Quantum patterns of genome size variation in angiosperms Liaofu Luo1 2 Lirong Zhang1 1 School of Physical Science and Technology, Inner Mongolia University, Hohhot, Inner Mongolia, 010021, P. R. China 2 School of Life Science and Technology, Inner Mongolia University of Science and Technology, Baotou, Inner Mongolia,014010, P. R. China Email to LZ: [email protected] ; LL: [email protected] Abstract The nuclear DNA amount in angiosperms is studied from the eigen-value equation of the genome evolution operator H. The operator H is introduced by physical simulation and it is defined as a function of the genome size N and the derivative with respective to the size. The discontinuity of DNA size distribution and its synergetic occurrence in related angiosperms species are successfully deduced from the solution of the equation. The results agree well with the existing experimental data of Aloe, Clarkia, Nicotiana, Lathyrus, Allium and other genera. It may indicate that the evolutionary constrains on angiosperm genome are essentially of quantum origin. Key words: Genome size; DNA amount; Evolution; Angiosperms; Quantum Introduction DNA amount is relatively constant and tends to be highly characteristic for a species. The nuclear DNA amounts of angiosperm plants were estimated by at least eight different techniques and mainly (over 96%) by flow cytometry and Feulgen microdensitometry. 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 (2-4). 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 (2, 3, 5). 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” (6, 7). The discontinuous distribution of nuclear DNA amounts in angiosperm genera is rather common. It was also observed in genus Rosa (8). Both mechanisms on genome size increase and on genome size reduction in plant evolution were proposed (2, 3, 9-12). It was suggested that in plants the genome sizes increase due to the proliferation of long terminal repeats (LTR) retrotransposons (9). While the mechanism of genome reduction may be due to many small deletions (10) or some unknown force removing LTR retrotransposons and excess DNA (11). However, all these works 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 eigen-functions of the operator to deduce the distribution of DNA amount for a species or genus. Therefore, the law obeyed 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. Results Genome evolution operator H(,)N To describe the discontinuous patterns of genome size N distribution we simulate the genome evolution by a physical system. The system is characterized by a genome evolution operator H(,)N , where N is the genome size. We assume the eigenvalue and N the eigen-function of H describe the organization of DNA amounts. Assume LL22 H 2()2 VNNN ()2() 2 (2) 2 (1) ccNN22 L 2 The Hamiltonian includes the “kinetic” term 2( )2 and the “potential” term cN 2 VNNN( ) (2 ) 2 . Lc is an evolution-related constant for a family or a genus of species. and are environment-related parameters. Without loss of generality one may assume much smaller than 2. The eigenvalue E and eigen-function ()N obey the Schrodinger equation H()()NEN (2) that describes the steady states of the chromosome organization of the evolving species. The solution of Eq(2) is generally discontinuous. The discrete eigenvalues of Hamiltonian characterize the quantum states of the genome size and the corresponding eigen-function gives the distribution of genome size, 2 namely ()N dN representing the occurring probability of size N in the state ()N . 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 2L22 { (2 )NNNEN 2 } ( ) ( ) (3) cN22 are 1L (2 )2 E E ( n ) 8 ( n 0, 1, 2, ...) (4) n 24c and 2 (NNCH ) ( ) exp( ) ( ) n n2 n 2 HHH0( ) 1, 1 ( ) 2 , 2 ( ) 4 2, ... Hn1 ( ) 2 H n ( ) 2 nH n 1 ( ) (5) 12c 14/ ()()()NNN 0 d L 22 22 L Nd, 0 2 c Cn is normalization constant. The function Hn () is called Hermite polynomial in mathematics. It is easily shown that the probability distribution of ground state (n=0) peaks at 0 or NN0 (2 ) 2 , the probability of the first excited state (n=1) having two peaks at 1 and -1, and the probability of the second excited state (n=2) having three peaks at 52, 0 and 52, etc. Discontinuous distribution of DNA amounts of long and short chromosomes in Aloe The Aloaceae is one of the most stable angiosperm families as far as gross chromosome morphology and basic number are concerned. The genome size 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 between long and short chromosome are very close to 4.6:1 for 20 species (5). 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 2 2 4 ()NN (N ) const . ( N N )2 exp[ 0 ] (6) n10dd It follows that the probability density peaks at N N N0 d() for long chromosomes (7) N N N0 d() for short chromosomes Therefore one has NN11dd ,() NNNtot (7.1) NNNNtot2 200 tot 2 2 where NN0 tot /2, d( N N ) / 2 . The two parameters have been defined by Eq(5). Note that N0 is also the DNA amount or genome size of the ground state n0 and d means the width of the size distribution in ground state. Eq (7.1) gives the DNA amount ratio between long and short chromosomes of n=1 state. From Eq (5) one has d 2/5/4Lc 3/4 N0 2 which is a constant for Aloe family. Therefore the theory predicts the DNA amounts of long and short DNA fall into two straight lines in the figure of N and N versus Ntot . The comparisons between theory and experiments are shown in Figure 1. We found the theory of nuclear DNA distribution is in good agreement with the experimental data. Moreover, the experimental data NN 4.6 gives 2 32 dN 0.413 0 that implies the relation of parameters Lc 0.292 (as assuming much smaller than 2) for Aloeceae family. [INSERT FIGURE 1 HERE] Discontinuous distribution of nuclear DNA amount in four genera Clarkia, Nicotiana, Lathyrus and Allium The discontinuous distribution of nuclear DNA amount also occurs in other angiosperms. The 2C DNA values of 125 species divided into four genera (Clarkia, Nicotiana, Lathyrus and Allium) were studied and they show the same discontinuous distribution of 4-8 groups (4 groups for Clarkia, 7 groups for Nicotiana and 8 groups for Lathyrus and Allium). It was reported that the group means occur at regular intervals averaging approximately 2pg in Clarkia and Nicotiana and approximately 4 pg in Lathyrus and Allium (7). The above empirical laws can be explained naturally in the present theory. [INSERT TABLE 1 HERE] ()n Set the distribution of genome size in quantum state n peaks at i (in 1,.., 1) . The expression of wave function n ()N in Eq (5) shows the DNA amount N in quantum state n always peaks at ()()nn Nii N0 d (8) ()()()()n n n n Ni N i11 d () i i ()n Since i is discrete in given quantum state n, the genome size always shows nodal distribution. From the eigen-function n ()N given by Eq (5) the positions of peaks of genome size are determined by an algebraic equation of parameter . The equations for n=0 to n=7 and their roots are listed ()n in Table 1. Since the adjacent roots and i+1 depart each other by a number of 1 to 2 ()()nn approximately one has NNii 1 approximately equal d to 2 d for each quantum state n. Because d is a constant for a given genus the group means of DNA amount should occur at regular ()n intervals of d to 2 d . Generally, the theoretical value of nuclear DNA amount Ni can be predicted by Eq (8) through two parameters N0 and d . Two parameters are estimated by 1 ()n NN0 i (9) n 1 i NN()()nn ()d ii1 (10) 1 ()()nn ii1 AV where means the average over i . The parameters N and can also be estimated through AV 0 d minimization of the deviation 21 (nn ) ( ) 2 [NNii (exp)] nterms i ()n ()n Here Ni (exp) represents the experiment value of Ni (7) and nterms means the number of predicted species.