## 2 Results  
### Information Theory Applied to Genomics  
Our mathematical-physical analysis of genomic sequences is based on [PERSON]'s information theory [5] and the concept of information entropy (IE). For a given set of m independent and distinctive events, \(Y=\{y_{1},\,y_{2},\,\ldots\,\,y_{m}\}\), which are described by a discrete probability distribution \(P=\{p_{1},p_{2},\,\ldots\,\,p_{m}\}\) on \(Y\), the information content per event measured in bits, or the IE extracted when observing the set of events \(Y\) once is:  
\[H(Y)=\sum_{i=1}^{m}p_{i}\cdot\log_{2}\frac{1}{p_{i}} \tag{1}\]  
where the function \(H(Y)\) is called the Shannon IE, and it is maximum when the events \(y_{j}\) have equal probabilities of occurring, \(p_{i}=1/m\), so \(H(Y)=\log_{2}m\). Observing N sets of events \(Y\), or N times the set of events \(Y\), the information content is N\(\cdot H(Y)\) bits. A DNA sequence can be represented as a long string of the letters A, C, G, and T. These represent the four nucleotides: adenine (A), cytosine (C), guanine (G), and thymine (T) (replaced with uracil (U) in RNA sequences). Hence, within [PERSON]'s information theory framework, a typical genome can be represented as a four-state probabilistic system, with \(m=4\) distinctive events, \(Y=\{\text{A},\text{C},\text{G},\text{T}\}\), and probabilities \(p=\{p_{\text{A}},p_{\text{C}},p_{\text{G}},p_{\text{T}}\}\). Assuming the events have equal probabilities of occurring, using digital information units, i.e., the base of the logarithm is 2, and Equation (1) for \(m=4\), we determine that maximum Shannon IE is 2, indicating that the maximum information encodes per nucleotide is 2 bits, i.e., A = 00, C = 01, G = 10, T = 11. When the events occur with different probabilities, Relation (1) must be used to calculate the encoded information content. In order to clarify the procedure, let us consider an RNA subset consisting of N = 103 characters, which is exactly the window containing the first mutation (see LC542809 in Table 1 and window index 108 in Table 2):  
CAATTTTGACTTTGACTTTGTTAGTTTGCCAGAGATACTAATGGCTTTTGTTTTTTGTATGAAAATGCCTTTTTACTTTTGCATGGGTATTATTGC  
If the letters within this subset would occur with equal probabilities (1/4), then the subset would have \(H(Y)=2\), and a total entropy of N\(\cdot H(Y)=206\) bits of information. However, counting the occurrences of each nucleotide, the above subset has the following probability distribution: \(P=\left\{p_{\text{A}}=\frac{20}{103},p_{\text{C}}=\frac{16}{103},p_{\text{C}}= \frac{16}{103},p_{\text{T}}=\frac{51}{103}\right\}\). This results in the following:  
\[H(X)=-\left(\frac{20}{103}\log_{2}\left(\frac{20}{103}\right)+\frac{16}{103} \log_{2}\left(\frac{16}{103}\right)+\frac{16}{103}\log_{2}\left(\frac{16}{103} \right)+\frac{51}{103}\log_{2}\left(\frac{51}{103}\right)\right)=1.7958\]  
Therefore, the Shannon information entropy of this window is 1.7958 bits instead of 2 bits, and the total IE of the subset is 184.96 bits instead of 206 bits. After the mutation occurs (G \(\rightarrow\) T), the new probability distribution is \(P=\left\{p_{\text{A}}=\frac{20}{103},p_{\text{C}}=\frac{16}{103},p_{\text{C}}= \frac{15}{103},p_{\text{T}}=\frac{52}{103}\right\}\) and the IE of the mutated window is reduced to 1.779 bits.  
Considering the genome as a non-equilibrium information coding system with a very long relaxation time, and given the highly complex nature of genome architecture, the main assumption here is that the IE profile of genetic sequences may represent a characteristic pattern able to highlight characteristic changes at sites where the nucleotide sequence is changed, such as single nucleotide polymorphisms (SNPs), deletions, and insertions, suggesting not only a role for IE in the determination of genomic variation as already proven [6] but also a role in developing a probabilistic quasi-deterministic approach to genetic mutations.