# An Accurate Estimate of the Free Energy and Phase Diagram of All-DNA Bulk Fluids

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## Abstract

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## 1. Introduction

## 2. Methods

#### 2.1. DNA Sequences

#### 2.2. Numerical Methods

#### 2.3. Theoretical Framework

## 3. Results

#### 3.1. Structural Properties of the Fluid

#### 3.2. Pressure of the Reference Fluid

#### 3.3. Critical Points and Coexistence

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A. Calculation of the Hybridization Free Energy

**Table A1.**Hybridization enthalpies $\Delta H$, entropies $\Delta S$ and free energies $\Delta G$ at $T=$ 37 °C for the sequence $\left({5}^{\prime}A\right)-CGATCG-{3}^{\prime}$ according to SantaLucia nearest-neighbor DNA model

Sequence | $\mathbf{\Delta}\mathit{H}$ (kcal/mol) | $\mathbf{\Delta}\mathit{S}$ (cal/K·mol) | $\mathbf{\Delta}\mathit{G}(37\xb0\mathbf{C})$ (kcal/mol) |
---|---|---|---|

CG/GC | −9.8 | −24.38 | −2.24 |

GA/CT | −7.8 | −21.02 | −1.28 |

AT/TA | −7.2 | −21.34 | −0.58 |

TC/AG | −7.8 | −21.02 | −1.28 |

CG/GC | −9.8 | −24.38 | −2.24 |

$\Delta {H}_{\mathrm{stack}}=-42.4$ kcal/mol $\Delta {S}_{\mathrm{stack}}=-112.139$ cal/K·mol. |

$\Delta {H}_{\mathrm{init}}=0.2$ kcal/mol $\Delta {S}_{\mathrm{init}}=-5.7$ cal/K·mol. |

$\Delta {H}_{\mathrm{sym}}=0$ kcal/mol $\Delta {S}_{\mathrm{sym}}=-1.4$ cal/K·mol. |

$\Delta {H}_{\mathrm{dangl}}=-5.9$ kcal/mol $\Delta {S}_{\mathrm{dangl}}=-16.38$ cal/K·mol. |

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**Figure 1.**(

**Top**) Strand sequences for the DNA constructs used in this work. Spacers (nucleotides that add flexibility to the structure but are not designed to pair up) are coloured in grey. Non-grey nucleotides that are part of complementary sequences share the same colour. (

**Bottom**) Simulation snapshots of a trimer and a tetramer, coloured according to the same color convention.

**Figure 2.**Radial distribution function of the centers of mass of the DNA tetramers as function of the inter-particle distance r, measured in units of the average arm length ${L}_{\mathrm{arm}}$, at fixed temperature $T=$ 25 °C and salt concentration $S=$ 0.5 M, for different reduced densities $\rho /{\rho}^{*}$.

**Figure 3.**Reduced pressure for the reference fluid as function of the reduced density ${B}_{2}\rho $, for different functionalities, salt concentrations and temperatures.

**Figure 4.**(

**a**) Reduced pressure of a reference fluid of tetramers, as function of the reduced density $\rho /{\rho}^{*}$. Points are data from numerical simulations; dashed lines are polynomial fits (Equation (4)). Inset: Pressure as function of $\rho /{\rho}^{*}$ for $S=0.5$ M: symbols and dashed line as in the main panel; the full line is the second-order virial approximation; (

**b**) Reference free energy as function of the reduced density $\rho /{\rho}^{*}$ computed using thermodynamic integration (TI) (full symbols) and a third-order virial approximation (dashed lines).

**Figure 5.**(

**a**) Critical temperature as function of salt concentration. (

**b**) Critical density as function of salt concentration. Circles and full line refer to tetramers ($f=$ 4); squares and dashed line refer to trimers ($f=$ 3). The data has been obtained with the second-order virial approximation [31] (black lines), the third-order virial approximation (full red symbols), thermodynamic integration (TI) (dashed green symbols) and experiments [8] (full blue symbols).

**Figure 6.**Main panel: Coexistence regions in the T–$\rho $ plane for tetramers at different salt concentrations. Symbols and lines refer to data obtained with second-order virial approximation (full lines) and third-order virial approximation (open symbols); full symbols refer to critical points. Inset: As in main panel, semi-log scale emphasizes the low-density branch.

**Figure 7.**Critical temperature as function of salt concentrations, estimated by taking into account the contribution of the the repulsive tails, as is done in [32]. Circles and solid lines refer to tetramers ($f=4$); squares and dashed lines refer to trimers ($f=3$). The data has been obtained with the second-order virial approximation [31] (orange and black lines for data obtained with and without tail corrections), the third-order virial approximation (full violet symbols) and experiments [8] (full blue symbols). Data obtained with thermodynamic integration (TI) overlap with the ${B}_{2}+{B}_{3}$ symbols and are therefore omitted for clarity.

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**MDPI and ACS Style**

Locatelli, E.; Rovigatti, L.
An Accurate Estimate of the Free Energy and Phase Diagram of All-DNA Bulk Fluids. *Polymers* **2018**, *10*, 447.
https://doi.org/10.3390/polym10040447

**AMA Style**

Locatelli E, Rovigatti L.
An Accurate Estimate of the Free Energy and Phase Diagram of All-DNA Bulk Fluids. *Polymers*. 2018; 10(4):447.
https://doi.org/10.3390/polym10040447

**Chicago/Turabian Style**

Locatelli, Emanuele, and Lorenzo Rovigatti.
2018. "An Accurate Estimate of the Free Energy and Phase Diagram of All-DNA Bulk Fluids" *Polymers* 10, no. 4: 447.
https://doi.org/10.3390/polym10040447