Kappa‐Frameshift Background Mutations and Long‐Range Correlations of the Single‐Stand DNA/RNA Base Sequences

1. Introduction

Informational entropy analysis is often used in biology. The meaning of information entropy in molecular biology and evolution, however, should be distinguished from that of thermodynamic entropy [2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26]. The genetic information in DNA supplies straightforward examples for the informational entropy concept. The characteristic nucleotide sequence of the four bases (Adenine A, Guanine G, Cytosine C, and Thymine T) in an organism’s DNA is the same in all cells of the organism, and results in the combination of random mutations and functional selection during biological evolution. There is no preference for a specific sequence; $4^N$ different ssDNA sequences are equally possible (with $\text{'}\text{'}N\text{'}\text{'}$ being the number of bases in the ssDNA sequence, ranging from one million in bacteria to billions in many animals and plants), and the actual ssDNA sequence in an organism has only a $4^{-N}$probability$(p)$. Therefore, the space of the possible configurations of molecules within an organism is astronomically large. We may assume that the system is “non-ergodic” in the sense that not every part of the configuration space can be explored [3,27,28,29,30]. There are several major reasons to expect convergence [3]. Firstly, all evolutionary trajectories occur under certain generic selective constraints, such as the laws of mathematics, physics, and chemistry; these should lead to some universal features [31,32]. For instance, a novel quantum system (quantum dots) recently proposed by the author (2024) [33], due its peculiar quantum state–a Diophantine Markov state, would be trapped in a sequence of self-replicating states. This system would never reach a classical limit, no matter how big the cube the since gaps between states are always finite). Secondly, many of these spaces are not explored randomly [34,35,36,37,38,39,40,41,42,43]. Finally, evolutionary trajectories happen within a certain system: Darwinian evolution will act on populations of entities which emerge within assembly spaces [44,45,46]. They are constructed from components over evolutionary timescales. These factors shape evolution, and shift the way one should think about sequence search conditioned on the past. The logic of assembly spaces may lead to certain types of universal convergence [3,46].

DNA replication can lead to the introduction of small insertions or deletions of base pairs (insertion or deletion mutations, respectively). As shown in [47], the loss or addition of a single nucleotide causes all of the subsequent three-letter codons to change [48]. These mutations are called frameshift mutations because the frame of the triplet reading is altered during translation. A frameshift mutation occurs when any number of bases are added or deleted, except multiples of three which reestablish the initial reading frame.

Kappa-distributions emerge within the framework of statistical mechanics by maximizing the associated kappa-entropy $S_{¯}$under the constraints of the canonical ensemble (e.g., Livadiotis G. and McComas D. J. (2009, 2014, 2023) [1,49,50]). Kappa-distributions are consistent with thermodynamics, but this consistency alone cannot justify their generation and existence. Examples of kappa-distribution include super-statistics [51,52,53,54,55,56], the effect of shock waves [57], turbulence [58,59,60], the effect of pickup ions [61,62,63], the pump acceleration mechanism [64], colloidal particles [65], and polytropic behavior [66,67]. For all these phenomena, there is a unique thermodynamic origin. The existence of correlations among the particles of a system does not permit the usage of the classical framework based on Boltzmann [68]—Gibbs [69] (BG) entropy and the Maxwell–Boltzmann (MB)[70] canonical distribution of particle velocity ($u$). The thermodynamic origin of kappa-distributions has recently been connected to the new concept of “entropy defect”, proposed by G. Livadiotis G. and D.J. McComas (2021) [71,72]. Recently, Livadiotis and McComas (2023) [1] introduced a specific algebra of the addition rule with entropy defect. The addition rule forms a mathematical Abelian group on the set of any measurable physical-quantity (including entropy), called the kapa-addition and referred to by the symbol${\oplus }_{¯}$. The physical meaning of the kappa index can be understood through particle (subsystems) correlations: the kappa index provides a measure of the loss (for $¯>0$ ) or gain (for $¯<0$) of entropy $\Delta S_{system}$after the mixing of the partial systems A and B to compose the combined system [1].

Shannon, Rényie and Tsallis entropy analysis of DNA have been cited by J.A. Tenreiro Machadoa et.al [73]. Dynamics of a single base point genome mutation (T into A) using non-equilibrium information theory have been stated by Melvin M. Vopson, Samuel C. Robson [74,75]. Following Livadiotis and McComas (2023) [1], we propose a new type of ssDNA frameshift mutations that occur spontaneously due to information exchange between the ssDNA sequence of length bases ($N$) and the mutation sequence of length base (M), and respect the kappa-addition symbol ${\oplus }_{¯}$. We call these proposed mutations Kappa-Frameshift Background (KFB) Mutations. We find that the origin of an entropy defect is based on the interdependence of the information length systems (or their interconnectedness, that is, the dependence, or connection, of systems with a significant number of constituents (information length bases). The proposed KFB mutations quantify the correlations among the information length bases–constituents of these systems–due to information exchanges. In the presence of entropy defects, the Landauer’s bound and the minimal metabolic rate for a biological system are modified. We observe that the different $N$ and $¯$ scales are manifested in the evolution of the proposed biological system through subsystems correlations. In this evolution, for short time scales and specific values of the kappa-parameter, we can expect deterministic laws associated with a single biological polymer.

2. Kappa-Frameshift Background Single-Stand DNA Mutations

Spontaneous DNA mutations occur suddenly and their origin is unknown. Such mutations, also called “background mutations”, have been reported in many organisms, such as Oenothera, maize, bread molds, microorganisms (bacteria and viruses), Drosophila, mice, humans, etc. (see Ref [48] for details). A frameshift mutation occurs when any number of bases are added or deleted during DNA replication, except multiples of three which reestablish the initial reading frame. Mutations of multiples of three nucleotides just add one or several reading frames into the sequence without altering the composition of those reading frames thereafter. It is possible, however, that the frameshift mutation causes early termination of the translation (e.g., when a 3-nucleotide stop codon sequence is introduced). Frameshift mutations can, therefore, have very severe results if they occur early in the coding sequence. Frameshift mutations can occur at microsatellites (tandem repeats of 1–6 base pairs per repeat unit) and also at non-iterated or short repetitive sequences. Proofreading can remove insertion or deletion mutations in short repetitive and non-iterated sequences with almost the same efficiency as that for point mutations; however, it is much less efficient in removing frameshift mutations in homopolymeric runs [47]. These mutations are removed efficiently by mismatch repair systems. The instability of microsatellite sequences is associated with many diseases, including cancer [48].

Since DNA is double-stranded (ds), bases should be base pairs. In the present paper we only use the 5'–3' sequence, while the other sequence is implied. The 5'–3' sequence denotes a single strand of a nucleic acid (ssDNA or RNA). Following Livadiotis and McComas (2023) [1], we propose a new type of ssDNA frameshift mutations that occur spontaneously due to information exchange between the ssDNA sequence of base lengths ($N$) and the mutation sequence of base lengths (M), and respect the kappa addition (${\oplus }_{¯}$). We call these proposed mutations Kappa-Frameshift Background (KFB) mutations. KFB mutations will occur when any number of bases$M$are added or deleted to the initial ssDNA sequence$N$while respecting ${\oplus }_{¯}$(except multiples of three, which will reestablish the initial reading frame). In the case of the kappa-addition of any number of bases $M\in \mathbb{N}$ to the initial ssDNA base sequence$N\in \mathbb{N}$, we define the set of kappa-natural numbers$\left(\mathbb{N},{\oplus }_{¯}\right)$ as the set of natural numbers$\mathbb{N}$ that satisfies kappa-addition law. The proposed KFB mutations are then given as follows:

$$Y_{¯}=N{\oplus }_{¯}M=N+M-{\displaystyle \frac{NM}{¯}},$$

(1)

where$Y_{¯}$are not multiples of three, $N$are multiples of three,$M$are not multiples of three, and$¯\in {\mathbb{Q}}^{*}$, which brings the usual sum as a particular case of ⊕_∞ ≡ +. Relation 1 can be expressed in the convenient product form [1]:

$$1-{\displaystyle \frac{N{\oplus }_{¯}M}{¯}}=\left(1-{\displaystyle \frac{N}{¯}}\right)\left(1-{\displaystyle \frac{M}{¯}}\right),$$

(2)

where$N$is the initial length of information (string length) of the ssDNA or biological polymer that is being copied and $M$ (no multiples of three) is the length of information (string length) of the mutation added to the initial base $N$. The κ-sum is commutative (N⊕_κM = M⊕_κN) and associative (N⊕_κ(M⊕_κX) = (N⊕_κM)⊕_κX), but not distributive in relation to the usual multiplication (α(N⊕_κM)≠(αN⊕_καM)). The neutral element of the κ-sum is zero, N⊕_κ0 = N.

Similarly, in the case of κ-deletion of any number of bases $M\in \mathbb{N}$ (no multiples of three) from the initial ssDNA base sequence$N\in \mathbb{N}$, the proposed KFB mutation (see Table.1) is given by the following:

$$\begin{array}{l}{\overline{Y}}_{¯}=N{\oplus }_{¯}\overline{M}=N+\overline{M}-{\displaystyle \frac{N\overline{M}}{¯}}={\displaystyle \frac{N-M}{\left(1-\frac{M}{¯}\right)}},\\ \overline{M}={\displaystyle \frac{-M}{\left(1-\frac{M}{¯}\right),}}\end{array}$$

(3)

where ${\overline{Y}}_{¯}$ is not multiples of three,$N$ is multiples of three, $M\ne ¯$ (not multiples of three), and$¯\in {\mathbb{Q}}^{*}$. Equation (3) is κ-subtraction, the inverse operation of κ-addition (1): physically, it means that some base(s) are deducted from the initial DNA or biological polymer.

The κ-difference obeys N⊕_κm̅=⊕_κm̅⊕_κN and N⊕_κ(m̅⊕_κX) = (N⊕_κm̅)⊕_κX, but α(N⊕_κm̅) ≠ (αN⊕_καm̅). For both κ-addition to and κ-subtraction from the initial base$N$of any bases $M$which are multiples of three, we obtain the KFB changes of the information length of the ssDNA or biological polymer being copied. The KFB mutation just adds one or several reading frames into the sequence without altering the composition of those reading frames thereafter. The meaning of the kappa index in the proposed KFB mutation can be understood through the subsystem (subbases) of correlations between$N$and$M$due to information exchanges within a given length of information (string length) in the ssDNA or biological polymer being copied. In fact, a simple relation exists between the Pearson correlation coefficient $R$ and the kappa index $¯$:

$$R¯=d_{¯}/2,$$

(4)

where $¯\in {\mathbb{Q}}^{*}$for subbases with $d_{¯}$-degrees of freedom [71].

The largest value of kappa,$¯\to \infty$, corresponds to an ssDNA or biological polymer characterized by the absence of any correlations between subbases of $N,M\in \mathbb{N}$:

$$Y_{\infty }=N{\oplus }_{\infty }M=N+M,$$

(5)

where $Y_{\infty }=Y$is not multiples of three ,$N$is multiples of three and$M$is not multiples of three. This condition is the frameshift mutation. The smallest possible kappa value, ${¯}_{\min }=d_{¯}/2$, corresponds to the state furthest from the ordinary frameshift mutation of an ssDNA or biological polymer, and is characterized by the strongest correlation between N and M due to information exchange. The parameter $¯$ provides a measure of the loss ($¯>0$ ) or gain ($¯<0$) of information after the mixing of the partial bases $N$ and $M$ to compose the combined finial ssDNA or biological polymer. To ensure that the values of Equation (1) are not divisible by 3, the positive integers $N$,$M$and $¯$must satisfy the following conditions:

$N$: Information length of the initial sequence; any nuclide “letter” of the initial ss -DNA/RNA sequence is multiple of 3.

$$\begin{gathered} N\equiv 0(\mathrm{mod}3), \\ N=3,6,9,12,\dots . \end{gathered}$$

$M$: Information length of the insert sequence; any nuclide “letter” of the adding ss DNA/RNA sequence is not divisible by 3.

$M\ne 0(\mathrm{mod}3)$, $M=1,2,4,5,\dots ,$$¯$: Any positive factor of $NM$such that either:

1. $¯$ has fewer factors of 3 than $N$does, or

2. $¯$ has exactly the same number of factors of 3 as $N$does, provided that $M\ne {\displaystyle \frac{NM}{¯}}(\mathrm{mod}3)$.The details of the insert KFB mutation are given in the Table.1

Table 1. Details of κ-frameshift background mutation (1). The traditional reading frame and κ-reading frame used for the definition of the KFB mutation are given in Table 2 and Table 3, respectively.

$N$	Information length of the initial sequence; Number of nuclide “letters” of the initial ss DNA/RNA sequence not divisible by 3.
$M$	Information length of the insert sequence; Number of nuclide “letters” of the adding mutation in the ss DNA/RNA sequence not divisible by 3.
$N+M$	Maximum of non-correlated number of nuclide “letters” of the mutated ss DNA/RNA sequence not divisible by 3.
${\displaystyle \frac{NM}{¯}}$	Information defect: the correlated number of nuclide “letters” of the mutated ss DNA/RNA sequence.
$Y_{¯}=N{\oplus }_{¯}M$	Number of nuclide “letters” of the mutated ss DNA/RNA sequence not divisible by 3, for specific values of the κ parameter.

Table 1. Details of κ-frameshift background mutation (1). The traditional reading frame and κ-reading frame used for the definition of the KFB mutation are given in Table 2 and Table 3, respectively.

$N$	Information length of the initial sequence; Number of nuclide “letters” of the initial ss DNA/RNA sequence not divisible by 3.
$M$	Information length of the insert sequence; Number of nuclide “letters” of the adding mutation in the ss DNA/RNA sequence not divisible by 3.
$N+M$	Maximum of non-correlated number of nuclide “letters” of the mutated ss DNA/RNA sequence not divisible by 3.
${\displaystyle \frac{NM}{¯}}$	Information defect: the correlated number of nuclide “letters” of the mutated ss DNA/RNA sequence.
$Y_{¯}=N{\oplus }_{¯}M$	Number of nuclide “letters” of the mutated ss DNA/RNA sequence not divisible by 3, for specific values of the κ parameter.

Table 2. Traditional reading frame.

$N$	$M$	$Y=N+M$	Traditional reading
3	1	3 + 1 = 4	frameshift ( Y is not div. by 3)
6	1	6 + 1 = 7	frameshift ( Y is not div. by 3)
3	1	3 + 1 = 4	frameshift ( Y is not div. by 3)
3	2	3 + 2= 6	In frame (Y is div. by 3)
6	1	6 + 1 =7	frameshift ( Y is not div. by 3)

Table 2. Traditional reading frame.

$N$	$M$	$Y=N+M$	Traditional reading
3	1	3 + 1 = 4	frameshift ( Y is not div. by 3)
6	1	6 + 1 = 7	frameshift ( Y is not div. by 3)
3	1	3 + 1 = 4	frameshift ( Y is not div. by 3)
3	2	3 + 2= 6	In frame (Y is div. by 3)
6	1	6 + 1 =7	frameshift ( Y is not div. by 3)

Table 3. κ-reading frame.

$¯$	$N$	$M$	$Y_{¯}=N{\oplus }_{¯}M$	κ-reading frame
1	3	1	3 + 1-3 = 1	κ=1-frameshift ( $Y_{¯=1}$is not div. by 3)
2	6	1	6 + 1-3 = 4	κ=2-frameshift ($Y_{¯=2}$ is not div. by 3)
3	3	1	3 + 1-1 = 3	In κ=3-frame ($Y_{¯=3}$is div. by 3)
3	3	2	3 + 2-2 = 3	In κ=3-frame ($Y_{¯=3}$ is div. by 3)
3	6	1	6 + 1-2 = 5	κ=3-frameshift ( $Y_{¯=3}$is not div. by 3)

Table 3. κ-reading frame.

$¯$	$N$	$M$	$Y_{¯}=N{\oplus }_{¯}M$	κ-reading frame
1	3	1	3 + 1-3 = 1	κ=1-frameshift ( $Y_{¯=1}$is not div. by 3)
2	6	1	6 + 1-3 = 4	κ=2-frameshift ($Y_{¯=2}$ is not div. by 3)
3	3	1	3 + 1-1 = 3	In κ=3-frame ($Y_{¯=3}$is div. by 3)
3	3	2	3 + 2-2 = 3	In κ=3-frame ($Y_{¯=3}$ is div. by 3)
3	6	1	6 + 1-2 = 5	κ=3-frameshift ( $Y_{¯=3}$is not div. by 3)

In-frame refers to the reading of the genetic code in the correct sequence of three-letter units (codons). Because DNA is read in sets of three nucleotides, maintaining the traditional reading frame ensures that the cell translates the genetic message into the proper sequence of amino acids to build a functional protein. A traditional reading frame (Table 2) is the specific starting point that dictates which triplets are read.

κ-frame refers to that reading of the genetic code in the correct sequence of three-letter units (codons) that is achieved only for specific values of the κ index. The DNA is read in sets of three nucleotides with respects to the κ-index. Maintaining the κ-reading frame may yield translations different to the traditional in-frame translations. κ-frame may thus translate the genetic message into a sequence of amino acids that build a function different to the know traditional function. We note that a κ-frame reading (Table 3) is the specific starting point that dictates which triplets are read only for specific values of the κ-index.

Example 1: Let a very large string of A, C, G, and T letters, which stand for the nucleotides adenine (A), cytosine (C), guanine (G), and thymine (T) (or uracil (U) replacing T in RNA). Let us consider a random ssDNA subset, consisting of N=15 letters:

$$(TCA)(GAC)(ATA)(TAC)(CAA)$$

(A) 

The M=1 base G frameshift mutation at index 6 (insert G) yields to the following mutated subset:

$$\begin{array}{l}(TCA)(GAC)(\stackrel{\downarrow (+)Insert}{G})(ATA)(TAC)(CAA)\\ \to (TCA)(GAC)(GTA)(CCA)(A\mathrm{..}).\end{array}$$

(Β) 

Now let the corresponding insert KFB-mutation diagram of the frameshift diagram (Β) be:

$$\begin{gathered} (TCA)(GAC)(\stackrel{\downarrow (\oplus ¯)Insert}{G})(ATA)(TAC)(CAA)\to \\ \underset{\begin{array}{l}\downarrow (-)Delete-correlation\\ [\left({\displaystyle \frac{G}{¯}}\right)\times (TCA)(GAC)(ATA)(TAC)(CAA)]\end{array}}{{\displaystyle \{(TCA)(GAC)(\stackrel{\downarrow (+)Insert}{G})(ATA)(TAC)(CAA)\}}} \\ \underset{[\left({\displaystyle \frac{G}{¯}}\right)\times (TCA)(GAC)(ATA)(TAC)(CAA)]}{{\displaystyle \underset{\downarrow (-)Delete-correlations}{{\displaystyle \to (TCA)(GAC)(GTA)(CCA)(A\mathrm{..}).}}}} \end{gathered}$$

(Γ) 

Let us assume that κ=1, meaning that only one letter is non-correlated – for example the insert letter G. Evaluating Diagram (Γ) yields the following KFB-mutated subset:

$$(TCA)(GAC)(\stackrel{\downarrow (\oplus 1)Insert}{G})(ATA)(TAC)(CAA)\to G.$$

(Δ) 

In the case of all the letters being non-correlated and κ having non-integer values, it is:

$$\begin{array}{l}¯\equiv {\displaystyle \frac{N+M}{NM}}\equiv {\displaystyle \frac{(TCA)(GAC)(CCA)(A\mathrm{..}).}{G(TCA)(GAC)(ATA)(TAC)(CAA),}},\\ N+M\equiv (TCA)(GAC)(CCA)(A\mathrm{..}).\\ NM\equiv G(TCA)(GAC)(CCA)(A\mathrm{..}).\end{array}$$

(Ε) 

Evaluating Diagram (Γ) using Diagram (Ε) yields the following subset:

$$(TCA)(GAC)(\stackrel{\downarrow ({\oplus }_{N+M/NM})Insert}{G})(ATA)(TAC)(CAA)\to 0.$$

(Z) 

Diagram (Ζ) is in the null κ-frame for same non-integer values of κ.

3. Information Defects in the ssDNA/RNA or Biological Polymer

Following Ricard Solé et al. (2024) [3], the size of the sequence space $\Omega (N)$ associated with a biological polymer of information length$N\in \mathbb{N}$, built from a molecular alphabet $\Sigma =\{A_1,A_2{A}_3,\dots ,A_N\}$of size $\left|\Sigma \right|$, is:

$$\Omega (N)=|\Sigma {|}^N.$$

(6)

For example, the space of possible proteins with a length base $N$of 1000 amino acids is $\Omega ={20}^{1000}$– a space too large to be explored [27]. The space of possible molecular configurations within an organism is astronomically large. The probability $(p)$ for such a protein to exist in nature is $p(\Omega )={20}^{-1000}$. We may assume that biological systems are “non-ergodic” in the sense that not every part of their configuration space can be explored [28,29,30].

In statistical physics, an ergodic system is one which, over sufficiently long time-scales, explores all possible microstates that are consistent with its macroscopic properties. Mathematically, for any physically observable $Q(q)$, where $q$ is the microstate that specifies all of the system’s coordinates $(q=(q_1,q_2,q_3,\mathrm{..}))$, the long-time average of $Q(q)$converges to the ensemble average, so:

$$〈O〉=\underset{\tau \to \infty }{\lim }{\displaystyle \frac{1}{\tau }}{\displaystyle {\displaystyle \underset{0}{\overset{\tau }{\int }}}O(q(t))dt}.$$

(7)

In intuitive terms, an ergodic system explores (over time) all the possible ways it can exist. Given enough time, the system will visit all its available configurations or states [3]. There are several major reasons to expect convergence [3]. Firstly, all evolutionary trajectories occur under certain generic selective constraints, such as the laws of mathematics, physics, and chemistry. These constrains should lead to universal features [31,32]. Secondly, many of these states are not explored randomly. Finally, evolutionary trajectories form within a certain system: Darwinian evolution acts on populations of entities, which emerge within the system in assembly spaces [44, 45, and 46].

The second law of thermodynamics states that a physical process must generate an overall increase$\Delta S_{total}$in the entropy of a system ($\Delta S_{system}$) and its environment ($\Delta S_{env}$):

$$\Delta S_{total}=\Delta S_{systen}+\Delta S_{env}.$$

(8)

Therefore, living systems can only perform processes that reduce entropy internally if, at the same time, they produce an even greater increase in entropy in the environment. Thus, the universal thermodynamic logic of life is that low-entropy input (“resource”) is turned into high-entropy output (“waste”), which can be expressed as:

$$\begin{gathered} \Delta S_{system}<0, \\ \Delta S_{env}>0, \\ |\Delta S_{env}|>|\Delta S_{system}|, \end{gathered}$$

(9)

where

$$\Delta S_{env}=Q/(k_{B}T)$$

(10)

is the entropy production in the environment, $Q$ is the amount of heat released, $T$is the thermal reservoir at temperature $k_B=1.38\times {10}^{-23}J/K$ Boltzmann’s constant.

This suggests that all lifeforms face the problem of maintaining low internal entropy in a race against the second law of thermodynamics.

In non-equilibrium thermodynamics [3], the rate of entropy production per unit time τ can be expressed as an integral:

$$\Delta S_{total}/\tau ={\displaystyle {\int }_V\sigma dV},$$

(11)

for a system of volume $V$, with the specific entropy production rate, $\sigma$, written as follows:

$$\sigma ={\displaystyle \sum _{k=1}^N{X}_k{J}_k}.$$

(12)

In Equation (12),$\{X_1,X_2,X_3,\dots ,X_N\}$indicates a set of “forces" (such as temperature gradients or chemical affinities) and $\{J_1,J_2,J_3,\dots ,J_N\}$is the set of conjugate fluxes (e.g., heat flows, reaction rates). According to Shannon’s formula [2,22,23,24], the evolution of the base sequences of modern organisms from an unspecified base sequence resulted in a gain of information (a decrease in entropy: negative$\Delta S_{system}$), as follows:

$$\Delta S_{system}(N)=S_f(N)-S_i(N).$$

(13)

For the case of writing a specific string from a set of unordered letters ($N=0$), we have:

$$S_f(N)=\ln (|\Sigma {|}^N)=N\ln |\Sigma |=0(\mathrm{if}\mathrm{and}\mathrm{only}\mathrm{if}N=0).$$

(14)

Then

$$\Delta S_{system}(N)=-S_i(N)=-\ln (|\Sigma {|}^N)=-N\ln |\Sigma |,$$

(15)

$\left|\Sigma \right|$ is the number of unique elements (or letters) in the informational system and $N$is the length of information (the string length) that is being copied.

Let us now examine the possible evolutionary events resulting from the combination of the proposed KFB mutations and functional selection during biological evolution in the presence of information exchanges between the length bases. This examination involves the following steps: Firstly, we introduce the kappa-multiplication (${\odot }_{¯}$) acting on the space of a possible ssDNA/RNA or biological polymer, $\Omega (N)$ and $\Omega (M)$, with a length base of $N$ and $M$, respectively. In a similar way that the κ-addition ${\oplus }_{¯}(N,M)$results from the usual addition$+(N,M)$in Equation (1), the kappa-addition ${\oplus }_{¯}$ of bases $N$ and $M$ in Equation (6) gives the following:

(16a)

for $N,M\in (\mathbb{N},{\oplus }_{¯})$and $¯\in {\mathbb{Q}}^{*}$. Equation (16α) shows the proposed KFB mutation space of the two possible ssDNA/RNA or a biological polymers (see Table 1 and Diagrams A–Z), with$\Sigma$ being a molecular alphabet of size$\left|\Sigma \right|$. Equation (16α) is the standard morphism from $\left(\mathbb{N},{\oplus }_{¯}\right)$ to$\left({\mathbb{N}}^{*},{\odot }_{¯}\right)$. Morphism (16α) is defined with the help of the morphisms: $\Omega (N)\times \Omega (M)=\Omega (N+M)$and $\Omega {(N)}^{-M/¯}$from $\left(\mathbb{N},+\right)$to $\left({\mathbb{N}}^{*},\times \right)$. We note that the standard morphism (16α) brings the standard morphism $\Omega (N)\times \Omega (M)=\Omega (N+M)$ as a particular case of ⊙_∞ ≡ × and ⊕_∞ ≡ + for (κ→∞). By reciprocity, the logarithm function is the standard morphism from$\left({\mathbb{N}}^{*},{\odot }_{¯}\right)$ to$\left(\mathbb{N},{\oplus }_{¯}\right)$; namely:

$$\begin{array}{l}{\log }_{\left|\Sigma \right|}\left(\Omega (N){\odot }_{¯}\Omega (M)\right)\equiv {\log }_{\left|\Sigma \right|}(\Omega (N)\times \Omega (M)\times \Omega {(N)}^{-M/¯})\\ ={\log }_{\left|\Sigma \right|}(\Omega (N+M)\times \Omega {(N)}^{-M/¯})\\ ={\log }_{\left|\Sigma \right|}\Omega (N+M)+{\log }_{\left|\Sigma \right|}\Omega {(N)}^{-M/¯}={\log }_{\left|\Sigma \right|}\Omega (N)+{\log }_{\left|\Sigma \right|}\Omega (M)\\ -{\displaystyle \frac{M}{¯}}{\log }_{\left|\Sigma \right|}\Omega (N)={\log }_{\left|\Sigma \right|}\Omega (N)+{\log }_{\left|\Sigma \right|}\Omega (M)\\ -{\displaystyle \frac{{\log }_{\left|\Sigma \right|}\Omega (N){\log }_{\left|\Sigma \right|}\Omega (M)}{¯}}={\log }_{\left|\Sigma \right|}\Omega (N){\oplus }_{¯}{\log }_{\left|\Sigma \right|,}\Omega (M)\end{array}$$

(16β)

Equation (16b) brings the standard morphism:

${\log }_{\left|\Sigma \right|}(\Omega (N)\times \Omega (M))={\log }_{\left|\Sigma \right|}\Omega (N)+{\log }_{\left|\Sigma \right|}(M)$from $\left({\mathbb{N}}^{*},\times \right)$ to $\left(\mathbb{N},+\right)$ as a particular case of ⊙_∞ ≡ × and ⊕_∞ ≡ + for (κ→∞).

We impose the condition:

$$N{\oplus }_{¯}M=N+M-{\displaystyle \frac{NM}{¯}}\ge 0,$$

(17)

for$N$, $M$(see Table.1), and$¯\in {\mathbb{Q}}^{*}$.This implies the following restriction to the values of the κ-parameter:

$$¯\le {\displaystyle \frac{H\mathrm{armonic}\mathrm{mean}(\mathrm{N},\mathrm{M})}{2}}={\displaystyle \frac{NM}{N+M}}.$$

(18)

For subbases with $d_{¯}$degrees of freedom [71], using Equation (4), we obtain the following restriction to the values of the correlation coefficient $R$:

$$R={\displaystyle \frac{d_{¯}}{2¯}}\le {\displaystyle \frac{d_{¯}(N+M)}{2NM}}.$$

(19)

The largest value of kappa,$¯\to \infty$, corresponds to ssDNA/RNA or biological polymer without any correlation between the subbases of information length $N$ and $M$ given by Equation (5). The space of the two possible proteins (Equation (16α)) then becomes:

$$\Omega (N){\odot }_{\infty }\Omega (M)=\Omega (N)\times \Omega (M)\times 1=|\Sigma {|}^{N{\oplus }_{\infty }M}=|\Sigma {|}^{N+M},$$

(20)

for $N,M\in (\mathbb{N},{\oplus }_{¯})$ and $¯\in {\mathbb{Q}}^{*}$. This is the frameshift mutation space of the two possible proteins. We may write Equation (16a) as follows:

$$\Omega (N){\odot }_{¯}\Omega (M)=\Omega (N)\times \Omega (M)\times \Omega {(N,M)}^{(corel)}=|\Sigma {|}^{N{\oplus }_{¯}M},$$

(21)

for$N,M\in (\mathbb{N},{\oplus }_{¯})$,$¯\in {\mathbb{Q}}^{*}$. In Equation (21), the correlation space of the two possible ssDNA/RNA or biological polymers, $\Omega (N)$ and $\Omega (M)$ (with a length base $N$ and $M$, respectively) due to information exchange is given by:

$$\Omega {(N,M)}^{(correl)}=\Omega {(N)}^{-M/¯}=|\Sigma {|}^{-{\displaystyle \frac{NM}{¯}}},$$

(22)

$N,M\in (\mathbb{N},{\oplus }_{¯})$,$¯\in {\mathbb{Q}}^{*},$The κ-product (16α) is commutative (Ω(N)⊙_κΩ(M)=Ω(N)⊙_κΩ(M)) and associative (Ω(N) ⊙_κ(Ω(M)⊙_κΩ(X))=(Ω(N)⊙_κΩ(M))⊙_κΩ(X)). The number 1 is the neutral element of the κ-product (Ω(N)⊙_κ1=Ω(N). This permits us to define the inverse multiplicative, 1⊙_κ Ω(Nˉ), by means of Ω(N)⊙_κ(1⊙_κΩ(Nˉ)) ≡ 1.

Similarly, in the case of the deletion of any number of bases $M$ (except multiples of three – see Table 1) from the initial ssDNA sequence of base$N$, the space of the two possible ssDNA/RNA or biological polymers, Ω(M) and Ω(M), respectively, is:

$$\begin{gathered} \Omega (N){\odot }_{¯}\Omega (\overline{M})=|\Sigma {|}^{(N-M)/1-M/¯)}=|\Sigma {|}^{N{\oplus }_{¯}\overline{M}}, \\ N,M\in (\mathbb{N},{\oplus }_{¯}), \\ ¯\in {\mathbb{Q}}^{*}. \end{gathered}$$

(23)

Using Equations (13)–(15), the kappa-gain of information (a decrease in kappa-entropy: negative$\Delta S_{¯-system}$) of the system (ssDNA or biological polymer) during the proposed KFB mutations (see Diagrams (A-Z)) is given by:

(24)

$\Delta S_{¯-system}$(N,M) is the entropy of a composite system (ssDNA or biological polymer) with correlations between N and M, where:

$$S(N)=-N\ln (|\Sigma |),$$

(25)

$$S(M)=-M\ln (|\Sigma |),$$

(26)

$$\Delta S_{Def}(N,M)={\displaystyle \frac{NM}{¯}}\ln (|\Sigma |).$$

(27)

Equation (27) is the proposed entropy defect for the ssDNA or biological polymer.

The characteristic non-additive rule of the kappa entropy [1] was first derived on a physical basis through the concept of entropy defect [1]. Then, it was demonstrated that the entropy defect can lead to the generalized, kappa-associated entropy [1].Here, we explain the origin of entropy defects based on the interdependence of information length systems (or the interconnectedness of these systems: the condition whereby systems with a significant number of constituents (information length bases) dependent on, and/or are connected to, each other). The proposed KFB mutation quantifies the correlations among the information length of the bases (ssDNA or polymer constituents) due to information exchanges. Interdependence induces order and reduces the entropy-information in ssDNA or biological polymer systems thus producing entropy-information defects. Interdependence is measured through the magnitude $1/¯$of the entropy-information defect, defined as:

$${\displaystyle \frac{1}{¯}}={\displaystyle \frac{1}{\Delta S_{¯-system}(N,M)}}{\displaystyle \frac{\partial S_{Def}(N,M)}{\partial S_{\infty -system}(N,M)}},$$

(28)

where$\Delta S_{¯-system}(N,M)$(see Equation (24)) is the entropy-information of the system with correlations among its constituents (e.g., information length bases), while

$$\Delta S_{\infty -system}(N,M)=-\left(N+M\right)\ln \left(\left|\Sigma \right|\right),\mathrm{for}(\Delta S_{Def}(N,M)\to 0\mathrm{hence}¯\to \infty )$$

(29)

denotes the entropy-information of this system if there was no correlation among the information length bases. The smallest possible kappa value,$¯=d_{¯}/2$, corresponds to the state furthest from the ordinary frameshift mutation of a ssDNA or biological polymer, characterized by the highest correlations between bases due to information exchange. The parameter $¯$provides a measure of the loss ($¯>0$ ) or gain ($¯<0$) of information after the mixing of the partial bases $N$ and $M$ to compose the combined final ssDNA or biological polymer.

The entropy of a biological system with correlations is lower than the entropy of the biological system without correlations, because the correlations add order to the biological system. In a composite biological system, its entropy differs from that of the sum of the entropies of its constituent biological subsystems by the entropy defect.

4. Properties of the Information-Entropy Defect in ssDNA/RNA or Biological Polymers

The information defect determines how the information of the ssDNA or biological polymer partitions into the entropy information of its constituents. The participation of each constituent’s entropy-information in the information defect must be (1) separable, (2) symmetric, and (3) (upper- or lower-) bounded for information loss ($¯>0$) or gain ($¯<0$). Below, we analyze the physical meaning and mathematical formulation of the three properties that constitute the foundations of information defect:

(1) Separable. The entropy-information defect relates to the constituent’s entropies (information) through a separable function:

$$S_{Def}(N,M)\propto S_{Def}(N,M=const.)\times S_{Def}(N=const.,M).$$

(30)

(for details see Ref [1])

(2) Symmetric. The information of the total system is symmetric to the $N$ and $M$bases:

$$S_{sys}(N,M)=S_{sys}(M,N).$$

The same holds for the information defect:

$$S_{Def}(N,M)=S_{Def}(M,N).$$

(31)

(for details see Ref [1])

(3) Bounded. This means that the entropic information has an upper ($S\le s_{\max }$) or lower ($s_{\min }\le S$) limit for information loss ($¯>0$) or gain ($¯<0$).

We note that the upper limit tends to infinity in the classical thermal equilibrium: $S\le s_{\max }=¯\to \infty$ or $¯\to -\infty =s_{\min }\le S$ for information loss ($¯>0$) or gain ($¯<0$), respectively. This indicates that, in the classical case, entropy changes can constantly increase or decrease tending to infinity, and eventually can attain any possible positive or negative values for information loss ($¯>0$) or gain ($¯<0$), respectively [1].

Example.2: Following Melvin M. Vopson and Samuel C. Robson [74,75] and applying information theory, our set of independent and distinctive events becomes Σ = {A, C, G, T}, so |Σ| = 4 and probabilities P ={P_A ,P_C ,P_G ,P_T }. Using digital bits b = 2, for |Σ| = 4, we need 2 bits per letter (H = log |Σ| = 2 bits) to encode the message: A = 00, C = 01, G = 10, T = 11. If the letters within this subset would have equal probabilities to occur (1/4), then the subset given by Diagram (A) in Example.1 would have H = 2b and a total entropy of NH = 30 bits of information. However, the original subset given by Diagram (A) in Example.1 has the following probability distribution and information entropy:

$$P_{original}=\left\{P_A={\displaystyle \frac{3}{15}},P_C={\displaystyle \frac{4}{15}},P_G={\displaystyle \frac{7}{15}},P_T={\displaystyle \frac{1}{15}}\right\},$$

(32)

$$H_{original}=-\left({\displaystyle \frac{3}{15}}{\log }_2\left({\displaystyle \frac{3}{15}}\right)+{\displaystyle \frac{4}{15}}{\log }_2\left({\displaystyle \frac{4}{15}}\right)+{\displaystyle \frac{7}{15}}{\log }_2\left({\displaystyle \frac{7}{15}}\right)+{\displaystyle \frac{1}{15}}{\log }_2\left({\displaystyle \frac{1}{15}}\right)\right).$$

(33)

Therefore, the information entropy of the original subset is 1.751bits and the total information encoded is NH_orig=26.26bits. The traditional insert frameshift-mutated subset as κ → ∞ (see Diagram (B)) has the following probability distribution and information entropy:

$$P_{frameshift-mutated}=\left\{P_A={\displaystyle \frac{3}{16}},P_C={\displaystyle \frac{4}{16}},P_G={\displaystyle \frac{7}{16}},P_T={\displaystyle \frac{2}{16}}\right\},$$

(34)

$$H_{frameshift-mutated}=-\left({\displaystyle \frac{3}{16}}{\log }_2\left({\displaystyle \frac{3}{16}}\right)+{\displaystyle \frac{4}{16}}{\log }_2\left({\displaystyle \frac{4}{16}}\right)+{\displaystyle \frac{7}{16}}{\log }_2\left({\displaystyle \frac{7}{16}}\right)+{\displaystyle \frac{2}{16}}{\log }_2\left({\displaystyle \frac{2}{16}}\right)\right).$$

(35)

The information entropy of the insert frameshift-mutated genome subset is 1.849bits (Equation (35)), and the total information encoded is NH_fsh= 29.59bits. Hence the information entropy difference between the original and insert frameshift-mutated subsets is ΔH_fsh=|H_fsh- H_orig|=0,099bits, and the difference in the total information encoded is NΔH_fsh= 1,584bits. Using Equation (16β), the KFB-mutated subset (Diagram (Γ)) has the following information entropy:

$$\begin{array}{l}{H}_{KFB}=H_{original}+H_G-\Delta H_{defect}=H_{original}+H_G-{\displaystyle \frac{H_{original}{H}_G}{¯}},\\ \Delta H_{defect}={\displaystyle \frac{H_{original}{H}_G}{¯}},\\ ¯\le {\displaystyle \frac{H\mathrm{armonic}\mathrm{mean}(H_{original},H_G)}{2}}={\displaystyle \frac{H_{original}{H}_G}{H_{original}+H_G.}}\end{array}$$

(36)

where$\Delta H_{defect}$ denotes the information-entropy defects. The information entropy difference between the original and insert-mutated subsets (see Diagrams (Β-Ζ)), and the differences in the total information encoded are calculated in Table 4.

Table 4 demonstrates the concept of using κ-information theory to study genetic mutations.

Example.3: For instance, biologically, there is symmetry between sequence N and mutation M. Does this symmetry imply that insertion and deletion KFB mutations are equivalent in terms of information entropy loss? To answer this question, we consider the M=1 base κ-frameshift mutation at index 6 (insert). This mutation yields the following delete KFB mutated subset:

$$(TCA)(GAC)(\stackrel{\downarrow (\oplus ¯)Insert}{\overline{G}})(ATA)(TAC)(CAA)$$

(H) 

The corresponding delete KFB-mutation diagram of the insert KFB-mutation is as follows (Diagram (Γ), Example, 1):

$$(TCA)(GAC)(\stackrel{\downarrow (\oplus ¯)Insert}{\overline{G}})(ATA)(TAC)(CAA)\to$$

(Θ) 

$$\underset{\begin{array}{l}\downarrow (-)Delete-correlation\\ [\left({\displaystyle \frac{\overline{G}}{¯}}\right)\times (TCA)(GAC)(ATA)(TAC)(CAA)]\end{array}}{{\displaystyle \{(TCA)(GAC)(\stackrel{\downarrow (+)Insert}{\overline{G}})(ATA)(TAC)(CAA)\}}}$$

From Equation (3) and using Equation (24), we obtain:

$$\begin{array}{l}{S}_{¯}(N,\overline{M})=-\left(N+\overline{M}-{\displaystyle \frac{N\overline{M}}{¯}}\right)\log \left|\Sigma \right|=-{\rm N}\log \left|\Sigma \right|-\overline{{\rm M}}\log \left|\Sigma \right|+{\displaystyle \frac{N\overline{M}}{¯}}\log \left|\Sigma \right|\\ =S(N)+S(\overline{M})-\Delta S_{def}^{(delete)},\\ \overline{M}={\displaystyle \frac{-M}{\left(1-\frac{M}{¯}\right),}}\end{array}$$

(37)

Where

$$\begin{array}{l}S(N)=-{\rm N}\log \left|\Sigma \right|,\\ S(\overline{M})=-\overline{{\rm M}}\log \left|\Sigma \right|,\\ \Delta S_{def}^{(delete)}={\displaystyle \frac{N\overline{M}}{¯}}\log \left|\Sigma \right|\end{array}$$

(38)

The information-entropy defect for the delete KFB mutation $\Delta S_{def}^{(insert)}$can be written as follows:

$$\begin{array}{l}\Delta S_{def}^{(delete)}={\displaystyle \frac{N\overline{M}}{¯}}\log \left|\Sigma \right|={\displaystyle \frac{-NM}{¯\left(1-\frac{M}{¯}\right)}}\log \left|\Sigma \right|={\displaystyle \frac{-\Delta S_{def}^{(insert)}}{\left(1-\frac{M}{¯}\right)}},\\ \Delta S_{def}^{(insert)}={\displaystyle \frac{NM,}{¯}}\log \left|\Sigma \right|\end{array}$$

(39)

where$\Delta S_{def}^{(insert)}$is the information-entropy defect for the insert KFB mutation Equation.(27). Now, from Equations (37) and (27) we derive the information entropy loss for insert and delete KFB-mutations in terms of information-entropy defects, respectively:

$$\begin{array}{l}\Delta S_{¯}(delete)=S_{¯}(N,\overline{M})-(S(N)+S(\overline{M}))=-\Delta S_{def}^{(delete)},\\ \Delta S_{¯}(insert)=S_{¯}(N,M)-(S(N)+S(M))=-\Delta S_{def}^{(insert)}\end{array}$$

(40)

By using Equation (39), the relation between the information entropy loss for insert and delete KFB-mutations is given as follows:

$$\Delta S_{¯}(delete)={\displaystyle \frac{-\Delta S_{¯}(insert)}{\left(1-\frac{M}{¯}\right)}}.$$

(41)

Example.4: Following J.A. Tenreiro Machado et al [73], we consider the counting of n-tuple sequences, n = {1,...,10}, along each initial ssDNA, and the counting of the m-tuple sequences, m = {1,...,10} inserted along each additional ssDNA to the initial system. Note that the insert of m-tuple sequences to the initial n-tuple sequences does not necessarily yield to the KFB mutation; therefore, after the m-tuple sequences are inserted to the initial n-tuple sequences the system under consideration could be either in the κ-frame or κ-frameshift. The states distribution of the system is given by

$$f_{¯}\left(n,m\right)=4^{n{\oplus }_{¯}m}=4^{n+m-{\displaystyle \frac{nm}{¯}}},$$

(42)

where

$$n+m-{\displaystyle \frac{nm}{¯}}\ge 0,$$

(43)

and

$$¯\le {\displaystyle \frac{H\mathrm{armonic}\mathrm{mean}(\mathrm{n},\mathrm{m})}{2}}={\displaystyle \frac{nm}{n+m}},$$

(44)

for some$¯\in {\mathbb{Q}}^{*}$. The states distribution (42) approaches the traditional states distribution as κ approaches large values:

$$f\left(n,m\right)=4^{n{\oplus }_{\infty }m}\approx 4^n.$$

(45)

Here, we consider a word length ranging from (n,m,κ) = (1,0,κ), representing merely a static counting of

$$f\left(1,0\right)=4^{1{\oplus }_{¯}0}=4^1,$$

(46)

states, up to (n,m,κ) = (10,10,κ), representing the dynamics of a system with

$$\begin{gathered} f\left(10,10\right)=4^{10{\oplus }_{¯}10}=4^{20-{\displaystyle \frac{100}{¯}}}, \\ ¯\le 5, \end{gathered}$$

(47)

states. Therefore, for (n,m,κ) = (1,0,κ), we adopt a state diagram with short bins, such as

$$\left[\left\{A\right\},\left\{C\right\},\left\{G\right\},\{T\}\right],$$

up to the diagram with long sequence-counting bins, such as

$$\begin{array}{l}\begin{array}{l}\left[\left\{AAAAAAAAAA\right\},\dots ,\{TTTTTTTTTT\}\right]{\oplus }_{¯}\left[\left\{AAAAAAAAAA\right\},\dots ,\{TTTTTTTTTT\}\right]\hfill \\ =\left[\left\{AAAAAAAAAA\right\},\dots ,\{TTTTTTTTTT\}\right]+\left[\left\{AAAAAAAAAA\right\},\dots ,\{TTTTTTTTTT\}\right]\hfill \\ -\left(\left[\left\{AAAAAAAAAA\right\},\mathrm{..},\left(\left\{TTTTTTTTTT\right\}\right]\times \right[\left\{AAAAAAAAAA\right\},\dots \left\{TTTTTTTTTT\right\}\right])/¯\hfill \\ \begin{array}{l}=\left[\left\{AAAAAAAAAAAAAAAAAAAA\right\},\dots ,\{TTTTTTTTTTTTTTTTTTTT\}\right]\\ -(\left[\left\{AAAAAAAAAA\right\},\mathrm{..},\left(\left\{TTTTTTTTTT\right\}\right]\times \right[\left\{AAAAAAAAAA\right\},\dots \left\{TTTTTTTTTT\right\}])/¯,\end{array}\hfill \end{array}\\ =\left[\left\{AAAAAAAAAAAAAAAAAAAA\right\},\dots ,\{TTTTTTTTTTTTTTTTTTTT\}\right]\\ -([\left\{AAAAAAAAAA,\dots A_{100}\right\},\mathrm{..},\left(\left\{TTTTTTTTTT,\dots {{\rm T}}_{100}\right\}\right])/¯\end{array}$$

for (n,m,κ) = (10,10,κ). The comparison plot of distribution functions (47), (45) is given in Figure 1.

The comparison plot of distribution functions (Figure 1a, b) shows the impact of varying correlation strengths on the spatial state distribution of sequences. The contrast between the traditional distribution as κ approaches large values (Figure 1b) with the distribution curve under small κ values (Figure 1a) demonstrates the core argument proposed here: that mutual dependence induces order. This comparison also reveals how the κ exponent constrains the range of exportable states in biological polymers by altering the concentration of κ-probability:

$$0\le p_{¯}\left(n,m\right)={\displaystyle \frac{1}{f_{¯}\left(n,m\right)}}={\displaystyle \frac{p\left(n)p(m\right)}{p_D\left(n,m\right)}}\le 1,$$

(48)

where$p\left(n\right)=4^{-n},p\left(m\right)=4^{-m}$ are the probabilities of the n-tuple and m-tuple sequences, and$p_D\left(n,m\right)=4^{-{\displaystyle \frac{nm}{¯}}}=p{(n)}^{{\displaystyle \frac{m}{¯}}}$; the probability of the informational defect.

5. Kappa Metabolic Rate in the Presence of Information Defects

Landauer’s bound [76] gives the minimal energy required to perform an abstract computation and many string-writing operations, including the copying and transformation of ssDNA or biological polymers [77]. We may ask if the available energy is sufficient for the string copying and processing of a very small amount of stored information. The fundamental bounds of information can be connected to metabolism via Landauer’s bound by calculating the minimal metabolic rate, W, needed to replicate genetic information for a given genome length and cellular growth rate [3,76,78]. In the presence of entropy defects, Landauer’s bound is modified due to the KFB mutations, as follows:

$$Q_{¯}\ge k_{B}T(S_{¯-I}-S_{¯-F}),$$

(49)

where $Q_{¯}$is the kappa heat released to a bath at temperature $T$, $k_B$is Boltzmann’s constant, and $S_{¯-I}$ and$S_{¯-F}$ are the initial and final kappa system entropy, respectively. This is nothing more than the expression of the second law of thermodynamics (Equations (13–15)), where$\Delta S_{¯-system}=S_{¯-I}-S_{¯-F}$. To write a specific string from a set of unordered letters ($N=0$), we have $S_{¯-F}=0$ and

$$S_{¯-I}\left(N,M\right)=\ln (|\Sigma {|}^{N{\oplus }_{¯}M})=S_I(N)+S_I(M)-\Delta S_{I-Def}\left(N,M\right).$$

(50)

In Equation (50), terms$S_I(N)$,$S_I(M)$and $\Delta S_{I-Def}\left(N,M\right)$ are calculated from Equations (25–27); $\left|\Sigma \right|$is the number of unique elements, or letters in the informational system (Equation(6)); $N$ is the length of information (the string length) that is being copied; and $M$ is the length of information (the string length) of the KFB mutation. Here, $Y_{¯}=N{\oplus }_{¯}M$ is the kappa-composed length of information (the composed string length) after the action of the KFB mutation (Equation (1)). Using Equations (25–27) and (50), the kappa-modified Landauer’s bound becomes:

$$Q_{¯}\ge Q-\Delta Q_{Def},$$

(51)

where

$$Q=k_{B}T{S}_I(N,M)=k_{B}T(N+M)\ln (|\Sigma |)$$

(52)

is the heat released to a bath at temperature $T$ independent of the kappa index, and

$$\Delta Q_{Def}=k_{B}T\Delta S_{I-Def}(N,M)=k_{B}T{\displaystyle \frac{NM}{¯}}(\ln |\Sigma |)$$

(53)

is the heat released to a bath at temperature $T$ dependent on the kappa index, and corresponding to the entropy-information defects of the ssDNA or biological polymer. The term $NM/¯$ in Equation (53) is the correlation between the information lengths N and M due to information exchange after the action of the KFB mutation.

We can convert the kappa-modified Landauer’s bound into a kappa-modified metabolic rate$W_{¯}$due to the presence of entropy defects by considering how fast the information is copied. Given a division time,$t_{div}$ (seconds), the kappa-modified minimal metabolic rate of copying is:

$$W_{¯}=W-\Delta W_{Def}=k_{B}T{\displaystyle \frac{(N{\oplus }_{¯}M)\ln (|\Sigma |)}{t_{div}}}={\displaystyle \frac{k_{B}T\ln (|\Sigma |)}{t_{div}}}\left(N+M-{\displaystyle \frac{NM}{¯}}\right),$$

(54)

where

$$W=k_{B}T{\displaystyle \frac{(N+M)\ln (|\Sigma |)}{t_{div}}},$$

(55)

is the minimal metabolic rate independent of the kappa index, and

$$\Delta W_{def}=k_{B}T{\displaystyle \frac{NM\ln (|\Sigma |)}{¯t_{div}}},$$

(56)

is the minimal metabolic rate dependent on the kappa index, and corresponding to the entropy-information defects of the ssDNA or a biological polymer. Again, at limit (29) Equations (51) and (54) become

$$Q_{\infty -system}(n,m)\ge k_{B}T\left(n+m\right)\ln \left(\left|\Sigma \right|\right),\mathrm{for}(\Delta Q_{Def}(n,m)\to 0\mathrm{hence}¯\to \infty ),$$

(57)

$$W_{\infty -system}(n,m)=k_{B}T{\displaystyle \frac{\left(n+m\right)\ln \left(\left|\Sigma \right|\right)}{t_{div}}},\mathrm{for}(\Delta W_{Def}(n,m)\to 0\mathrm{hence}¯\to \infty ).$$

(58)

Equation (57) denotes Landauer’s bound whereas Equation (58) shows the minimal metabolic rate for a biological system without any correlation among the information length bases. The minimal metabolic rate formula (54) is highly dependent on environmental temperature T. By Equation (24), we have:

$$\Delta W_{¯}=-\Delta W_{Def}=k_{B}T{\displaystyle \frac{nm}{¯t_{div}}}\log \left|\Sigma \right|,$$

(59)

For$T\to 0{\rm K}$, the entropy defects $\Delta W_{¯}=-\Delta W_{Def}\to 0$are diminished on minimal metabolic rate while for$T\to \infty$, the entropy defects $\Delta W_{¯}=-\Delta W_{Def}\to \infty$are amplified on minimal metabolic rate.

6. Discussion

In the MB probability distribution for ssDNA/RNA or a biological polymer,$p_{\infty }(\Omega )=|\Sigma {|}^{-N}$, entropy is zero $S_{\infty }(\Omega )=0$only when a specific information string is written from a set of unordered letters ($N=0$). The probability value for such an unordered information string is$p_{\infty }(\Omega )=1$. Furthermore, in the MB statistical mechanics for a specific information string from a set of ordered letters ($N>0$), there is no preference for a specific sequence: if ${\Omega }_{\infty }(N)=4^N$, different ssDNA sequences are equally possible, and the actual ssDNA/RNA sequence has only a probability$p_{\infty }(\Omega )=4^{-N}$ in the ssDNA sequence, ranging from one million in bacteria to billions in many animals and plants. For example, the space of possible proteins ${\Omega }_{\infty }(N)$ with a base length $n$ of 1000 amino acids is $\Omega (1000)={20}^{1000}$– a space so large that it could never be explored in our universe [27]. The space of possible molecular configurations within an organism is astronomically larger. The probability $p$ of such a protein (ordered information string) to exist in nature is $p_{\infty }(\Omega )={20}^{-1000}$. The requirements of longer evolutionary time and a larger number of individuals are even more conducive to the observation of deterministic laws associated with the base length$N$. The evolutionary fitness of organisms relates to the progressive reduction of the entropy production rate. This, and the Boltzmann’s ergodic principle, allow us to consider organisms as evolving chemical entities (Sabater 2006, 2009 [2,4]). Accordingly, the time-average state of a single individual over a long period (evolutionary time) coincides with the average state of a sufficiently large number of individuals for shorter time periods [5]. We note that Sabater (2009[5]) worked with an alternative theory of biological time arrows, showing that deterministic behavior can arise in a biological system.

Let us now examine the evolutionary events that could result from the combination of functional selection during biological evolution and information exchange between the base lengths of biological polymers, represented by the kappa parameter. For ssDNA/RNA or a biological polymer with the kappa-probability distribution (Equation (48) derived from the kappa entropy (Equation (24) under the constraint of normalized${\displaystyle \sum p_{¯}(n,m)=1}$, we observe that the different $n$ and $¯$ scales in Equations (24) and (48) are manifested in the double evolutionary emergence of the proposed biological polymer through correlations between its bases. For specific values of the kappa parameter: ($¯=nm/n+m$), deterministic laws associated with a single biological polymer can be expected for short timescales. Thus, for the kappa-probability distribution, we have the short-timescale possibilities for an individual. Let us now consider a general example for a ($¯=nm/n+m$) state characterized by the highest correlation between bases due to information exchange. By writing a specific initial information string from a set of unordered letters for kappa values ($(n,m)=(0,0)$,$\forall ¯\in {\mathbb{Q}}^{*}\backslash \left\{nm/n+m\right\}$), we have:

$$S_{¯-I}(n,m)=\left(n+m+{\displaystyle \frac{nm}{¯}}\right)\ln \left|\Sigma \right|=0bit,$$

(60)

if and only if,$(n,m)=(0,0)$,$\forall ¯\in {\mathbb{Q}}^{*}\backslash \left\{nm/n+m\right\}$.We then obtain the entropy changes:

$$\Delta S_{system}(n,m)=S_{¯=nm/n+m}^{(F)}(n,m)-S_{¯}^{(I)}(0,0)=0bit,$$

(61)

where $S_{¯=nm/n+m}^{(F)}(n,m)=0bit$is the final entropy of the “mathematical” biological polymer. Equation (61) is the iso-entropic process for a given “mathematical” biological polymer, and is adiabatic and reversible [83–86], with equal initial and final probability (that is, $p_{¯}^{(I)}(0,0)=p_{¯=nm/n+m}^{(F)}(n,m)=1$). Using Equations (54) and (61), the iso-entropic changes for the proposed “mathematical’’ biological polymer yield to the zero kappa-minimal metabolic rate. This implies that, at this optimum state ($¯=nm/n+m$), to protect itself from damage it cannot endure, the “mathematical” biological polymer must not waste energy (produce entropy) in reactions. We call this state of the “mathematical” protein the kappa-functional optimum state. This state is given by Equation (61). In nature, it may be possible for a real biological polymer to approximate the kappa-functional optimum state with increasing correlation between its parts. Due to the strongest correlation between its parts, such a protein would need to waste much less energy (produce less entropy) in reactions, and would be less susceptible to damage than what the classical information theory predicts.

7. Conclusions

Following Livadiotis G. and McComas D. J, (2023) [1], we propose a new type of ssDNA/RNA frameshift mutations: Kappa-Frameshift Background (KFB) mutations. KFB mutations occur spontaneously due to information exchange between the ssDNA/RNA sequence of length base ($N$) and the mutation sequence of length base (M), and respect the kappa-addition symbol ${\oplus }_{¯}$. KFB mutations occur when any number of $M$ bases are added to or deleted from the initial ssDNA sequence of $N$ bases. This repeats the kappa-addition ${\oplus }_{¯}$, except in the case of multiples of three, which reestablish the initial reading frame. In the proposed KFB mutation, the kappa index can be understood through the subsystem (subbase) of information length correlation between $N$ and $M$ due to information exchange within the given length of ssDNA or biological polymer (the string length) that is being copied.

We find that entropy defects originate in the interdependence of information length systems (or their interconnectedness: the state at which systems with a significant number of constituents (information length bases) depend on, or are connected with, each other) by the proposed KFB mutations. In general, interdependence induces order in the systems involved (ssDNA or biological polymers) and reduces their entropy-information, thus producing entropy-information defect. The entropy of a biological system with correlations is lower than the entropy of a biological system without correlations, because correlations add order to the biological system. In a composite biological system, its entropy differs from the sum of the entropy of the biological subsystems that constitute it by the entropy defect.

In the presence of entropy defects, the Landauer’s bound and minimal metabolic rate for a biological system is modified. We observe that that the different n-tuple sequences and $¯$ scales in Equations (24) and (48) are manifested in the double evolutionary emergence of proposed biological polymers due to subsystem correlations. In this process, for specific values of the kappa parameter, we can expect deterministic laws associated with a single biological polymer for short timescales. We find that the possible evolutionary events resulting from the combined effects of functional selection during biological evolution and information exchange between the length bases of the biological polymer are represented by the kappa parameter. Using a proposed “mathematical” biological polymer, we find that iso-entropic changes in this biological polymer yield to the zero kappa-minimal metabolic rate. This implies that, at the optimum state given by the half of the harmonic means of n-tuple and m-tuple sequences, ($¯=nm/n+m$), to protect itself from damage it cannot endure, the “mathematical” biological polymer must not waste energy (produce entropy) in reactions. We call this state of the “mathematical” biological polymer the kappa-functional optimum state. In nature, with increasing correlation between its parts, a real protein may approximate the kappa-functional optimum state. Due to the strongest correlation between its parts, such a protein would waste much less energy (produce less entropy) in reactions, and would be less susceptible to damage, than what the classical information theory predicts.