17. Estimation of recombination values between loci C and wx from distorted F2 segregation data

H. I. OKA

National Institute of Genetics, Mishima, 411 Japan

Recombination itself is not affected by segregation distortion caused by gametic selection, but the effect of distortion must be considered in estimating recombination values from distorted ratios. First, let us consider a case in which genes A/a are subjected to gametic selection and the A:a ratio in F1 gametes becomes 1:x. If the factor causing distortion affects male and female gametes equally, the x value can be obtained from the F2 phenotypic ratio (A):(a) as: x=

1/(A)[(a) + SQRT((a)2 + (a)(A))] 

                                                  2(a)
If it affects male or female gametes only, x= _____________.
                                               (A)  -  (a)
When A or a is linked with an F1-sterility gene s with recombination value q, letting the rate of gamete abortion by s be 1 -k (normal gametes: k), it will be expected that:


a     1-q(1-k)           1-xk
__=x=_________  and q = _________               (1)
A     k+q(1-k)         (1-k)(1+x)
When there is a set of duplicate gametic lethals and A is linked with s\1\, it may be assumed that k= 1/2 as gametes having S\1\ S\2\ deteriorate (cf. Oka 1988, p.191). Then, x = (2 - q)/(1 + q) and A: a = (1 + q) : (2 - q). When there is a locus of sporogametophytic interaction alleles and A is linked with Sa, it may be assumed that k=O as gametes with Sa are eliminated. Then X=(1-q)/q and A:a=q: (1-q). When a is linked with s\1\ or Sa, the reciprocals of respective ratios are expected. Letting the recombination value between loci A and B be p and k=0.5, the F1 gametic series is expected as shown by the following example.


             s-A-b                               s-a-b
Gamete  F1:_________(Repulsion)               F1:________(Coupling)
            + -a-B                               + -A-B

             k=0.5        k=O               k = 0.5             k=0

A-B       1/3p(1 +q)      pq               1/3(1-p)(2-q)  (1-p)(1-q)

A-b       1/3(1-p)(1+q)  (1-p)q            1/3p(2 - q)      p(1- q)

a-B       1/3(1-p)(2-q)  (1-p)(1-q)        1/3p(1+q)            pq

a-b       1/3p(2-q)      p(1-q)            1/3(1-p)(1+q)    (1 -p)q     (2)
In this model, genes B/b are influenced by s indirectly through the linkage with A/a. Letting the recombination value between B and s be r, r=p+q-2pq. When the s gene is located between A and B, there can be different combinations of genes. For instance, when the F1 genotype is 

A-s-b
_____
a-+-B
and k=0.5,letting the recombination values between A and B, A and s, and between s and b be P, q and r, respectively (p=q+r-2qr), the gametic series will be (A -B): p+q-qr, (A-b): 1-p+qr, (a-B): 2-p-q-r+qr, (a-b): p+r-qr (3)

(all must be multiplied by 1/3 to make the sum equal to unity).

As an example, the data for F2 from 563 X 143 (Tables 1 and 2) will be dealt with. The x/1 value for C:c(=1:x\1\) was obtained to be 1.47 from the numbers of apiculus-colored and colorless F2 plants, 93:51. The x\2\ value for + : wx was estimated to be 0.644 from the numbers of + +, +wx and wx wx F2 plants (1 : 2x\2\ : x\2\2 = 53:69:22, as the mean of values from 53:69, 69:22 and 53:22). Its reciprocal, 1.55, is close to x\1\(= 1.47). This suggests that an s gene is located in-between C and wx and the F1 genotype is 

 C-s-wx ______ c-+-+wx (k=0.5).
On this assumption,the q and r values were obtained to be 0.215 and 0.175, respectively, from the x\1\ and x\2\ values using formulas (1). Substituting these values for q and r in formulas (3), the frequencies of four kinds of gametes were obtained: C-+wx as 0.177+p; C- wx as 1.038-p; c-+wk as 1.647 -p; c- wx as 0.138 +p. Then, the frequencies of 6 types of F2 segregants were expected as follows:

                (Frequency)   (Obs.) (Exp.) (Neglecting distortion) (Exp.)
Apiculus colored,

+wx+wx       0.614+3.294p-p2  24    24.2   0.5p-0.25p2             18.4
+wxwx        3.835-3.018p+2p2 50    49.7   0.5-0.5p+0.5p2          56.9
wx wx        1.364-0.276p-p2  19    19.1   0.25 -0.25p2            33.8
Colorless,
+wx+wx       2.713-3.294p+p2  29    29.0   0.25-0.5p+0.25p2        17.6
+wxwx        0.455+3.081p-2p2 19    18.9   0.5p-0.5p2              15.1
wx wx        0.019+0.276p+p2   3     3.1       0.25p2               3.2
(Total)       9.0              144  X2=0.01   1.0         X2=16.8
The maximum likelihood estimate was p=0.299(+/-)0.060, where the standard error was computed following Allard (1956). When distortion was not considered, P= 0.302+-0.044 was obtained, which was close to the above value, p = 0.299. However, when distortion was taken into account, the observed and expected numbers showed a close agreement (X2 =0.01), and when distortion was disregarded, the chi-square value between the observed and expected number was significantly large (X2=16.8).

Secondly, let us consider the effect of certation on F2 ratios. When the relative fertilizing capacity of pollen grains with =wx and wx is 1:x, the F2 plant

Table 1. Parental strains
________________________________________________________________
Acc.no. Origin    Type     Vernacular name           Genotype
________________________________________________________________
001  Vietnam      Indica    70a som cau            +wx cAP
108  Taiwan         "       Peh-kuh                +wx cAP
143    "            "       Liu-tou-tzu            +wx cAP
160    "            "       Hong-ka-chiu           wx CAP
219  Philippines  Japonica  Garumbalay             +wx CAP
221    "            "       Inakupa                wx CAP
242    "            "       Malagkit pirurutong    wx cAP
414  India        Indica    P.T.B. 10              +wx cAP
521  Japan        Japonica  Kisshin                +wx caP
532    "            "       Gaisen-mochi           wx CAp
563    "            "       Kinoshita-mochi        wx CAP
571    "            "       Mansaku                +wx caP
________________________________________________________________
with +wx+wx, +wxwx and wxwx will be 1:1 + x:x(= a: b: c). The x value may be estimated as the mean of (b-a)/a, c/(b-c) and c/a. Letting the C-wx recombination value be p, the pollen series from a 

 C-wx
______
c-+wx
plant will be expected: C-+wx as p(1/1+x);C-wx as (1-p)(x/1+x);c-+wx as (1-p)(1/1+x); c-wx as p(x/1 + x).

As an example, the F2 data from 414 x 221 will be taken up, in which the +wx+wx:+wxwx: wxwx ratio (84: 116: 32) deviated from 1 : 2: 1 significantly, but the ratio of apiculus-colored: colorless plants (164: 68) fitted 3: 1 (Tables 1 and 2). The x value was estimated to be 0.381. Then, the relative fertilizing capacities of +wx and wx pollen grains 


(1/(1 + x) : x/(1 + x))
was found to be (1/2)1.448 : (1/2)0.552. From this, the frequencies of 6 types of F2 plants were assumed as follows:

                 (Frequency)          (Obs.) (Exp.)    Neglecting
                                                       certation
Apiculus colored,                                         
+wx+wx         1.448( 0.5p-0.25p2)     38   36.8          25.4
+wxwx          (0.5-0.5p+0.5p2)        95   94.2          94.2
wx wx          0.552(0.25-0.25p2)      31   30.0          54.3

Colorless,
+wx+wx         1.448(0.25-0.5p+0.25p2) 46   47.2          32.7
+wxwx          (0.5p-0.5p2)            21   21.8          21.8
wx wx          0.552(0.25p2)            1    2.0           3.6
(Total)        1.0                      232  X2=0.64  X2=23.6
Certation does not affect the female gametes. The frequency formulas for 6 F2 types show that their modification by considering certation does not affect the

Table 2. Estimation of recombination values between loci C and wx in 15 F2 populations, (A) taking segregation distortion into account and (B) disregarding distortion



estimate of recombination value. The maximum likelihood estimate was 0.247+- 0.030, equally when certation was and was not taken into account. However, when certation was not taken into account, the deviation between observed and expected numbers was quite large (X2=23.6).

When the F2 segregation for apiculus coloration involved two or three genes and is considered to be 9: 7 or 27 :37, if distortion is assumable to be due to gametic selection for C: c, the same method of estimation as above can be used even though the computation is time-consuming. The recombination values between C and wx were thus estimated in 15 F2 populations (Table 2). The three modes of distortion considered, S\1\ (an F1-sterility gene s is linked with C as s- C-wx), S\2\ (s is located between C and wx as C-s-wx), and C (certation), were chosen so as to best elucidate the observed pattern of segregation (Table 2).

Out of the 15 F2 populations, three showed no significant distortion allowing the use of ordinary frequency formulas. In other 12 populations, distortion was significant and was taken into account (Table 2). In general, the use of formulas for distorted ratios did not alter the estimates of recombination value much, but improved the agreement between observed and expected numbers greatly. The largest difference in p value between the estimates obtained by considering and neglecting distortion was 5% (108 X 532 and 414 X 532, in opposite directions; Table 2).

The recombination value thus estimated between C and wx ranged from 20.4% to 35.0% with a mean of 28.0+-5.15%. The phase of linkage was repulsion in most of crosses with an exception in 219 X 242. The linkage between C and wx has been known since the early days of genetic studies of rice (Yamaguchi 1926), and the recombination value has been reported by different workers. In crosses between Japanese varieties, Takahashi (1964) reported 22.8%. In the present study, the p values obtained in three Japonica crosses (excluding 219 X 242) were 21.7 to 24.8%, being close to Takahashi's report. But the values from Indica X Indica and many of Indica-Japonica crosses were much higher, with the exception of 108 X 532 and 414 X 532 (20.4 and 22.3%, respectively). Some were close to 36.0% reported by Seetharaman (1964) in crosses of Indica varieties. From the standard errors of p values and plant numbers recorded in respective crosses, the least significant difference (5%) was estimated to be 13.2%. Then, some of differences in p value found between crosses reach the significance level, although more elaborate statistical investigations are necessary for conclusion. It may be suggested that recombination value differ according to crosses as affected by parental genotypes.

References

Allard, R. W., 1956. Formulas and tables to facilitate the calculation of recombination values in heredity. Hilgardia (Calif. Agr. Exp. Sta.) 24(10): 235-278.

Oka, H. I.,, 1988. Origin of Cultivated Rice. Elsevier/Jpn. Sci. Soc. Press, Amsterdam/Tokyo, 254 pp.

Seetharaman, R., 1964. Certain considerations on genic analysis and linkage groups in rice. In IRRI (ed.), Rice Genetics and Cytogenetics, p. 205-214. Elsevier, Amsterdam.

Takahashi, M., 1964. Linkage groups and gene schemes of some striking morphological characters in Japanese rice. In IRRI (ed.), Rice Genetics and Cytogenetics, p. 215-236. Elsevier, Amsterdam.

Yamaguchi, Y., 1926. Kreuzungs-untersuchungen an Reispflanzen, 1. Genetische Analyse der Speltzenfarbe und Endosperm-beschaffenheit by einigen Sorten des Reises. Bericht. Ohara Inst. Landwrt. Forsh. 3: 1-126.