Genomic τ — DNA Replication, Repair, and Energetic Accounting

Abstract

We express DNA replication and repair energetics in a τ-first formalism, where τ ≡ E/c³ ≡ m/c. Nucleotide incorporation is driven by dNTP → dNMP + PPi with subsequent PPi hydrolysis, supplying free energy per base on the order of 10⁻²⁰ J. We map these energies to τ per base, relate copying to information-theoretic limits, and propose τ-consistency tests using calorimetry, single-molecule force spectroscopy, and enzymatic ATPase assays.

1. Introduction

DNA replication converts chemical potentials of nucleotides into ordered sequence, consuming energy and exporting heat. In τ units, every incorporated base changes the system’s τ by Δτ = ΔE/c³. A τ-first view tightens the link between biochemical work (polymerization, helicase unwinding, ligation), mass/energy balance (reagents and products), and information storage (bits per base).

2. DNA Biophysics & Energetics

2.1 Polymerization drive

Incorporation step: DNAₙ + dNTP → DNAₙ₊₁ + PPi followed by pyrophosphatase: PPi + H₂O → 2 Pi. The coupled reaction supplies a net free energy ΔG_inc per base (order −5×10⁻²⁰ to −7×10⁻²⁰ J per nucleotide, depending on conditions and base/sequence context).

2.2 Unwinding & processing

Replicative helicases consume ATP to unwind duplex DNA (order-of-magnitude ~1 ATP per base pair) and ligases consume ATP to seal nicks on lagging strands. Stacking interactions and ionic conditions (e.g., Mg²⁺) modulate ΔG of melting/hybridization.

2.3 Single-molecule work scale

Unzipping work per bp from force–extension is roughly W ≈ F·Δx ≈ (10–15 pN)·(0.34 nm) ≈ (3–5)×10⁻²¹ J, comparable to thermal energies per degree of freedom and below the chemical drive from dNTP hydrolysis.

3. τ-Formulation for Genomic Processes

For ΔG_inc ≈ 5×10⁻²⁰ J, Δτ_base ≈ 5×10⁻²⁰ / c³ ≈ 1.9×10⁻⁴٥ (SI units of τ). Replication τ-flux at rate r bases·s⁻¹ is \dot τ_rep ≈ r·Δτ_base. Helicase ATP turnover adds \dot τ_hel ≈ \dot n_ATP·ΔG_ATP/c³. τ-consistency requires that calorimetric energy release matches the sum of chemical τ inputs minus stored τ in products.

4. Information & Landauer Limits

A DNA base encodes 2 bits (A/T/G/C). Landauer’s bound for erasing one bit is E ≥ k_B T ln 2; at 310 K this is ~3×10⁻²¹ J per bit → ~6×10⁻²¹ J per base. The chemical drive per incorporation (~5×10⁻²⁰ J) comfortably exceeds this minimum, allowing high-fidelity copying and error correction overhead while remaining thermodynamically consistent.

5. Quantitative Benchmarks

• Polymerase rate: order 10²–10³ nt·s⁻¹ (organism/enzyme dependent) → \dot τ_rep in the 10⁻⁴³–10⁻⁴² range (using Δτ_base above).
• Helicase ATPase: ~1 ATP·bp⁻¹ (order-of-magnitude); ΔG_ATP ≈ 5×10⁻²⁰ to 1×10⁻¹٩ J per hydrolysis → comparable τ contribution to polymerization per base.
• Ligation: 1 ATP per nick sealed; cost accumulates with Okazaki fragment number and repair load.
• Melting curves: DNA Tₘ shifts reflect ΔH, ΔS; integrating yields per-base ΔG consistent with τ accounting.

6. Implications

• Unified accounting: τ ties sequence copying (information) to biochemical energy and measurable heat.
• Error correction budget: Fidelity mechanisms (proofreading/mismatch repair) consume additional τ, bounded above by available chemical drive.
• Biotech design: PCR/qPCR and isothermal methods can be optimized by explicit τ budgets (buffer ions, dNTP levels, enzyme turnover).

7. Conclusion

Genomic processes are τ-flows: chemical potentials of dNTPs and ATP are converted into ordered sequence and heat. τ makes the connections between mass/energy balance, measurable calorimetry, single-molecule work, and information limits explicit and testable.

References

1. Alberts et al., Molecular Biology of the Cell.
2. Phillips, Kondev, Theriot, Garcia, Physical Biology of the Cell.
3. SantaLucia & Hicks (2004), Annu. Rev. Biophys. Biomol. Struct. — DNA nearest-neighbor thermodynamics.
4. Landauer (1961); Bennett (2003) — Information erasure thermodynamics.
5. Single-molecule DNA unzipping/optical tweezer literature for force–extension work per bp.

Appendix A — τ-First Genomic Dictionary

A.1 Core identities

A.2 Reactions

A.3 Energetic links

A.4 Information

A.5 τ-consistency (experiment)

Appendix B — Test Protocols (Checklist)

B.1 Calorimetry & Enzyme Assays (in vitro)

B.2 Single-Molecule Mechanics

B.3 PCR / qPCR Energetics

B.4 Reporting

• Report per-base ΔG and τ = ΔG/c³ with buffer/temperature/ionic conditions.
• Provide τ-consistency: inputs (dNTP, ATP) vs outputs (heat + order).
• Include uncertainty budgets and replicate counts.
• Publish raw force–extension and calorimetry traces for reanalysis.

B.5 Worked Example Template