Life has been a mystery for most physicists since the question of Maxwell’s demon. In the present paper, the self-reproduction, which is characteristic of the organism, is shown to be essential in resolving the paradox of Maxwell’s demon. For this purpose, a new thermodynamic quantity of biological activity is introduced to represent the molecular events in an organism recently revealed by molecular biology. This quantity gives a measure that the entropy production arising from the difference between acquired energy and stored energy compensates for the negative entropy of systematizing the organism, and is considered to be proportional to the self-reproducing rate of the organism as the first approximation. The equation of replicator dynamics consisting of self-reproducing rate, death rate and mutation terms contains all known types of evolution of unicellular organisms. When the mutation term is restricted to the point mutation mainly due to the nucleotide base changes in genes, this equation automatically leads to Darwinian evolution that the mutant with the higher increase rate is selected to become prevailing in the population. Throughout this evolution, the nucleotide bases in genes are converged to the special arrangement exhibiting the optimal increase rate of the organism. Moreover, the mutants having experienced gene duplication first decline to the minor members in the population but some of the descendants recover as a new style of organisms by generating new gene(s) from the counterpart of duplicated genes. This evolutionary process to expand the repertoire of genes is mathematically formulated by solving the equation of replicator dynamics up to the higher order of mutation terms. The present theoretical approach can be not only extended to the multicellular diploid eukaryotes but also applied to explain the origin of genes in the self-reproducing proto-cells formed anciently.
When the first and second laws of thermodynamics have been established, Maxwell has raised the question about the compatibility of life with the second law in terms of a sorting demon [
According to the suggestion by Szilard [
Recently, the studies of molecular biology have revealed the central dogma in the free-living organism; the proteins are translated from messenger RNAs (mRNAs) by the aid of ribosomal RNAs (rRNAs) and transfer RNAs (tRNAs), and the three kinds of RNAs are all transcribed from the DNA genes, which are replicated upon self-reproduction [
A free-living organism is generally characterized by two internal variables; the size N of its genome (a set of genes) and the systematization SN of the genome and its products. The systematization is the degree of negative entropy, ―SN, which should be measured for the arrangement of deoxynucleotide bases in genes, the arrangement of amino acid residues in the proteins which is transmitted from the genes, the metabolic pathways formed by the catalytic reactions of enzyme proteins, the regulation and control of translation, transcription and replication of DNAs, the cell structure constructed by the intrinsic property of lipid molecules to form a vesicle of lipid bilayer and by the cell wall to support the lipid vesicle, and for furthering the communication between differentiated cells in the case of a multicellular organism.
The energy acquired by such an organism depends on its genome size N and systematization SN as well as on the material and energy source M available from the environment. Thus, the acquired energy is expressed to be E a ( M ; N , S N ) , which is an increasing function of N and SN as well as M. On the other hand, the organism utilizes the material and energy to produce the biomolecules for its growth and self-reproduction. The energy Es(N, SN) stored in the biomolecules such as polynucleotides, proteins, lipids and cell wall is also another increasing function of N and SN. These biomolecules to exhibit biological functions have the higher energy than the inorganic molecules, into which they are finally decomposed, and the energy stored in these biomolecules should be measured in comparison with the energy of the decomposed state. The difference between the acquired energy and the stored energy, E a ( M ; N , S N ) − Es(N, SN), is released as heat. If the entropy production by the heat compensates for the entropy reduction―SN due to the systematization, this is consistent with the second law of thermodynamics. In the organism, the acquired energy is transiently trapped in ATP and NADPH molecules as chemical energy, and it is gradually consumed in the syntheses of other biomolecules under the guidance of enzyme proteins without drastic change in temperature T. Thus, the following quantity will be proposed as biological activity BA.
B A ≡ E a ( M ; N , S N ) − E s ( N , S N ) − T S N > 0. (1)
The positive value of this quantity has been illustrated for simple organisms and also estimated for higher organisms except for the developmental stage of the latter [
To illustrate simply that organisms maintain and further extend their negative entropy, we consider the behaviour of unicellular organisms that self-reproduce, sometimes mutate, and die. The set of internal variables (N, SN) of the organism will be simply denoted by a single variable x, unless the description of changes in the genome is necessary. In the population of such organisms taking a common material and energy source M from the outside, the number nxi(t) of i-th variants xi changes with time t according to the following equation of replicator dynamics.
d d t n x i ( t ) = { Q x i ( t ) R ( M ; x i ) − D ( x i ) } n x i ( t ) + ∑ j ( ≠ i ) q x i , x j ( t ) R ( M ; x j ) n x j ( t ) . (2)
Here, the self-reproducing rate and death rate of the variant xi are denoted by R(M; xi) and D(xi), respectively. The apparent decrease factor Qxi(t) for the self-reproducing rate of variant xi means the mutation of variant xi to other kinds of variants and is related with the mutation term qxj,xi(t) in the following way.
Q x i ( t ) = 1 − ∑ j ( ≠ i ) q x j , x i ( t ) . (3)
The population behaviour becomes transparent by transforming Equation (2) into two types of equations; one concerning the total number B(t) of all kinds of variants defined by
B ( t ) = ∑ i n x i ( t ) . (4)
and another concerning the fraction fxi(t) of variants xi defined by nxi(t)/B(t). By simple calculation, these equations are obtained from Equation (2) to the following forms, respectively, using the relation (3).
d d t B ( t ) = W a v ( M ; t ) B ( t ) . (5)
and
d d t f x i ( t ) = { W ( M ; x x i ) − W a v ( M ; t ) } f x i ( t ) + ∑ j q x i , x j ( t ) R ( M ; x j ) f x j ( t ) . (6)
Here, qxi,xi(t) is defined by Qxi(t) − 1. The increase rate W(M; xi) of the variant xi is defined by
W ( M ; x i ) ≡ R ( M ; x i ) − D ( x i ) . (7)
and the average increase rate of organisms in the population is defined by
W a v ( M ; t ) ≡ ∑ i W ( M ; x i ) f x i ( t ) . (8)
If the mutation term is the point mutation such as the nucleotide base change in a gene, this only changes SN in a definite size of genome and the time change in fraction can be evaluated in the following way by the first order of mutation term. When the increase rate of an occasionally arisen mutantxi is larger than the average increase rate of the population, that is, W(M; xi) − Wav(M; t) > 0, the fraction fxi(t) of the mutant xi increases with time according to the first term on the right side of Equation (6). This raises the average increase rate Wav(M;t), resulting in the increase in the total number B(t) of organisms according to Equation (5), although this increase in B(t) is ultimately stopped by the decrease in available material and energy source M. On the other hand, the fraction fxi(t) decreases when W(M;xi) − Wav(M;t) < 0. Thus, the organisms taking a common material and energy source M are elaborated by the mutation and above selection, and most of them reach the ones xopt with the optimum increase rate.
This is Darwinian evolution of unicellular organisms. This evolution is first proposed qualitatively to explain the generation of new species from the observation of unique species in a geographically isolated region and of domestic animals and plants [
The gene and genome sequencing started in the latter half of last century has brought new information about the evolution of organisms. The amino acid sequence similarities of paralogous proteins strongly suggest that the repertoire of protein functions has been expanded by gene duplication and by the succeeding changes in the counterpart of duplicated genes due to the nucleotide base changes, partial deletion and/or insertion, and domain shuffling [
For formulating mathematically the above evolutionary route, the fraction of variants with the lower increase rate has to be considered more accurately than in Darwinian evolution, after the organisms xopt become dominant in the population, because the gene duplication occurs less frequently than the nucleotide base change. For this purpose, Equation (6) will be formally integrated with respect to time t, i.e.,
f x i ( t ) = exp [ ∫ 0 t { W ( M ; x i ) − W a v ( M ; τ ) } d τ ] [ ∫ 0 t ∑ j q x i , x j ( τ ) R ( M ; x j ) f x j ( τ ) exp [ − ∫ 0 τ { W ( M ; x i ) − W a v ( M ; τ ′ ) d τ ′ } ] d τ + f x i ( 0 ) ] (9)
Among the fractions fxi(t)’s in this expression, we focus on the fraction fxi1(t) of the variants xi1, which have arisen from the dominant organism xopt by the duplication of one kind of gene. Then, the average increase rate, Wav(M; τ) and Wav(M; τ’), is approximated to be W(M, xopt) and the fractions of other variants expect for xopt are neglected on the right side of Equation (9). The mutation term qxi1,xopt(τ) is averaged over a sufficiently long time to be regarded as the rate of gene duplication;
q x i 1 , x o p t = 1 t ∫ 0 t q x i 1 , x o p t ( τ ) d τ . (10)
This quantity qxi1,xopt is used as the occurrence rate of gene duplication. Even in this large time scale, the fraction f(xi1) of variants xi1 is present with the following relation to the fraction f(xopt) of dominant organisms xopt as a semi-stationary state.
f ( x i 1 ) = q x i 1 , x o p t R ( M ; x o p t ) W ( M ; x o p t ) − W ( M ; x i 1 ) f ( x o p t ) (11-1)
The fraction f(xi2) of the variants xi2, which have further experienced the second kind of gene duplication, is also obtained from Equation (9). By focusing on the mutation term qxi2,xi1(τ) of fxi1(τ) on the right side of this equation, the fraction f(xi2) of variants xi2 is shown to be related with the fraction f(xi1) in the following way.
f ( x i 2 ) = q x i 2 , x i 1 R ( M ; x i 1 ) W ( M ; x o p t ) − W ( x i 2 ) f ( x i 1 ) (11-2)
By repeating the similar procedure, the fraction f(xin) of variants xin, which have experienced n kinds of gene duplication, is obtained as
f ( x i n ) = q x i n , x i n − 1 R ( M ; x i n − 1 ) W ( M ; x o p t ) − W ( x i n ) f ( x i n − 1 ) . (11-n)
A new style of organisms y can appear from such a minor member xin by changing the counterparts of n kinds of duplicated genes into new genes. When this change probability is denoted by q y , x i n , the probability P n ( y ← x o ) that a new style organism y appears from the original style organism xo is expressed as
P n ( y ← x o ) = q y , x i n ∏ m = 1 n q x i m , x i m − 1 R ( M ; x i m − 1 ) W ( M ; x o ) − W ( M ; x i m ) . (12)
Here, xopt in Equations (11-1) - (11-n) is replaced by x o or x i o , with the meaning of original.
If the new style organisms y are elaborated to utilize the material and energy source M more efficiently by Darwinian evolution, they compel the original style organisms x to be extinct. If the new style organisms y utilize a new material and energy source L other than M, on the contrary, they form a new population, where the fraction fyk(t) of variant yk obeys the following equation, apart from the population of original style organisms xi in Equation (6).
d d t f y k ( t ) = { W ( L ; y k ) − W ¯ ( t ) } f y k ( t ) + ∑ l q y k , y l ( t ) R ( L ; y l ) f y l ( t ) . (13)
Here, the average increase rate of new and original styles of organisms is defined by
W ¯ ( t ) ≡ ∑ i W ( M ; x i ) f x i ( t ) + ∑ k W ( L ; y k ) f y k ( t ) . (14)
and the total number B(t) of new and original style organisms obeys
d d t B ( t ) = W ¯ ( t ) B ( t ) . (15)
This divergence of new and original styles of organisms can occur without geographical isolation and/or climate change. Generally, the material and energy sources L and M are not completely independent to each other but are connected through the circular flow of materials in the global scale. This gives the fundamentals for forming an ecological system, which is also considered to be the systematization of organisms and environment [
The evolution by gene duplication is investigated systematically at the molecular level about the O2-releasing photosynthesis and O2-respiration in eubacteria, which produce the circular flow of oxygen molecules. In addition to the amino acid sequence similarities of the proteins constituting these systems to the ubiquitous proteins [
In an organism, the entropy production due to the heat released from the difference between acquired energy and stored energy compensates for the negative entropy, which is so designed as to self-reproduce itself by the special arrangement of nucleotide bases in DNA genes. This is realized by the difference in stability and function between DNAs, RNAs and proteins. These molecular events in such an organism are represented by a thermodynamic quantity of biological activity. This quantity more directly reflects the genome change in an organism than the “characters” such as shape, colour, size etc. used customarily in evolutionary biology and the change in this quantity is the better measure to formulate any type of evolutionary process of organisms. The negative entropy in an organism is maintained through the selection of self-reproduced organisms. Moreover, the organisms have the potential to extend the range of negative entropy, even if their increase rate is transiently lowered. This is illustrated for unicellular organisms in the present paper, using the concept of biological activity and the equation of replicator dynamics containing the mutation terms. This concept and the equation are specific to the organism, and they correspond to the answer to the opinion (B) among the three opinions summarized by Brillouin [
The present mathematical formulation of self-reproducing unicellular organisms also has a possibility to explain the origin of genes. Although the RNA replicase has been proposed as the start of life [
Such self-reproducing proto-cells also obey the equation of replicator dynamics with the following mutation terms. If the first formed proto-cells are specified by xi’s on their RNA contents, these RNA contents would have changed upon their replication by the primitive RNA polymerase as well as RNA replicase activity. Thus, most of proto-cells are elaborated to xopt by Darwinian evolution, especially with respect to the primitive rRNA and tRNAs to catalyze the polymerization of amino acids. However, this evolution increases the concentration of primitive RNA polymerases and thus increases the concentration of non-functional RNAs in the cell. This yields the variant proto-cells xi1, in which the increased non-functional RNAs interfere with the primitive tRNAs and rRNAs by the hydrogen bonds transiently formed between their nucleotide bases. Such variant cells are declined to the minor members in the population but produce the translation apparatus after the following steps of variation. A part (anticodon) of tRNA originally embracing the side chain of amino acid residue alternatively attaches to the complementary bases in non-functional RNAs which become later the codon of RNA genes of proteins. Such ancestral RNA gene then comes into the contact with the other type of RNAs which become later the initiation complex. The primitive rRNA is enlarged to form the sites for the successive acceptance of amino acid residues carried by tRNAs aligned on ancestral genes of proteins. After the above series of variant cells, xi2, xi3, ⋯ , xin, a new style of self-reproducing cells, in which L-type of amino acids are polymerized according to the sequence of codons in ancestral RNA genes, appear with the probability such as Equation (12). In this new style of cells, the polymerization rate of amino acids is raised in comparison with the random collision of charged tRNA and primitive rRNA, and the substituted bases in the ancestral RNA genes are selected so as to encode the proteins for raising the increase rate of the cell by Darwinian evolution, compelling the original proto-cells xo to be extinct.
The deoxidization of RNAs would have then occurred in some of such new style cells that began the glycolysis releasing protons. The DNAs thus generated would have been first rubbish and decreases the increase rate of such variant cells. If the optimum RNA content of the self-reproducing cell with the translation apparatus is rewritten into xopt, the internal variable of the variant cell suffering deoxidization corresponds to xi1. For utilizing the DNAs as genes, the variant cells xi1 must have further experienced the succeeding steps of variation xi2, xi3, ⋯ , xin to induce the genes of proteins for the transcription and replication of DNAs as well as of auxiliary proteins for the unwinding of double-stranded DNAs. Such induction of DNA-associated genes from the RNA-associated genes is suggested from the fact that DNA-dependent DNA polymerases and RNA polymerases form a protein superfamily together with the RNA-dependent RNA polymerases in RNA viruses [
In fact, the analyses of neutral base changes in rRNAs reveal that all free-living organisms are traced back to the ancient divergence of prokaryote and eukaryote, probably occurred 4 × 109 years ago [
While the organisms in the DNA-RNA-protein world have evolved overcoming the decrease in organic compounds synthesized non-biologically, the organisms in the RNA-protein world would have turned to utilize the translation apparatus as well as nucleotides and amino acids in the prokaryotes and eukaryotes, and survive as RNA phages and viruses under the common codon usage. In this connection, it should be noted that the biological activity is degenerate, that is, almost the same strength of biological activity can be attained by either a small genome, low systematization and a small amount of acquired energy or a large genome, high systematization and a large amount of acquired energy.
The author declares no conflicts of interest regarding the publication of this paper.
Otsuka, J. (2018) The Negative Entropy in Organisms; Its Maintenance and Extension. Journal of Modern Physics, 9, 2156-2169. https://doi.org/10.4236/jmp.2018.912136