The Origin of Digital Information

I. INTRODUCTION

In modern society, digitization is progressing in every field. Computers are used in every area, and information is digitized in every field. What is the origin of digital information?

In considering this problem, life's genetic information must be digital. Furthermore, we need to combine information theory and thermodynamics. Then, we will be able to consider life from the viewpoint of information.

Historically, Schrödinger (1992) was the first to consider this issue. In 1944, his "What is life?" marked the beginning of molecular biology. He described the thermodynamic properties of life and suggested the structure of DNA with the term 'aperiodic crystal,' but at the time of "What is life?" the concept of information did not exist.

Shannon (1948) published his "A Mathematical Theory of Communication" in 1948, and Watson and Crick (1953) solved the double-helix structure of DNA in 1953.

After that, Ben-Naim (2008) claimed that entropy can be substituted with missing information. Furthermore, Avery (2022, pp. 129-148) asserts the identity of entropy and missing information and describes the relationship between information and energy.

Furthermore, we shall consider life from the perspective of information. As previously mentioned, life uses energy to maintain order, i.e., the amount of information (Alberts, 2015, p8). Furthermore, natural selection eliminates lethal mutations by the death of the mutant (Kotani, 2019).

Next, we shall describe Platonic one, an essential concept in considering the relationship between life and genetic information. First, digital information corresponds to natural numbers; binary natural numbers in digital computers can represent all digital information. Second, as previously mentioned, life has properties as the natural number one (Kotani, 2017).

Moreover, Plato said of the natural number one as follows. "O my friends, what are these wonderful numbers about which you are reasoning, in which, as you say, there is a unity such as you demand, and each unit is equal, invariable, indivisible,—what would they answer?" (The Republic, VII, 526a).

In this paper, we shall consider the relationship between life and information. Astounding is that life stores and accumulates information despite the second law of thermodynamics. Let us consider the mechanism of information storage in life.

II. THERMODYNAMIC PROPERTIES OF LIFE

In "What is Life?" Schrödinger (1992, p. 69) described that the remarkable characteristic of life is that life escapes from the law of entropy. In other words, life maintains internal order and escapes collapse into disorder.

However, if living organisms do nothing, the order of the living organisms will be lost according to the second law of thermodynamics. Thus, animals need energy from food to maintain their bodies. On the other hand, plants get their energy from sunlight.

In thermodynamics, life uses Gibbs free energy not to increase entropy in the living body (Alberts, 2015, p. 8). Equation 1 shows Gibbs free energy (Alberts, 2015, p. 103).

In Equation 1, G is Gibbs free energy, H is enthalpy, T is absolute temperature, and S is entropy. Gibbs free energy is beneficial in isothermal and isobaric systems such as a cell in the body of homeothermic animals.

As an example of the usefulness of Gibbs free energy, consider the proliferation of unicellular organisms in the sea. They obtain energy from food and use the power to grow their bodies and replicate their DNA. In this case, the total entropy of the outer ocean plus their bodies necessarily increases.

However, the entropy in their bodies does not increase because the heat is discarded outside, and the entropy of the outer ocean rises. In their bodies, the reaction proceeds in the direction that minimizes Gibbs free energy, so entropy does not grow in their bodies.

Next, since the word order is unclear, let us return to molecular biology's starting point: "What is life?" Schrödinger used the term 'order,' but he used 'order' in the sense of information content rather than order. For example, periodic crystals seem more ordered than DNA. However, the amount of information in DNA is more significant than that in regular crystals.

So, he said, "the 'aperiodic solids,' the chromosome molecules, which doubtless represent the highest degree of well-ordered atomic association we know of – much higher than the ordinary periodic crystal" (Schrödinger, 1992, p. 185). The aperiodic solid is DNA, which has much more information than the ordinary periodic crystal, so the highest degree of well-ordered atomic association means that DNA has a large amount of information.

Next, we shall consider the cells' thermodynamic properties from the information perspective. In this case, traditional terms are replaced by information terms. First, entropy is replaced by missing information. Next, the order is replaced by the amount of information. As a result, the overall outlook is much better.

The concept of the amount of information is essential in biology. So, living organisms are precisely constructed based on DNA blueprints, and living organisms have a large amount of information. On the other hand, when a living organism is disrupted, order is destroyed, and its amount of information decreases. Eventually, when it dies and reaches a state of equilibrium, its amount of information is reduced to zero.

Let us consider the previous example with the terminology replaced. When unicellular organisms ingest food and increase in the sea, they consume Gibbs free energy to synthesize macromolecules such as proteins and DNA. As a result, they increase. Then, their amount of information increases. Therefore, living organisms can use Gibbs free energy to increase their amount of information.

Similarly, human civilization uses energy to increase the amount of information. In car manufacturing plants, energy is consumed to produce cars based on blueprints. Also, cities with roads and buildings are always maintained using energy and blueprints. Therefore, human civilization also uses energy to store information.

III. NATURAL SELECTION PROTECTS INFORMATION AGAINST ENTROPY

The next problem is how to maintain the DNA sequence. According to the second law of thermodynamics, a perfectly exact copy is impossible (Kotani, 2019). If living organisms copy the DNA base sequence, errors necessarily occur. No matter how low the probability of miscopying is, errors accumulate if they repeat copying many times. When the copying accuracy is r, and the number of copies is n, equation 2 is established. As n increases without limit, $r^n$ will approach o. Since r is always less than 1, information will eventually be lost.

Let us take E. coli as an example. Without natural selection, E. coli would become extinct. Under laboratory conditions, E. coli divides about once every 30 minutes, and its mutation rate is about three nucleotide changes per ${10}^{10}$ nucleotides (Alberts, 2015, pp. 237-238). That is, r is 0.9999999997 in equation 2. If E. coli continued to divide at this rate without natural selection, more than 99% of the DNA bases in E. coli would mutate within one million years. Therefore, E. coli cannot survive without natural selection.

However, ribosomal RNA genes have remarkably similar base sequences in all organisms(Alberts, 2015, pp. 237-238).

2015, pp. 14-16). All living organisms are classified into three domains: bacteria, archaea, and eukaryotes. Since living organisms branched into domains about 3.5 billion years ago, ribosomal RNA genes have been highly conserved. The mutants were eliminated because many mutations in the ribosomal RNA gene are fatal. Similarly, genes for proteins and DNAs necessary for survival are highly conserved. In these cases, equation 3 is established. Equation 3 corresponds to r = 1 in Equation 2. Natural selection will conserve the base no matter how many times it is copied.

Natural selection eliminates lethal mutations by irreversible death of life, but the second law of thermodynamics applies only to reversible phenomena. Natural selection retains information regardless of the second law of thermodynamics.

Next, we shall confirm that the second law of thermodynamics applies only to fundamentally reversible phenomena. For example, consider the phenomenon of salt dissolving in water. We get a salt solution if we add salt to the water and mix it. At first glance, the above phenomenon seems irreversible. Indeed, it is irreversible in an isolated system, as in an insulated container.

However, we can reverse the above phenomenon using Gibbs free energy in the open system. If we heat the saline, it is possible to separate the saline solution into salt and distilled water. Therefore, the phenomenon that the system reaches the equilibrium state is not an irreversible phenomenon in an open system. Thus, the second law of thermodynamics does not apply to the irreversible death of life in an open system.

IV. INFINITE MONKEY THEOREM

The prerequisite for natural selection is the irreversibility of death. Then, we shall consider why the end of life is irreversible. From a thermodynamic point of view, the cell is the smallest unit of life that can prevent an increase in entropy and maintain order. Thus, the minimal cell is the reference in thinking about this problem.

Craig Venter created a synthetic minimal cell with a 1079 kb genome (Gibson, 2010). After that, research continued aiming at minimal cells, and the new synthetic minimal cell, JCVI-syn3A, has a 543 kb genome (Pelletier, 2021). The probability of randomly synthesizing bases to produce the same sequence is about one in $3.8\times {10}^{326918}$. How much effort and time does it take to get the same result by chance?

Such problems are known as the infinite monkey theorem, which is the theorem that if a monkey keeps making random strings long enough, eventually, the monkey will create any document. In reality, however, the probability that a monkey randomly type-writes and finally completes Hamlet is infinitely close to zero.

Recently, Anderson (2011) considered this problem. According to him, since Hamlet has 130,000 characters, the probability of a monkey randomly typing out complete Hamlet is 1 in ${26}^{130000}$: one in $3.4\times {10}^{183946}$.

He said the probability is as follows. "If we took the same number of monkeys as the number of particles in the universe $({10}^{80})$, and each type 1000 keystrokes per second for 100 times the life of the universe $({10}^{20}$ seconds), we would still find the probability of the monkeys replicating even a short book to be impossibly small."

If we followed his hypothetical experiment, we would undoubtedly reach 72 characters, but no matter how lucky we were, we would not get 100 characters. Therefore, monkeys cannot complete Hamlet even with all the currently conjectured resources of the universe.

Furthermore, the probability of accidentally reproducing the base sequence of artificial bacteria is lower than that of monkeys typing Hamlet. Therefore, we believe that anyone cannot revive dead life.

The irreversibility of death of life is the basis for natural selection. Of course, the same applies to literary works such as Hamlet. In contrast, literary works that are no longer read disappear.

Similarly, in scientific texts, even Euclid's Elements was not read for a while, so some parts do not make sense today. Documents survive only when read repeatedly.

V. DISCUSSION

As mentioned above, we can now speculate about the beginnings of cellular life. We recognize the vital nature of life. Most importantly, life is indivisible. Indeed, it is possible to divide multicellular organisms into cells, but cells are indivisible. Therefore, we can regard life as having the properties of the natural number one.

Moreover, we regard life as a gene vehicle, as Dawkins (2006, p. 47) says. Then, all differences in life are genetic, and lives are equal. Therefore, life has Platonic one's properties: indivisibility, invariability, and equality.

Next, by being subjected to natural selection, DNA bases acquire the properties of Platonic one. If a mutation in a DNA base is fatal, life will try to copy the DNA base exactly and repair it. Furthermore, since natural selection eliminates lethal mutations, ribosomal RNA gene bases are conserved for 3.5 billion years. As a result, conserved bases have properties close to Platonic one.

In this way, DNA bases acquire properties close to Platonic one by natural selection. On the contrary, the nature of life depends on genes. Therefore, life and genetic information are interdependent. Likewise, concepts and languages are interdependent.

The biological basis for this conclusion is Quiroga's concept cell. Quiroga and colleagues (2005) found neurons in the human brain that are selectively activated by strikingly different pictures of given individuals, landmarks or objects and, in some cases, even by letter strings with their names. For example, a cell responding to pictures of actress Halle Berry also responds to Halle Berry in a Catwoman costume and the letter string of 'Halle Berry.'

Quiroga (2019) also claims concept cells exist in the human brain. Furthermore, elaborate concept cells have only been found in humans. Then, he says that language allows us to have complicated abstractions.

Indeed, all concepts are defined by words in dictionaries. On the other hand, the concept cell has the property of life as Platonic one. Furthermore, language, which is digital information, defines concepts. This relationship is isomorphic to the relationship between life and genes.

Finally, we shall consider the difference between Platonic one and digital information. In the case of the alphabet, each letter has close properties to Platonic one, but each letter has unique visual properties.

In the case of DNA, the four types of bases have different physical properties. However, any base sequence can maintain the double helix structure of DNA. Therefore, we can regard that the four bases of DNA also have close properties to Platonic one.

The advantages of digital information are explained using the alphabet as an example. An alphabet document contains 26 different characters in a sequence. Each letter is the same size and compatible. The typewriter uses these properties.

If there is a typewriter, no matter who hits the keys in any order, the information content in the character sequence will increase exponentially. Similarly, DNA polymerase, an enzyme replicating DNA, can use four bases as equal substrates (Alberts, 2015, pp. 239-244).

As mentioned above, the advantage of digital information is that enormous amounts of information can be obtained through simple repetitive tasks. In this way, even the minimal cell has an information content far beyond Avogadro's number. The enormous amount of genetic information causes the irreversibility of death of life, which is the necessary condition for natural selection.

Therefore, the DNA replication system must be complete for evolution to begin. In conclusion, we regard the completion of life's DNA replication machinery as the origin of digital information. Moreover, it is challenging to consider earlier, but the presence of RNA viruses suggests the existence of an earlier stage.