# Changes of Conformation in Albumin with Temperature by Molecular Dynamics Simulations

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## Abstract

**:**

## 1. Introduction

## 2. Results

## 3. Discussion

## 4. Materials and Methods

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Structure of the albumin protein in a ribbon-like form. The colors on the molecular surface indicate the secondary structure: blue—$\alpha $-helix, green—turn, yellow—3–10 helix, cyan—coil.

**Figure 2.**The bivariate histogram of chosen mean end-to-end signals in the 0–30 ns simulation time window.

**Figure 3.**The bivariate histogram of chosen mean end-to-end signals in the 70–100 ns simulation time window.

**Figure 4.**Logarithms of angle energies vs. the simulation time (0–10 ns), exemplary outcome of $T=300K$ and $T=309K$.

**Figure 5.**Logarithms of angle energies vs. the simulation time (90–100 ns), exemplary outcome of $T=300K$ and $T=309K$.

**Figure 6.**Dihedral energies for each temperature vs. simulation time. We use the double logarithmic scale. Green lines represent linear regression.

**Figure 8.**Numerically obtained distribution of angles $\varphi $ and $\psi $ for $T=300K$ and 30 ns of simulation.

**Figure 10.**Value of conformational entropy for various temperatures and 70–100 ns of simulation time.

**Figure 11.**The bivariate histogram of the chosen mean RMSD signals in the 0–30 ns simulation time window.

Parameter | $300\mathit{K}$ | $303\mathit{K}$ | $306\mathit{K}$ | $309\mathit{K}$ | $312\mathit{K}$ |
---|---|---|---|---|---|

${\langle {R}^{2}\rangle}^{1/2}$ [Å] for 0 ns–30 ns | 47.88 | 46.69 | 46.72 | 47.09 | 47.81 |

${\langle {R}^{2}\rangle}^{1/2}$ [Å] for 70 ns–100 ns | 47.12 | 46.67 | 47.72 | 47.13 | 47.64 |

**Table 2.**Values of H statistic in the Kruscal–Wallis test for two time windows for a root mean square end-to-end vector.

Window | Value of H |
---|---|

0–30 ns | $H=0.62$ |

70–100 ns | $H=1.22$ |

**Table 3.**Kullback–Leibler divergence for root mean square length of mean end-to-end signals in the time window 0–30 ns. The initial data is the same as for the Kruscal–Wallis test.

Temperature of the Reference Probability | $300\mathit{K}$ | $303\mathit{K}$ | $306\mathit{K}$ | $309\mathit{K}$ | $312\mathit{K}$ |
---|---|---|---|---|---|

$300K$ | 0 | 0.9324 | 1.3858 | 0.8946 | 0.7604 |

$303K$ | 0.9324 | 0 | 1.4628 | 0.8176 | 1.6165 |

$306K$ | 0.2802 | 0.3703 | 0 | 0.4096 | 0.1532 |

$309K$ | 0.9062 | 0.8161 | 1.4816 | 0 | 2.4139 |

$312K$ | 0.2040 | 0.3710 | 0.7023 | 1.1887 | 0 |

**Table 4.**Symmetric Kullback–Leibler distance for a root mean square length of end-to-end signals in the time window 0–30 ns. The initial data is the same as for the Kruscal–Wallis test.

$300\mathit{K}$ | $303\mathit{K}$ | $306\mathit{K}$ | $309\mathit{K}$ | $312\mathit{K}$ | |
---|---|---|---|---|---|

$300K$ | 0 | 0.9324 | 0.8330 | 0.9004 | 0.4822 |

$303K$ | 0.9324 | 0 | 0.9165 | 0.8169 | 0.9938 |

$306K$ | 0.8330 | 0.9165 | 0 | 0.9456 | 0.4278 |

$309K$ | 0.9004 | 0.8169 | 0.9456 | 0 | 1.8013 |

$312K$ | 0.4822 | 0.9938 | 0.4278 | 1.8013 | 0 |

**Table 5.**Kullback–Leibler divergence for a root mean square length of end-to-end signals in the time window 70–100 ns. The initial data is the same as for the Kruscal–Wallis test.

Temperature of the Reference Probability | $300\mathit{K}$ | $303\mathit{K}$ | $306\mathit{K}$ | $309\mathit{K}$ | $312\mathit{K}$ |
---|---|---|---|---|---|

$300K$ | 0 | 0.1983 | 0.1983 | 0.1409 | 0.2883 |

$303K$ | 0.7474 | 0 | 0.8555 | 1.6675 | 2.6390 |

$306K$ | 0.7474 | 0.8555 | 0 | 1.6675 | 1.1576 |

$309K$ | 0.1351 | 0.4423 | 0.4423 | 0 | 0.1925 |

$312K$ | 0.8113 | 1.2084 | 1.8656 | 0.7532 | 0 |

**Table 6.**Symmetric Kullback–Leibler distance for a root mean square length of end-to-end signals in the time window 70–100 ns. The data is the same for the Kruscal–Wallis test.

$300\mathit{K}$ | $303\mathit{K}$ | $306\mathit{K}$ | $309\mathit{K}$ | $312\mathit{K}$ | |
---|---|---|---|---|---|

$300K$ | 0 | 0.4728 | 0.4728 | 0.1380 | 0.5498 |

$303K$ | 0.4728 | 0 | 0.8555 | 1.0549 | 1.9237 |

$306K$ | 0.4728 | 0.8555 | 0 | 1.0549 | 1.5116 |

$309K$ | 0.1380 | 1.0549 | 1.0549 | 0 | 0.4728 |

$312K$ | 0.5498 | 1.9237 | 1.5116 | 0.4728 | 0 |

**Table 7.**Kullback–Leibler divergence between distributions of mean end-to-end signals in the time window 0–30 ns.

Temperature of the Reference Probability | $300\mathit{K}$ | $303\mathit{K}$ | $306\mathit{K}$ | $309\mathit{K}$ | $312\mathit{K}$ |
---|---|---|---|---|---|

$300K$ | 0 | 0.1056 | 0.3194 | 0.1367 | 0.1689 |

$303K$ | 0.0939 | 0 | 0.2370 | 0.1261 | 0.1425 |

$306K$ | 0.2062 | 0.1450 | 0 | 0.1528 | 0.1841 |

$309K$ | 0.1276 | 0.1144 | 0.1877 | 0 | 0.2319 |

$312K$ | 0.1288 | 0.1275 | 0.2126 | 0.1848 | 0 |

$300\mathit{K}$ | $303\mathit{K}$ | $306\mathit{K}$ | $309\mathit{K}$ | $312\mathit{K}$ | |
---|---|---|---|---|---|

$300K$ | 0 | 0.0997 | 0.2628 | 0.1321 | 0.1489 |

$303K$ | 0.0997 | 0 | 0.1935 | 0.1202 | 0.1350 |

$306K$ | 0.2628 | 0.1935 | 0 | 0.1703 | 0.1984 |

$309K$ | 0.1321 | 0.1202 | 0.1703 | 0 | 0.2084 |

$312K$ | 0.1489 | 0.1350 | 0.1984 | 0.2084 | 0 |

Temperature of the Reference Probability | $300\mathit{K}$ | $303\mathit{K}$ | $306\mathit{K}$ | $309\mathit{K}$ | $312\mathit{K}$ |
---|---|---|---|---|---|

$300K$ | 0 | 0.0920 | 0.1605 | 0.1824 | 0.1183 |

$303K$ | 0.1155 | 0 | 0.2969 | 0.1642 | 0.1474 |

$306K$ | 0.1108 | 0.1324 | 0 | 0.1621 | 0.1206 |

$309K$ | 0.1628 | 0.1266 | 0.2719 | 0 | 0.1206 |

$312K$ | 0.0886 | 0.0979 | 0.1356 | 0.1122 | 0 |

$300\mathit{K}$ | $303\mathit{K}$ | $306\mathit{K}$ | $309\mathit{K}$ | $312\mathit{K}$ | |
---|---|---|---|---|---|

$300K$ | 0 | 0.1038 | 0.1356 | 0.1726 | 0.1034 |

$303K$ | 0.1038 | 0 | 0.2147 | 0.1454 | 0.1226 |

$306K$ | 0.1356 | 0.2147 | 0 | 0.2170 | 0.1281 |

$309K$ | 0.1726 | 0.1454 | 0.2170 | 0 | 0.1164 |

$312K$ | 0.1034 | 0.1226 | 0.1281 | 0.1164 | 0 |

**Table 11.**Kullback–Leibler distance for raw data end-to-end signals in the time window 70–100 ns and $306K$.

Number of Molecule | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
---|---|---|---|---|---|---|---|---|---|

1 | 0 | 0.2927 | 0.2073 | 0.2156 | 0.2450 | 0.3016 | 0.1751 | 0.2072 | 0.4298 |

2 | 0.2927 | 0 | 0.1661 | 0.0919 | 0.0931 | 0.1589 | 0.2367 | 0.1310 | 0.2765 |

3 | 0.2073 | 0.1661 | 0 | 0.1347 | 0.2135 | 0.3059 | 0.1282 | 0.1013 | 0.1843 |

4 | 0.2156 | 0.0919 | 0.1347 | 0 | 0.1203 | 0.1353 | 0.1499 | 0.1338 | 0.2585 |

5 | 0.2450 | 0.0931 | 0.2135 | 0.1203 | 0 | 0.1478 | 0.2635 | 0.1434 | 0.3083 |

6 | 0.3016 | 0.1589 | 0.3059 | 0.1353 | 0.1478 | 0 | 0.2812 | 0.1978 | 0.3122 |

7 | 0.1751 | 0.2366 | 0.1282 | 0.1499 | 0.2635 | 0.2812 | 0 | 0.1318 | 0.2281 |

8 | 0.2072 | 0.1310 | 0.1013 | 0.1338 | 0.1434 | 0.1978 | 0.1318 | 0 | 0.1614 |

9 | 0.4298 | 0.2765 | 0.1843 | 0.2585 | 0.3083 | 0.3122 | 0.2281 | 0.1615 | 0 |

**Table 12.**Size exponent obtained according to Formula (5).

Parameter | $300\mathit{K}$ | $303\mathit{K}$ | $306\mathit{K}$ | $309\mathit{K}$ | $312\mathit{K}$ |
---|---|---|---|---|---|

Size exponent ($\mu $) for 0 ns–30 ns | 0.3986 | 0.3947 | 0.3948 | 0.3960 | 0.3984 |

Size exponent ($\mu $) for 70 ns–100 ns | 0.3961 | 0.3946 | 0.3981 | 0.3961 | 0.3978 |

**Table 13.**Values of H statistic in the Kruscal–Wallis test for $\mu $ parameter for two time windows.

Window | Value of Parameter H |
---|---|

0–30 ns | $H=0.48$ |

70–100 ns | $H=1.47$ |

$300\mathit{K}$ | $303\mathit{K}$ | $306\mathit{K}$ | $309\mathit{K}$ | $312\mathit{K}$ | |
---|---|---|---|---|---|

p-value | $0.849$ | $0.601$ | $0.009$ | $0.536$ | $0.143$ |

**Table 15.**Probability (p-value) for entropy of single trajectory over time in the time interval 0–30 ns.

Number of Signal | $300\mathit{K}$ | $303\mathit{K}$ | $306\mathit{K}$ | $309\mathit{K}$ | $312\mathit{K}$ |
---|---|---|---|---|---|

1 | 0.439 | 0.737 | 0.367 | 0.754 | 0.643 |

2 | 0.153 | 0.660 | 0.961 | 0.922 | 0.835 |

3 | 0.320 | 0.483 | 0.012 | 0.558 | 0.0551 |

4 | 0.263 | 0.372 | 0.700 | 0.223 | 0.323 |

5 | 0.544 | 0.657 | 0.057 | 0.130 | 0.066 |

6 | 0.114 | 0.385 | 0.232 | 0.416 | 0.360 |

7 | 0.499 | 0.529 | 0.008 | 0.006 | 0.007 |

8 | 0.013 | 0.871 | 0.610 | 0.692 | 0.456 |

9 | 0.245 | 0.891 | 0.871 | 0.832 | 0.694 |

mean | 0.288 | 0.621 | 0.424 | 0.504 | 0.382 |

**Table 16.**Probability (p-value) for entropy of a single trajectory over time in the time interval 70–100 ns.

Number of Signal | $300\mathit{K}$ | $303\mathit{K}$ | $306\mathit{K}$ | $309\mathit{K}$ | $312\mathit{K}$ |
---|---|---|---|---|---|

1 | 0.328 | 0.720 | 0.460 | 0.078 | 0.041 |

2 | 0.183 | 0.890 | 0.146 | 0.713 | 0.898 |

3 | 0.092 | 0.378 | 0.219 | 0.464 | 0.264 |

4 | 0.764 | 0.800 | 0.440 | 0.629 | 0.297 |

5 | 0.030 | 0.802 | 0.275 | 0.323 | 0.313 |

6 | 0.220 | 0.108 | 0.330 | 0.440 | 0.540 |

7 | 0.279 | 0.121 | 0.382 | 0.253 | 0.046 |

8 | 0.582 | 0.299 | 0.976 | 0.324 | 0.457 |

9 | 0.438 | 0.550 | 0.438 | 0.951 | 0.642 |

mean | 0.324 | 0.519 | 0.407 | 0.464 | 0.389 |

Set of Molecule | $300\mathit{K}$ | $303\mathit{K}$ | $306\mathit{K}$ | $309\mathit{K}$ | $312\mathit{K}$ |
---|---|---|---|---|---|

$300K$ | - | 1.2222 | 16.1111 | 12.5556 | 10.3333 |

$303K$ | 1.2222 | - | 17.3333 | 13.7778 | 11.5556 |

$306K$ | 16.1111 | 17.3333 | - | 3.5556 | 5.7778 |

$309K$ | 12.5556 | 13.7778 | 3.5556 | - | 2.2222 |

$312K$ | 10.3333 | 11.5556 | 5.7778 | 2.2222 | - |

**Table 18.**Kullback–Leibler divergence for the mean RMSD in the time window 0–30 ns. The data are the same as for the Kruscal–Wallis test.

Temperature of Reference Distribution | $300\mathit{K}$ | $303\mathit{K}$ | $306\mathit{K}$ | $309\mathit{K}$ | $312\mathit{K}$ |
---|---|---|---|---|---|

$300K$ | 0 | 0.4045 | 2.0767 | 2.917 | 0.1932 |

$303K$ | 1.6280 | 0 | 1.6853 | 2.4913 | 0.6891 |

$306K$ | 3.3967 | 2.6057 | 0 | 1.8597 | 2.3939 |

$309K$ | 2.2053 | 0.5003 | 1.1502 | 0 | 1.1444 |

$312K$ | 0.1598 | 0.1343 | 1.6845 | 2.5806 | 0 |

**Table 19.**Symmetric Kullback–Leibler distance for the mean RMSD in the time window 0–30 ns. The data are the same as for the Kruscal–Wallis test.

$300\mathit{K}$ | $303\mathit{K}$ | $306\mathit{K}$ | $309\mathit{K}$ | $312\mathit{K}$ | |
---|---|---|---|---|---|

$300K$ | 0 | 1.0162 | 2.7367 | 2.5600 | 0.1765 |

$303K$ | 1.1016 | 0 | 2.1455 | 1.4958 | 0.4117 |

$306K$ | 2.7367 | 2.1455 | 0 | 1.5049 | 2.0392 |

$309K$ | 2.5600 | 1.4958 | 1.5049 | 0 | 1.8625 |

$312K$ | 0.1765 | 0.4117 | 2.0392 | 1.8625 | 0 |

**Table 20.**Kullback–Leibler divergence for a mean RMSD in the time window 70–100 ns. The data are the same as for the Kruscal–Wallis test.

Temperature of Reference Distribution | $300\mathit{K}$ | $303\mathit{K}$ | $306\mathit{K}$ | $309\mathit{K}$ | $312\mathit{K}$ |
---|---|---|---|---|---|

$300K$ | 0 | 0.2941 | 2.7930 | 2.1527 | 3.9428 |

$303K$ | 0.8374 | 0 | 5.1756 | 2.2304 | 3.7127 |

$306K$ | 1.3305 | 2.4804 | 0 | 0.8902 | 2.8295 |

$309K$ | 4.0173 | 3.3975 | 3.3421 | 0 | 2.7354 |

$312K$ | 3.6680 | 2.2372 | 3.3862 | 0.8271 | 0 |

**Table 21.**Symmetric Kullback–Leibler distance for the mean RMSD in the time window 70–100 ns. The data are the same as for the Kruscal–Wallis test.

$300\mathit{K}$ | $303\mathit{K}$ | $306\mathit{K}$ | $309\mathit{K}$ | $312\mathit{K}$ | |
---|---|---|---|---|---|

$300K$ | 0 | 0.5657 | 2.0617 | 3.0850 | 3.8054 |

$303K$ | 0.5657 | 0 | 3.8280 | 2.8139 | 2.9749 |

$306K$ | 2.0617 | 3.8280 | 0 | 2.1161 | 3.1078 |

$309K$ | 3.0850 | 2.8139 | 2.1161 | 0 | 1.7812 |

$312K$ | 3.8054 | 2.9749 | 3.1078 | 1.7812 | 0 |

$300\mathit{K}$ | $303\mathit{K}$ | $306\mathit{K}$ | $309\mathit{K}$ | $312\mathit{K}$ | |
---|---|---|---|---|---|

$300K$ | 0 | 0.3686 | 0.7090 | 1.7109 | 0.1870 |

$303K$ | 0.3686 | 0 | 0.3051 | 1.2909 | 0.1838 |

$306K$ | 0.7090 | 0.3051 | 0 | 0.7468 | 0.5706 |

$309K$ | 1.7109 | 1.2909 | 0.7468 | 0 | 1.4513 |

$312K$ | 0.1870 | 0.1838 | 0.5706 | 1.4513 | 0 |

$300\mathit{K}$ | $303\mathit{K}$ | $306\mathit{K}$ | $309\mathit{K}$ | $312\mathit{K}$ | |
---|---|---|---|---|---|

$300K$ | 0 | 0.1056 | 6.3487 | 5.5232 | 3.9800 |

$303K$ | 0.1056 | 0 | 6.4065 | 5.7737 | 4.3824 |

$306K$ | 6.3487 | 6.4065 | 0 | 0.9648 | 1.7829 |

$309K$ | 5.5232 | 5.7737 | 0.9648 | 0 | 0.4260 |

$312K$ | 3.9800 | 4.3824 | 1.7829 | 0.4260 | 0 |

**Table 24.**Symmetric Kullback–Leibler distance for raw signals of the RMSD in the time window 70–100 ns for $306K$.

Number of Molecule | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
---|---|---|---|---|---|---|---|---|---|

1 | 0 | 1.669 | 0.388 | 0.277 | 0.137 | 0.500 | 3.260 | 0.560 | 0.249 |

2 | 1.669 | 0 | 1.974 | 2.565 | 2.074 | 1.615 | 0.660 | 1.498 | 1.410 |

3 | 0.388 | 1.974 | 0 | 0.241 | 0.617 | 0.246 | 3.589 | 0.299 | 0.352 |

4 | 0.277 | 2.565 | 0.241 | 0 | 0.250 | 0.494 | 3.867 | 0.628 | 0.406 |

5 | 0.137 | 2.074 | 0.617 | 0.250 | 0 | 0.540 | 3.593 | 0.597 | 0.275 |

6 | 0.500 | 1.615 | 0.246 | 0.494 | 0.540 | 0 | 3.338 | 0.080 | 0.140 |

7 | 3.257 | 0.660 | 3.589 | 3.867 | 3.593 | 3.337 | 0 | 3.118 | 2.795 |

8 | 0.560 | 1.498 | 0.300 | 0.628 | 0.597 | 0.080 | 3.118 | 0 | 0.165 |

9 | 0.249 | 1.410 | 0.352 | 0.406 | 0.275 | 0.140 | 2.795 | 0.165 | 0 |

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**MDPI and ACS Style**

Weber, P.; Bełdowski, P.; Domino, K.; Ledziński, D.; Gadomski, A.
Changes of Conformation in Albumin with Temperature by Molecular Dynamics Simulations. *Entropy* **2020**, *22*, 405.
https://doi.org/10.3390/e22040405

**AMA Style**

Weber P, Bełdowski P, Domino K, Ledziński D, Gadomski A.
Changes of Conformation in Albumin with Temperature by Molecular Dynamics Simulations. *Entropy*. 2020; 22(4):405.
https://doi.org/10.3390/e22040405

**Chicago/Turabian Style**

Weber, Piotr, Piotr Bełdowski, Krzysztof Domino, Damian Ledziński, and Adam Gadomski.
2020. "Changes of Conformation in Albumin with Temperature by Molecular Dynamics Simulations" *Entropy* 22, no. 4: 405.
https://doi.org/10.3390/e22040405