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Higher-order corrections on the denaturation of homogeneous DNA thermodynamics

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Published/Copyright: February 7, 2025
Zeitschrift für Naturforschung A
From the journal Zeitschrift für Naturforschung A Volume 80 Issue 3

Abstract

DNA denaturation, the process of separating double-stranded DNA into single strands, plays a critical role in fundamental biological processes such as transcription, replication, and repair. Despite extensive studies on its thermodynamic properties, the effects of thermal fluctuations on DNA denaturation have not yet been explored. This paper addresses this gap by developing a statistical mechanical model to analyze homogeneous DNA denaturation thermodynamics with thermal fluctuations. Using the partition function framework, this study introduces two major corrections to the entropy of the system induced by thermal fluctuations: (1) a logarithmic correction of the leading order and (2) a higher-order correction term proportional to the inverse of the entropy. Analytical calculations and numerical analysis reveal that these corrections significantly influence the thermodynamic properties, including specific heat and free energy, leading to a more nuanced understanding of the DNA melting process. The corrected entropy modifies the specific heat profile, resulting in a sharp peak that reflects a first-order phase transition during DNA denaturation. The inclusion of higher-order corrections introduces asymmetry in the specific heat curve, highlighting the cooperative behavior of DNA melting. Furthermore, the free-energy analysis suggests the presence of intermediate states during strand separation, which are critical for understanding the initiation and propagation of the denaturation process. The results align well with experimental DNA melting profiles, particularly in the transition region, and provide insights into the microscopic mechanisms underlying DNA melting. This study not only advances the theoretical framework for DNA denaturation by explicitly incorporating thermal fluctuations but also offers a platform for future experimental validation and applications in biological systems. These findings have broader implications for understanding DNA stability under physiological conditions, cellular processes such as transcription initiation, and the role of ionic environments in modulating DNA thermodynamics.

Keywords: DNA denaturation; higher-order corrections; homogeneous DNA sequences; thermal fluctuations; free energy; phase transition
Corresponding author: İzzet Sakallı, Physics Department, Eastern Mediterranean University, Famagusta, 99628, North Cyprus via Mersin 10, Türkiye, E-mail:

Funding source: Anatolian University Libraries Consortium

Funding source: Sponsoring Consortium for Open Access Publishing in Particle Physics

Funding source: European Cooperation in Science and Technology

Award Identifier / Grant number: CA22113 and CA21106

Funding source: The Scientific and Technological Research Council of Türkiye

Acknowledgments

I. Sakalli expresses gratitude to TÜBİTAK, ANKOS, and SCOAP3 for their financial support. He also acknowledges COST Actions CA22113 and CA21106 for their contributions to networking.

  1. Research ethics: Not applicable.

  2. Informed consent: Not applicable.

  3. Author contributions: The authors have accepted responsibility for the entire content of this manuscript and approved its submission.

  4. Use of Large Language Models, AI and Machine Learning Tools: None declared.

  5. Conflict of interest: The authors state no conflict of interest.

  6. Research funding: TÜBİTAK, ANKOS, and SCOAP3.

  7. Data availability: Not applicable.

Appendix A: Detailed derivation of the partition function

A.1 Partition function

Let us recall the partition function (3.1), which is expressed as:

(A.1) Z = 1 2 λ ∫ − U 0 U 0 d x 0 ∏ m = 1 M F 1 π ∫ − U U e − s m 2 d s m ∫ − U U e − t m 2 d t m E m ,

where:

  1. λ = π k B T K is the thermal wavelength,

  2. s m 2 = m 2 π 3 a m 2 λ 2 and t m 2 = m 2 π 3 b m 2 λ 2 are Fourier coefficients,

  3. E m = e − ∫ 0 β d τ [ V M + V S ] , with V M and V S representing the Morse and stacking potentials, respectively.

A.2 Gaussian integral approximation

The integrals over s m and t m are Gaussian-like integrals with finite limits (−U to U):

(A.2) ∫ − U U e − s m 2 d s m   and   ∫ − U U e − t m 2 d t m .

The exact solution for these integrals is:

(A.3) ∫ − U U e − x 2 d x = π erf ( U ) ,

where erf(U) is the error function, with erf(U → ∞) = 1:

(A.4) erf ( U ) = 2 π ∫ 0 U e − x 2 d x .

For simplicity, we approximate the Gaussian integrals as:

(A.5) ∫ − U U e − s m 2 d s m ≈ π , ∫ − U U e − t m 2 d t m ≈ π .

A.3 Contribution from a single mode

The contribution from a single mode is:

(A.6) I m = 1 π ∫ − U U e − s m 2 d s m ∫ − U U e − t m 2 d t m e − ∫ 0 β d τ [ V M + V S ] .

Substituting the Gaussian integral results:

(A.7) I m = e − ∫ 0 β d τ [ V M + V S ] .

A.4 Total contribution from M F modes

For M F independent modes, the total contribution is the product of contributions from all modes:

(A.8) ∏ m = 1 M F I m = e − ∫ 0 β d τ [ V M + V S ] M F .

A.5 Modified partition function

Substituting this result into the original partition function, we obtain:

(A.9) Z = 1 2 λ ∫ − U 0 U 0 d x 0 e − ∫ 0 β d τ [ V M + V S ] M F .

A.6 Saddle point approximation

The saddle point approximation is used to simplify the integral over x 0. Expanding f(x 0) around its maximum x 0 * :

(A.10) f ( x 0 ) ≈ f ( x 0 * ) exp − 1 2 f ″ x 0 * x 0 − x 0 * 2 ,

where x 0 * is the maximum point, and f ″ x 0 * is the second derivative at x 0 * .

The integral becomes:

(A.11) ∫ − U 0 U 0 f ( x 0 ) d x 0 ≈ f ( x 0 * ) 2 π f ″ x 0 * .

The Gaussian integral over dx 0 then simplifies to:

(A.12) ∫ − U 0 U 0 f ( x 0 ) d x 0 ≈ λ π λ μ ,

where:

(A.13) λ μ = λ π f ( x 0 * ) 2 π | f ″ x 0 * | .

Here, λ μ accounts for the curvature at the saddle point.

A.7 Final partition function

The final partition function becomes:

(A.14) Z = 1 2 λ μ 1 π exp − ∫ 0 β d τ [ V M + V S ] M F ,

which is nothing but Eq. (3.10).

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Received: 2024-09-06
Accepted: 2025-01-18
Published Online: 2025-02-07
Published in Print: 2025-03-26

© 2025 Walter de Gruyter GmbH, Berlin/Boston

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https://doi.org/10.1515/zna-2024-0200
Keywords for this article
DNA denaturation; higher-order corrections; homogeneous DNA sequences; thermal fluctuations; free energy; phase transition

Articles in the same Issue

  1. Frontmatter
  2. Chemical Physics
  3. Mathematical modeling for the potential energy of the aminophenol derivative azomethine molecule via Bezier surfaces and fuzzy inference system
  4. Topological descriptors and connectivity analysis of coronene fractal structures: insights from atom-bond sum-connectivity and Sombor indices
  5. Fundamental Concepts of Physical Science
  6. Stability analysis of semi-analytical technique for time-fractional Cauchy reaction-diffusion equations
  7. Dynamics of closed-form invariant solutions and formal Lagrangian approach to a nonlinear model generated by the Jaulent–Miodek hierarchy
  8. Solid State Physics & Materials Science
  9. Temperature-dependent elastic, mechanical, thermal, and acoustic behavior in alkaline earth semiconductors
  10. Investigation of the effect of Si content on the structural, mechanical, and tribological properties of TiAlN/AlSi protective multilayers
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