{'fraction_non_alphanumeric': 0.03741072440766353, 'fraction_numerical': 0.014510826436911916, 'mean_word_length': 4.583544303797468, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 1, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Quantum patterns of genome size variation in angiosperms

The discontinuous DNA distribution in angiosperms is deduced from the eigen-value equation of the genome evolution operator H . The results agree well with the existing experimental data of Aloe, Nicotiana, Lathyrus and other genera. It may indicate that the evolutionary constrains on angiosperm genome are essentially of quantum origin.


Introduction
DNA amount is relatively constant and tends to be highly characteristic for a species. The nuclear DNA amounts of angiosperm plants were measured by flow cytometry and other techniques. The C-values for more than 6000 angiosperm species were reported till 2011 [1]. It is found that these C-values are highly variable, differing over 1000-fold. Changes in DNA amount in evolution often appear non-random in amount and distribution [2] [3]. One big surprise is the discovery of the large differences in DNA amounts between related plant species, while preserving the strictly proportional change in all chromosomes [4] [2]. As early as in 1980's Narayan discovered the discontinuous DNA distribution in four angiosperm genera and named it as "the quantum mode of C-value progression " [5] [6]. Both mechanisms on genome size increase and on genome size reduction in plant evolution were proposed [2] [7][8] [9]. But they cannot explain the law on the quantum pattern of the discontinuous DNA distribution in angiosperms. In the manuscript we shall introduce a genome evolution operator H, called Hamiltonian in physics, and use the eigenfunctions of the operator to deduce the distribution of DNA amount for a species or genus. Therefore, the equation satisfied by the statistical distribution of genome size is similar to the eigen-equation of observables in quantum mechanics and this equation can easily be generalized to the time-dependent form and used as a guard for studying the plant genome evolution.

Theoretical methods
To describe the discontinuous patterns of genome size distribution we simulate the genome evolution by a physical system characterized by the genome evolution operator H(N, N is the genome size. We assume the eigenvalue and the eigen-function of H describe the steady state of the evolving species. Assume The Hamiltonian includes the "kinetic " term -2(L/c) 2 2 2 N   and the "potential " term V(N)=(2+α)N -βN 2 . L/c is an evolution-related constant for a family or a genus of species. α and β are environment-related parameters. No loss of generality one may assume α much smaller than 2. The eigenvalue E and eigen-function Ψ(N) obey the Schrodinger equation that describes the steady states of the chromosome organization of the evolving species. The solution of (2) is generally discontinuous. The discrete eigenvalue of Hamiltonian describes the genome size of the steady state and the corresponding eigen-function gives the distribution of genome size, namely representing the occurring probability of N-value in the state.
The probability distribution may peak at one value or several values. As the probability peaks at two or more values it means the nuclear DNA amounts varying in that species.
The solutions of the eigenvalue equation (2) or its explicit form Cn is normalization constant. The function Hn(ξ) is called Hermite polynomial in mathematics.

Results and discussions
The Aloaceae is one of the most stable angiosperm families as far as gross chromosome morphology and basic number are concerned. The complement of all species is bimodal, with several long chromosomes and several short ones in the diploid. The 4C DNA amounts were studied for 20 Aloe species and it was found the overall DNA amounts can be divided into long and short chromosome categories. The ratios of DNA amount of long and short chromosome are very close to 4.6:1 for 20 species [4]. This provides a striking example of karyotypic orthoselection but its mechanism is unknown. We shall explain it as the quantum pattern of genome size. Suppose the genomes of the studied 20 Aloe species are all in the first excited states, n=1. From Eq (5) one has the probability density of genome size of the first excited state It follows that the probability density peaks at Therefore one has 0 0 . The two parameters have been defined by Eq (5). Note that N0 is also the DNA amount or genome size of the ground stateΨn=0 and d means the width of the size distribution in ground state. Eq (7.1) gives the DNA amount ratio of long and short chromosomes of n=1 state. From Eq (5)

Figure 1 DNA amounts of long and short chromosomes in Aloe
Experimental DNA amounts of long and short chromosomes of 20 Aloe species are plotted by ■ and ▲ respectively. Data are taken from [4]. The theoretical N+ and N-versus Ntot are calculated from Eq (7) and plotted by the straight lines. The abscissa gives the total 4C DNA (pg) and the coordinate -the long or short DNA amount (pg).
The discontinuous distribution of nuclear DNA amount also occur in other angiosperms.
Since ξi is discrete in given quantum state n the genome size always shows nodal distribution. From the eigen-function Ψ n given by Eq (5)  Generally, the theoretical value of nuclear DNA amount Ni (n) can be predicted by Eq (8) through two parameters N0 and√d . Two parameters are estimated by where av means the average over i. The parameters √ d can also be estimated through minimization of the deviation The comparisons between theoretical prediction and experimental data of nuclear DNA amounts for four genera are shown in Table 2. All nuclear DNA amounts listed are given in unit pg. Experimental data N ( i n) (exp) are group means of 2C DNA taken from [6]. Theoretical values N ( i n) (th1) and N ( i n) (th2) are calculated by use of Eq 8 with parameters estimated from Eq 9 and 10 (for N ( i n) (th1)) or Eq 9 and 11 (for N ( i n) (th2)) respectively. The experimental data were incomplete (namely, for i=3,5) for Allium and the corresponding theoretical calculations N ( i n) (th2) for this genus were obtained by manual regulation of parameter (√d)2 instead of using Eq (11).
From Table 2 we found the theoretical predictions N ( i n) (th1) and N ( i n) (th2) agree well with experimental data if the possible errors existing in group means of 2C DNA (see ref [6]) are taken into account. The experimental data were incomplete for Allium and the corresponding results of the calculations N ( i n) (th1) are not reliable. By manual regulation of parameter (√d)2 the calculations N ( i n) (th2) are consistent with experiments. The nuclear DNA distribution is a property fundamental to the complex organization of angiosperm genomes. Above studies proves that in these four genera the amounts of DNA form regular "nodes" along the entire sequence. The agreement between theory and experiments shows that the quantum mechanism governing the discontinuous pattern of DNA distribution is an acceptable assumption and this mechanism may be common to many plant genera.
The parameters N0 and (√d) are related to more fundamental evolutionary parameters L/c and β. The β value reflects the environmental influence on the genus and L/c reflects the inherent characteristics of the genus. From the values of N0 and (√d)2 listed in Table 2, by using Eq (5) where f >0 is a dissipation coefficient representing the effect of fluctuation force. Note that ∑xi=N (genome length or size). Both the dimension of N and xi should be zero. However, the genome size N can also be given as the mass (in picogram, pg, 1pg=0.98 ×10 9 base pairs) of DNA [2]. x . It has dimension of (time) 2 , [T] 2 (in the first dimensional system) or [T] 2 [M] -1 (in the second dimensional system). Eq (12) was applied in the discussion of avian In literature D is called diversity measure which was firstly introduced by Laxton [14]. In their studies the geographical distribution of species (the absolute frequencies of the species in different locations) was used as a source of diversity. Recently, the method was developed and applied successfully to various bio-informatics problems, for example, the intron splice site recognition [15], the protein structural classification [16], the nucleosome positioning prediction [17], etc. Now we use it to study evolutionary problem since many examples have shown that the genome always becomes as diverse as possible and expands their own dimensionality continuously in the long term of evolution. Moreover, assume the environmental potential env W in the simple case of k=1 is only the function of N Inserting ( When xi=N/4 one has D(x)=2N and the wave function Ψ is a function of variable N only . In this case, Ψ(N) satisfies Eq (3) exactly. Therefore, we have proved that the basic equation (3), the constraints on genome organization, is essentially the evolutionary equation on the steady states of genome.