**4. The monoploid eukaryotes that exchange homologous chromosomes through conjugation**

Some monoploid eukaryotes with the plural number of chromosomes conjugate to form a zygote during their life cycle, and the zygote produces monoploid descendants by exchanging homologous chromosomes upon the meiosis. Although the conjugation also occurs in prokaryotes, it only takes place to exchange plasmids and partial genes. Originally, the conjugation would have evolved to avoid the accumulation of disadvantageous mutations in a special lineage and to maintain the stability of a population by weakening the influence of such mutations. However, the conjugation in the eukaryote with the plural number of chromosomes makes it possible to produce the descendant receiving two or more new genes, even if these new genes are relatively large. Thus, the conjugation of such eukaryotes is considered to be the strategy to overcome the difficulty of generating many and large new genes from the successive gene duplication in a single lineage of monoploid organisms. For this illustration, several examples will be first listed in the following subsections 4.1 to 4.3, and they are used to estimate the probabilities of producing the descendant received more new genes by the conjugation of variants, each carrying a smaller number of new genes.

### **4.1 The probability of producing the descendant received two new genes**

Such a descendant is produced from the conjugation of two types of variants, one carrying a new gene *I* on a chromosome *C1* and another carrying a new gene *J* on another kind of chromosome *C2*. The genome of the variant carrying the new gene *I* is denoted by *(C1I, C20)* and the genome of another variant carrying the new gene *J* is by (C10, C2J). The conjugation of these two types of variants yields the zygote, whose genome constitution is represented by *(C1I, C10; C2J, C20)*. If the homologous chromosomes are randomly partitioned into two daughter cells, the probability *Pc2* of producing the new monoploid descendant received the genome *(C1I, C2J)* is calculated to be *Pm12/2*.

### **4.2 The probability of producing the descendant received three new genes**

The descendant received three new genes *I*, *J* and *K* can be produced from the conjugation of variants, one carrying one new gene *I* and another carrying two new genes *J* and *K*. Two cases are considerable for this production.

On the basis of this expression (20), the probabilities *Pmn*'s for several values of *n* are plotted against the reduction factor *s* in Fig. 1. In the case of *n = 1*, the reduction factor *s* is permitted in a whole range of *0 < s < 1* and the probability *Pm1* of generating a new gene is present in this range. This means that the monoploid organism is suitable to create a new gene step by step, testing the biological function of the new gene product, even if the gene size is large. As the value of *n* increases, however, the reduction factor *s* is restricted to the narrower range of *0 < s < 1/n*. When the monoploid organism creates simultaneously multiple kinds of new genes from different origins of gene duplication, therefore, these genes are obliged to be of a smaller size. Moreover, the probability *Pmn* is also decreased as the value of *n* increases. This is because *Qn* becomes smaller for the larger value of *n*. Thus, it is difficult for the monoploid organism to evolve a new character which requires the expression of many kinds of new and large genes. This result is common to the prokaryote with a single DNA molecule and the lower eukaryote with the plural number of chromosomes, if the latter does

**4. The monoploid eukaryotes that exchange homologous chromosomes** 

**4.1 The probability of producing the descendant received two new genes**

**4.2 The probability of producing the descendant received three new genes**

Such a descendant is produced from the conjugation of two types of variants, one carrying a new gene *I* on a chromosome *C1* and another carrying a new gene *J* on another kind of chromosome *C2*. The genome of the variant carrying the new gene *I* is denoted by *(C1I, C20)* and the genome of another variant carrying the new gene *J* is by (C10, C2J). The conjugation of these two types of variants yields the zygote, whose genome constitution is represented by *(C1I, C10; C2J, C20)*. If the homologous chromosomes are randomly partitioned into two daughter cells, the probability *Pc2* of producing the new monoploid descendant received the

The descendant received three new genes *I*, *J* and *K* can be produced from the conjugation of variants, one carrying one new gene *I* and another carrying two new genes *J* and *K*. Two

Some monoploid eukaryotes with the plural number of chromosomes conjugate to form a zygote during their life cycle, and the zygote produces monoploid descendants by exchanging homologous chromosomes upon the meiosis. Although the conjugation also occurs in prokaryotes, it only takes place to exchange plasmids and partial genes. Originally, the conjugation would have evolved to avoid the accumulation of disadvantageous mutations in a special lineage and to maintain the stability of a population by weakening the influence of such mutations. However, the conjugation in the eukaryote with the plural number of chromosomes makes it possible to produce the descendant receiving two or more new genes, even if these new genes are relatively large. Thus, the conjugation of such eukaryotes is considered to be the strategy to overcome the difficulty of generating many and large new genes from the successive gene duplication in a single lineage of monoploid organisms. For this illustration, several examples will be first listed in the following subsections 4.1 to 4.3, and they are used to estimate the probabilities of producing the descendant received more new genes by the conjugation of variants, each carrying a smaller

not conjugate to exchange homologous chromosomes.

**through conjugation**

number of new genes.

genome *(C1I, C2J)* is calculated to be *Pm12/2*.

cases are considerable for this production.

One is the case that the new gene *I* is encoded on the chromosome *C1* and both new genes *J* and *K* are encoded on another kind of chromosome *C2*. Then, the genome of the variant carrying the new gene *I* is denoted by *(C1I, C20)* and the genome of another variant carrying the new genes *J* and *K* is denoted by *(C10, C2JK)*. The conjugation of these two variants forms the zygote *(C1I, C10; C2JK, C20)*, which can produce four types of monoploid descendants, *(C1I, C2JK)*, *(C1I, C20)*, *(C10, C2JK)* and *(C10, C20)*. If the homologous chromosomes are equivalently partitioned into two daughter cells, regardless of carrying new genes or not, the new monoploid descendant *(C1I, C2JK)* is produced with the probability of *Pm1Pm2/2*.

In the second case, the new genes *J* and *K* are encoded on separate chromosomes. If the chromosome carrying the new gene *K* is denoted by *C3*, the genome of the variant carrying new genes *J* and *K* is represented by *(C10, C2J, C3K)*. The conjugation of this variant and the variant *(C1I, C20, C30)* forms the zygote *(C1I, C10; C2J, C20; C3K, C30)*. Under the random partition of homologous chromosomes, this zygote yields a new monoploid descendant *(C1I, C2J, C3K)* with the probability of *Pm1Pm2/4* .

As a whole, *3Pm1Pm2/4* is obtained for the probability *Pc3* of producing a new monoploid organism received three new genes by conjugation.

### **4.3 The probability of producing the descendant received four new genes**

The highest probability of producing the descendant received four new genes is obtained by the conjugation of two variants, one carrying two new genes *I* and *J*, and another carrying other two new genes *K* and *L*. The following three cases (i) ~ (iii) are considerable. (i) The new genes *I* and *J* are encoded on the chromosome *C1* in one variant, while the new genes *K* and *L* are encoded on the chromosome C*2* in another variant. The conjugation of these two variants forms the zygote *(C1IJ, C10; C2KL, C20)*, which yields four types of monoploid descendants, *(C1IJ, C2KL)*, *(C1IJ, C20)*, *(C10, C2KL)* and *(C10, C20)*. If the homologous chromosomes are randomly partitioned into two descendants, the probability of producing the monoploid descendant *(C1IJ, C2KL)* is calculated to be *Pm22/2*. (ii) The new genes *I* and *J* are encoded on the chromosome C*1* in one variant but the new genes *K* and *L* are encoded on the chromosomes *C2* and C*3*, respectively, in another variant. The conjugation of these two variants forms the zygote *(C1IJ, C10; C2K, C20; C3L, C30)*. If the homologous chromosomes in each kind of *1*, *2* and *3* are randomly partitioned into two daughter cells, the probability of producing the monoploid descendant *(C1IJ, C2K, C3L)* is calculated to be *Pm22/4*. (iii) The new genes *I* and *J* are encoded on the chromosomes *C1* and *C2*, respectively, in one variant, while the new genes *K* and *L* are encoded on the chromosomes *C3* and *C4*, respectively, in another variant. The conjugation of these two variants forms the zygote *(C1I, C10; C2J, C20; C3K, C30; C4L, C40)*, and yields the monoploid descendant *(C1I, C2J, C3k, C4L)* with the probability *Pm22/8*.

The monoploid organism receiving four new genes can be also produced by the conjugation of a variant with one new gene *I* on the chromosome *C1* and another variant with three new genes *J*, *K* and *L*. The following three cases (iv) ~ (vi) are considerable for the location of the three new genes *J*, *K* and *L*. (iv) The three new genes are encoded on the same chromosome *C2*. In this case, the conjugation of the two variants forms the zygote *(C1I, C10; C2JKL, C20)* and yields the monoploid descendant *(C1I, C2JKL)* with the probability of *Pm1Pm3/2*. (v) The new gene *J* is encoded on the chromosome *C2* and the other two new genes *K* and *L* are encoded on the chromosome *C3*. The conjugation of these variants forms the zygote *(C1I, C10; C2J, C20; C3KL, C30)* and yields the descendant monoploid *(C1I, C2J, C3KL)* with the probability of

A Theoretical Scheme

eukaryotes.

severely incompatible upon the meiosis in the zygote.

**5. Higher eukaryotes in the diploid state**

,

*j l*

∑

', '( , ) ,

+ ←

*i k ik jl*

≠

of the Large-Scale Evolution by Generating New Genes from Gene Duplication 13

reduction factor *s* for some values of *n* in Fig. 2. The probability *Pc2n* is present in the same range of reduction factor as the probability *Pmn* is present and the probability *Pc2n+1* is present in the same range of reduction factor as the probability *Pmn+1* is. This indicates that the larger size of new genes not generated from the successive gene duplication in a single lineage of monoploid organisms can be assembled into an organism through conjugation. Although the values of *Pc2n* and *Pc2n+1* are smaller than those of *Pmn* and *Pmn+1*, respectively, due to the relations of *Q2n < Qn* and *Q2n+1 < Qn+1*, the smaller value of the probability only means the longer time for the monoploid organism to receive *2n* or *(2n+1)* new genes through the conjugation of variants than the time for a single lineage of monoploid organisms to generate *n* or *(n+1)* new genes from gene duplication. If these larger new genes assembled by conjugation endow the descendant with a superior new character, such descendants increase their fraction as a new style of organisms. In this sense, it should be also noted that the descendant receiving *n( = 3, 4, 5,……..)* kinds of new genes can be produced with the lower probability of *(1/2)n(n-1)/2Pm1n* by the successive conjugation of variants having experienced gene duplication on different kinds of chromosomes. Such successive hybridization of different variants, each of them carrying one new gene, may become the main course to yield a new style of organisms carrying three or more new genes, if the homologous chromosomes different in carrying two or more new genes, such as those appeared in the first case of subsection 4.2 and in (i), (ii), (iv) and (v) of subsection 4.3, are

At any rate, the eukaryote with the plural number of chromosomes is suitable to create new characters each expressed by many kinds of new genes, through the conjugation exchanging homologous chromosomes. This explains the diversity of various living styles of eukaryotes, ranging from the unicellular organisms called the *Protoctista* evolving various intracellular organs to the multicellular organisms evolving cell differentiation. As will be discussed in the last section, it is evident from the phylogeny of eukaryotes that the multicellularity and cell differentiation have also started in the monoploid eukaryotes, although the higher hierarchy of cell differentiation has developed in the diploid

The higher eukaryote in the diploid state is characterized by the pairs of homologous chromosomes, and its large-scale evolution contains the process to establish the homozygote of new genes as well as their generation from gene duplication. Although the number of homologous chromosome pairs is different depending on the species of diploid organisms, a specific pair of homologous chromosomes *(xi, xk)* will be first focused for simplicity, where the suffixes *i* and *k* denote different mutations on the respective chromosomes. The number *n(xi,xk;t )* of variants carrying such a pair *(xi,xk)* obeys the following time-change equation in the population of organisms exchanging the homologous chromosomes upon reproduction.

' ' ' ' ' '' ' ' '

∑ ∑ (23)

( , , ;) ( ; , ) ( , ;)( , ;)

*<sup>d</sup> nx x t Qx x t RMx x nx x tnx x t Dx x nx x t dt*

= −

*i k i k ij kl i k ij kl i j k l*

*qx x x x t RMx x nx x tnx x t* × ×

( , ;) ( , ;) ( ; , ) ( , ;)( , ;) ( , )( , ;)

*i k i k ijxkl i k ijxkl i j k l ik ik*

*Pm1Pm3/4*. (vi) The new genes *J*, *K* and *L* are encoded on the chromosomes *C2*, *C3* and *C4*, respectively. In this case, the probability of producing the monoploid descendant *(C1I, C2J, C3K, C4L)* is further decreased to be *Pm1Pm3/8*.

As illustrated in the above examples in subsections 4.1 to 4.3, the probability *Pc2n* of producing the monoploid descendant received the even number *2n* of new genes through one time of conjugation is generally expressed as

Fig. 2. The probabilities of producing the descendants received multiple kinds of new genes by the conjugation of monoploid organisms. The probability *Pc2n* of producing the descendant received *2n* kinds of new genes is simply expressed as the square of the probability *Pmn*, i. e., *Pc2n = Pmn2*. In the same way, the probability *Pc2n+1* of producing the descendant received *(2n+1)* kinds of new genes is expressed as the product of the probabilities *Pmn+1* and *Pmn*, i. e., Pc2n+1 = *Pmn+1Pmn*. Using the relations of *Q2n = Qn2* and *Q2n+1 = QnQn+1*, *Pc2n/Q2n* and *Pc2n+1/Q2n+1* are plotted against the twelve- fold reduction factor *12s* for *n = 1* and *2*. It should be noted that the probabilities *Pc2n* and *Pc2n+1* are present in the wider range of reduction factor than the probabilities *Pm2n* and *Pm2n+1* shown in Fig. 1, respectively.

$$P\_{c\cdot 2m} = a\_{n,n} P\_{mn}{}^2 + b\_{n+1,n-1} P\_{mn+1} P\_{mn-1} + \dots + \dots + \dots \tag{21}$$

and the probability *Pc2n+1* of producing the monoploid descendant received the odd number *(2n+1)* of new genes is expressed as

$$P\_{c2n+1} = a\_{n,n+1} P\_{mn} P\_{mn+1} + b\_{n+2,n-1} P\_{mn+2} P\_{mn-1} + \dots + \dots \tag{22}$$

Although the coefficients *an,n, an,n+1, bn+1,n-1, bn+2, n-1 etc.* depend not only on the number of new genes but also on the distribution of new genes over chromosomes in a complex way, the first terms are most important on the right sides of Eqs. (21) and (22), respectively. This is because the probabilities *Pmn* and *Pmn+1* in these terms are present in the wider range of reduction factor than those in other terms, as indicated in the preceding section. Thus, *Pc2 ~ Pmn2* and *Pc2n+1 ~ PmnPmn+1*, without the coefficients *an,n* and *an,n+1*, are plotted against the

*Pm1Pm3/4*. (vi) The new genes *J*, *K* and *L* are encoded on the chromosomes *C2*, *C3* and *C4*, respectively. In this case, the probability of producing the monoploid descendant *(C1I, C2J,* 

As illustrated in the above examples in subsections 4.1 to 4.3, the probability *Pc2n* of producing the monoploid descendant received the even number *2n* of new genes through

Pc2/Q2

Pc3/Q3 Pc4/Q4

Pc5/Q5

1 2 3 4 5 6 7 8 9 10 11 12

12s

by the conjugation of monoploid organisms. The probability *Pc2n* of producing the descendant received *2n* kinds of new genes is simply expressed as the square of the probability *Pmn*, i. e., *Pc2n = Pmn2*. In the same way, the probability *Pc2n+1* of producing the descendant received *(2n+1)* kinds of new genes is expressed as the product of the

2

Fig. 2. The probabilities of producing the descendants received multiple kinds of new genes

probabilities *Pmn+1* and *Pmn*, i. e., Pc2n+1 = *Pmn+1Pmn*. Using the relations of *Q2n = Qn2* and *Q2n+1 = QnQn+1*, *Pc2n/Q2n* and *Pc2n+1/Q2n+1* are plotted against the twelve- fold reduction factor *12s* for *n = 1* and *2*. It should be noted that the probabilities *Pc2n* and *Pc2n+1* are present in the wider range of reduction factor than the probabilities *Pm2n* and *Pm2n+1* shown in Fig. 1, respectively.

and the probability *Pc2n+1* of producing the monoploid descendant received the odd number

Although the coefficients *an,n, an,n+1, bn+1,n-1, bn+2, n-1 etc.* depend not only on the number of new genes but also on the distribution of new genes over chromosomes in a complex way, the first terms are most important on the right sides of Eqs. (21) and (22), respectively. This is because the probabilities *Pmn* and *Pmn+1* in these terms are present in the wider range of reduction factor than those in other terms, as indicated in the preceding section. Thus, *Pc2 ~ Pmn2* and *Pc2n+1 ~ PmnPmn+1*, without the coefficients *an,n* and *an,n+1*, are plotted against the

2 , 1, 1 1 1 ................ *P aP b P P c n n n mn n n mn mn* =+ + +− + − (21)

2 1 , 1 1 2, 1 2 1 ........... *P a PP b P P c n n n mn mn n n mn mn* + + + +− + − =+ + (22)

*C3K, C4L)* is further decreased to be *Pm1Pm3/8*.

0

*(2n+1)* of new genes is expressed as

10

20

Pc2n/Q2n,

Pc2n+1/Q2n+1

30

40

one time of conjugation is generally expressed as

reduction factor *s* for some values of *n* in Fig. 2. The probability *Pc2n* is present in the same range of reduction factor as the probability *Pmn* is present and the probability *Pc2n+1* is present in the same range of reduction factor as the probability *Pmn+1* is. This indicates that the larger size of new genes not generated from the successive gene duplication in a single lineage of monoploid organisms can be assembled into an organism through conjugation. Although the values of *Pc2n* and *Pc2n+1* are smaller than those of *Pmn* and *Pmn+1*, respectively, due to the relations of *Q2n < Qn* and *Q2n+1 < Qn+1*, the smaller value of the probability only means the longer time for the monoploid organism to receive *2n* or *(2n+1)* new genes through the conjugation of variants than the time for a single lineage of monoploid organisms to generate *n* or *(n+1)* new genes from gene duplication. If these larger new genes assembled by conjugation endow the descendant with a superior new character, such descendants increase their fraction as a new style of organisms. In this sense, it should be also noted that the descendant receiving *n( = 3, 4, 5,……..)* kinds of new genes can be produced with the lower probability of *(1/2)n(n-1)/2Pm1n* by the successive conjugation of variants having experienced gene duplication on different kinds of chromosomes. Such successive hybridization of different variants, each of them carrying one new gene, may become the main course to yield a new style of organisms carrying three or more new genes, if the homologous chromosomes different in carrying two or more new genes, such as those appeared in the first case of subsection 4.2 and in (i), (ii), (iv) and (v) of subsection 4.3, are severely incompatible upon the meiosis in the zygote.

At any rate, the eukaryote with the plural number of chromosomes is suitable to create new characters each expressed by many kinds of new genes, through the conjugation exchanging homologous chromosomes. This explains the diversity of various living styles of eukaryotes, ranging from the unicellular organisms called the *Protoctista* evolving various intracellular organs to the multicellular organisms evolving cell differentiation. As will be discussed in the last section, it is evident from the phylogeny of eukaryotes that the multicellularity and cell differentiation have also started in the monoploid eukaryotes, although the higher hierarchy of cell differentiation has developed in the diploid eukaryotes.
