Introduction

What is life? Technically, it could be defined as cells with evolutionary potential, make up by organic matter and getting on in an autopoyetic metabolism¹. Organisms use energy flow gradients at different organization levels in accord with physical laws. The living systems have three principal functions as self-organizing units: a) compartmentalization, b) metabolism, and c) regulation of input and output information flux². The layer between internal and external environment of organisms controls matter, energy and information flows; metabolism regulates epigenetic and autopoyetic processes; biological information is a program (and a set of programs) that operates both physiology and functionality, codified in the DNA³.

Darwinian theory of evolution by means of variation, natural selection and reproductive success of heritable variation in populations and organisms, can be defined as an ecological process that change the covariance of phenotypic traits (as expression of genetic, epigenetic, ontogenic and environmental factors) in living organisms or biological systems grouped at different organization levels⁴. Natural selection operates on biological systems that have three features: a) variability, b) reproductivity, and c) heritability; one result of natural selection is a tendency toward to an increase in fitness and functionality of biological systems in an environmental stochasticity (both biotic and abiotic). Fitness is a measure of reproductive success of changes in allele frequency of organisms on a determinate ecological noise, conditioned on the phenotype or genotype⁵.

In this paper, an analogy between a biological system and a message was made; it was also considered the environmental stochasticity as the noise when a message is transmitted. Thus, the biological system is analogous to the amount of information of a message (for example: genetic information) that is transported to the next generation inside an ecological noise. This favors the use of the Norbert Wiener model of message's transmission in a telephonic line with noise⁶. In our model, it has been considered that information could be as simple as a binary unit (bit, 0 and 1), but it can grow by the additive property^{7 8}.

The amount of information defined by Wiener⁷ (p. 62) is the negative of the quantity usually defined as statistical entropy. This principle of the second law of thermodynamics can be understood for a closed system as the negative of its degree of restrictions⁹ (p. 23), i.e., its structuration level. Also, G.N. Lewis in 1930 (cited by Ben-Naim¹⁰ p. 20) quotes: "Gain in entropy always means loss of information".

In relation to information theory, Léon Brillouin in his book "La información y la incertidumbre en la ciencia"¹¹ (p. 22-25) wrote that entropy is connected with probabilities as is expressed in equations by Ludwig Eduard Boltzmann and Max Planck, and suggested that information is a negative entropy^{12 13} or negentropy¹². But instead of using the term of entropy, Arieh Ben-Naim¹⁰ (p. 21) proposes to replace it by "missing information".

Nevertheless, the second law of thermodynamics neither implies one-way time, nor has a statistical probabilities model. For this reason, in this paper it was employed a time series tool and the Brownian motion as a model to simulate the dynamics of the amount of information of a message, as an analogous of a biological system. This approach shows that biological information could be carried by "some physical process, say some form of radiation" as Wiener⁷ wrote (p. 58). What wavelength? Brillouin¹¹ (p. 132), in his scale mass-wavelength equals a mass of 10⁻¹⁷ g to a wavelength of 10⁻²⁰ cm; for this scales, it could be important to consider k , the Boltzmann constant, because its value is 1.38 x 10⁻¹⁶ erg/°K, in unities of the system cm, g, second. Likewise, a quantum h can be expressed as h = 10⁻³³ cm; this smaller length "plays a fundamental role in two interrelated aspects of fundamental research in particle physics and cosmology"¹⁴ and "is approximately the length scale at wich all fundamental interactions become indistinguishable"¹⁴. Moreover, it is necessary to mention that Brownian motion is a thermic noise that implies energy of the level k T (where T is temperature) by degree of freedom¹¹ (p. 135). It was hypothesized that this radiation could be the wavelength of photons incoming on Earth's surface, irradiated by the photosphere of the Sun at a temperature of 5760 °K; after the dissipation of high energy photons to low energy ones, through irreversible processes that maintain the biosphere, the temperature of the outgoing photons that Earth radiates to space is 255 °K¹⁵. This incoming radiation support plant photosynthesis, evapotranspiration flow, plant water potential, plant growth, energy for carbon-carbon bonds (or C-H, C-O, O=O, etc.), and food of the trophic chains in ecosystems, among other energy supported processes.

The question to answer in this paper is: What is the driven force of biological evolution? The possible answer is that driven force is the dynamics of the amount of information in a biological system (genetic and epigenetic messages). Antoine Danchin¹⁶, in a similar approach, proposes that mechanical selection of novel information drives evolution. In the next section, it will be described the mathematical model of the dynamics of message transmission, as a proposal to explain the nature of biological evolution.

The model

Shannon entropy formula is¹⁷ (p. 291):

Where Ω is the number of microstates or possible arrangements of a system, for N particles that can occupy m states with occupations numbers (n ₁, n₂ , ....., n _m ) and k is the Boltzmann constant, that is R (the ideal gas constant) divided by the Avogadro′s number:

and Brillouin information (I) formula¹⁸ (p. 1) is:

Where K is a constant and P_o is the number of possible states of a system, keeping in reserve that the states have a priori the same probability. The logarithm means that information has the additive property.

For example, if the configuration states of two independent systems are coupled in this way: P₀ = P₀₁ · P₀₂

with

and

The constant K can be equal to 1/(ln2) if information is measured in binary units (bits) (op. cit ., p. 2); nevertheless, if information of Brillouin's formula is compared with Shannon entropy formula, and K is replaced by k , the Boltzmann constant, then information and entropy have the same unities: energy divided by temperature¹⁸ (p. 3). The relationship, between entropy and information, if temperature -in centigrade scale- is measured in energy units is:

The problem to associate entropy and information as concepts and in unities is that information is not a change between an initial and a final state, and it lacks of statistical treatment of possible configurations as in thermodynamical statistics. Thus, Norbert Wiener⁷ developed the next model, where information has a time series and their distribution follows the configuration of the Brownian motion (chapter III).

For this, Wiener⁷ (p. 61) established that the amount of information of a system, where initially a variable x lies between 0 and 1, and in a final state it lies on the interval (a, b ) inside (0, 1), is:

The a priori knowledge is that the probability that a certain quantity lies between x and x + dx is a function of x in two times: initial and final or a priori and a posteriori . In this perspective, the a priori probability is f ₁(x) dx and the a posteriori probability is f ₂(x) dx.

Thus, it is valid to ask: "How much new information does our a posteriori probability give us?"⁷ (p. 62). Mathematically this means to bind a width to the regions under the curves y=f ₁(x) and y=f ₂(x), that is initial and final conditions of the system; also, it is assumed that the variable x have a fundamental equipartition in its distribution. Since f ₁(x) is a probability density, it could be established⁷ (p. 62):

and that the average logarithm of the breadth of the region under f ₁(x) may be considered as an average of the height of the logarithm of the reciprocal of f ₁(x), as Norbert Wiener⁷ wrote using a personal communication of J. von Neumann. Thus, an estimate measure of the amount of information associated with the curve f ₁(x) is:

This quantity is the amount of information of the described system and it is the negative of the quantity usually defined as entropy in similar situations⁷ (p. 62). It could also be showed that the amount of information from independent sources is additive⁷ (p. 63). Besides: "It is interesting to show that, on the average, it has the properties we associate with an entropy"⁷ (p. 64). For example, there are no operations on a message in communication engineering that can gain information, because on the average, there is a loss of information, in an analogous manner to the loss of energy by a heat machine, as predicted by the second law of thermodynamics.

In order to build up a time series as general as possible from the simple Brownian motion series, it is necessary to use functions that could be expanded by Fourier series developments, as it is showed by Norbert Wiener in his equation 3.46, and others, in his book on cybernetics⁷ (pp: 60-94).

It is possible to apply equation (5) to a particular case if the amount of information of the message is a constant over (a , b ) and is zero elsewhere, then:

Using this equation to compare the amount of information of a point in the region (0, 1), with the information that is the region (a , b ), it can be obtained for the measure of the difference:

It is important to say that this definition of the amount of information can be also applicable when the variable x is replaced by a variable ranging over two or more dimensions. In the two dimensional case, f(x, y) is a function such that:

and the amount of information is:

if f ₁(x, y) is the form φ(x) ψ(y) and

then

and

this shows that the amount of information from independent sources is additive⁷ (pp: 62-63).

For this model, it would be interesting to know the degrees of freedom of a message. It would be convenient to take the proposal of Léon Brillouin¹⁸ (pp: 90-91), who made the next development:

A certain function f(t) has a spectrum that does not include frequencies higher than a certain maximum limit: ν_M and the function can be expanded over a time interval τ. The outstanding question is: How much parameters (or degrees of freedom) are necessary for to define the function?

If it is established that there is N degrees of freedom and

when

To choose the independent parameters for the function, it was considered the interval 0 < t < τ and that it is suitable to know the function values before 0 and after τ , without any addition of information to the message f(t) . Thus, it is chosen a periodic function that reproduces in an indefinite way the solution curve of the function f(t) between 0 and τ as:

q is an integer

and use a periodic function with a period τ; thus, applying a development of a Fourier series, it is obtained:

with

as was stated by Brillouin¹⁸ (pp: 90-91).

This means that the degrees of freedom bound to a message could be a periodic function that is dependent on the amount of information and the time interval.

Discussion

It has been demonstrated a mathematical model that correlates the accumulation of information with biological evolution, by means of additive information (it could be genetic and epigenetic). Biological systems can be assimilated as messages transmitted to the next generation in the middle of an environmental noise, biotic and abiotic. What kind of information? It could be as simple as binary units (bits, 0 and 1), which can be a set in a wavelength of a photon and that can be assembled into integrative structures. For example, in computer technology the hardware is the physical support of an ensemble of structured information (algorithms hierarchically organized); this informatics programs run on a binary system where an alpha-numeric character is formed with a set of binary units, i. e. , a byte (byte is equals to 8 bit series), and then, they can form data (words), concepts, sentences, routines, algorithms, programs and sets of programs as in computers software. The DNA is a genetic information code and it must have epigenetic, ontogenic and autopoyetic programs that regulate (by expression or restriction) its development. The genetic pleiotropy in organisms is when a gene affects more than one phenotypic trait, and it exhibits the gene modulation by an epigenetic process.

The mathematical model also shows that changes in information and entropy could be the same process: an increase in entropy normally means a loss of structure or restrictions of a system. On the contrary, the evolution of a biological system usually means an enlargement in its amount of information and complexity that drives to an increase in its fitness and functionality. Biological systems are information building systems^{19 20 21}, i.e., genetic and epigenetic capacity to generate developmental functional complexity (phenotype), as is quoted by Elsheikh²² in his abstract, but organisms also face to environmental networks, where there are stochasticity, random processes (positive or negative) and, sometimes, chaos^{23 24}.

Two emergent attributes of biological organisms are phenotype and behavior. Phenotype is a synthesis of equilibrium between internal and external environments, and behavior is a driven force in individual evolution^{25 26 27}. A path frequently transited by evolutionary changes is the mutualistic symbiosis¹. In this sense, the transmission of biological messages (organisms) to the next generation increases its possibilities to improve the amount of information, if the message is redundant and symbiosis is a way to form additive messages. For example, if the organism A has three traits and become a functional unity with the organism B, which has other three different traits and both can share all traits; then, they have 9 binomial sets of traits.

Finally, it is necessary to point out that information create programs and a set of programs that drive toward power-law behavior of organisms. Behavior also has two sources: a) phylogenetic memory (i.e., genetically codified routines and conducts), and b) algorithms induced by epigenetic modulation and environmental noise or stochasticity. It is already known that ecological behavior of organisms is the driven force to its evolution (accumulation of information) or its degradation (loss of structures and information).

Concluding remarks

Figure 1 makes a synthesis of the role of information on the evolution of biological systems and on changes in entropy. Mathematically, the origin of matter in the universe could have begun above zero entropy¹⁵, and matter evolution drives to form organization levels with increasing complexity and information. The transition between both is called abiotic matter, and living systems must be carried by the degrees of freedom due to the product of the Boltzmann constant by temperature (k T). The big source of heat on the Earth is the Sun that is at 5760 Kelvin degrees at the photosphere and the Earth average temperature in the biosphere actually is ca. 288 °K. Then, organisms absorb high energy solar photons and they use that energy for photosynthesis or as food in the trophic chain by means of dissipative processes. Living organism "eat" high energy photons and they distribute them among photons with low energy. The entropy of photons is proportional to the number of photons¹⁵ in a system and on the Earth's energy balance, Lineweaver and Egan¹⁵ (p. 231) say: "when the Earth absorbs one solar photon, the Earth emits 20 photons with wavelengths 20 times longer".

Figure 1 A time-information graphic showing changes in matter evolution toward an increase or loss in its amount of information. The discontinuous line suggests the transition between what is called abiotic matter (below) and living systems (above). The information axis runs from 0 to ∞. 

Likewise, Karo Michaelian²⁸ (p. 43) wrote: "Direct absorption of a UV photon of 260 nm on RNA/DNA would leave 4.8 eV of energy locally which, given the heat capacity of water, would be sufficient energy to raise the temperature by an additional 3 Kelvin degrees of a local volume of water that could contain up to 50 base pairs". In this discerment, solar photons increase the degrees of freedom, in structure and functionality, of living organisms. One degree of freedom is an independent parameter of a system and it could be considered as novel information of it.

Finally, it is important to say that information can be the driven force of biological evolution. Life, as a state, can be defined as a dissipative system that has a structural biomass or hardware , genetic and epigenetic programs or software and metabolic-ontogenic interface that regulates flows of matter, energy and information, in order to have an autopoyetic homeostasis, behavior and increases in its fitness and functionality. Furthermore, a living system is an unique set of programs reservoir that evolves in face to ecological noise, stochasticity and, sometimes, chaos.