.2 \text{ kcal/mol per amino acid}$$

Since peptide bond formation itself is only slightly endergonic ($\Delta G \approx +0.5$ kcal/mol), the process is driven far from equilibrium, making translation essentially irreversible.

Step 6: Cost of a complete protein and cellular energy budget

For a typical 300-amino-acid protein:

$$\text{Cost} = 300 \times 4 = 1{,}200 \text{ NTP equivalents} \approx 8{,}760 \text{ kcal/mol}$$

An E. coli cell growing with a 30-minute doubling time synthesizes ~$2 \times 10^6$ proteins per generation, consuming ~$2.4 \times 10^9$ ATP equivalents. This represents ~75% of cellular energy expenditure, explaining why translation is the dominant energy sink and why translational regulation is crucial for cellular economy.