Anii. IJum. Genet., Lond. (1979), 43, 61
61
Priiitcd i n Great Britain
An alternative model of recombination and interference K. LANGE ATezu York State Psychiatric Institute, Department of Medical Genetics, 722 W . 16Sth Street, New York, N . Y . 10032 and Department of Biomathematics, University of California, Los Angeles, California 90024 BY N. RISCH
AND
INTRODUCTION
The correspondence between visually observed chiasmata and genetically observed recombinution has long been a subject of controversy. The correspondence is now generally accepted as being precise, particularly in light of additional recent cytological evidence (Tease, 1978). Cytological observations have shown that the number of chiasmata along a chromosome arm has R non-random (non-Poisson) distribution, while observations on genetic recombination hare revealed a non-random distribution of crossing-over points. These observations have led to the interpretation that already existent chiasmata somehow ‘interfere ’ with the forinat ion of subsequent ones, snd hence produce the non-randomness. The term ‘positive chiasma interference ’ has been used to denote an inhibitory effect, while ‘negative chiasma intcdkence ’ implies an enhancing effect. The non-randomness in the points of crossing-over has been quantified by the notion of coincidence. The coefficient of coincidence is defined on two disjoint segments as the ratio of the probability of double recombination in the two seginen ts to the product of the probabilities of recombination in the individual segments. The coefficient of coincidence is usually observed to be less than unity, which is taken as evit1rnc.e of ‘positive chiasma interference ’. l r i a recent paper, Sturt (1976) shows that it is unnecessary to assume an interference meclianism in order to account for both the cytological and genetic observations. The nonrandomness may be due to the existence of an obligatory chiasma. She proposes that an obligatory chiasma occurs randomly with respect t o map length, and subsequent chiasmata occur according to a Poisson process. In this paper, we present a more general model of chiasma formation, of which the Sturt proposal is a submodel. Our fundamental assumption is that the non-randomness in crossingover is in the numbers of chiasmata that occur, not in their locations. Coincidence is calculated in terms of the model, and its range of values as a function of the underlying parameters is examined. THE MODEL
Single-armed chromosome The points of exchange along a bivalent may be considered t o constitute a point process (Jagers, 1974). As stated in the introduction, we assume all non-randomness occurs in the distribution of numbers of points. Hence, we assume a general prior probability distribution for the total number of points qi = Probti points} with the associated probability generating function (0
Q(s) =
c qisi a==O
with mean ,u and variance cr2. 003-4800/79/0000-4308 $02.00 @ 1979 Universlty College London
62
N. RISCHAND K. LANCE
Once the number of points is given, the points are distributed independently along the chromosome arm according to some non-atomic probability measure w . If A is any subset of the arm, then w(A) gives the probability that a given point occurs within A . Note that if A is a finite union of n disjoint intervals I l , .. . ,I,,,then
We next define the random variable X4 as the nuiiiber of points that occur in the subset
A . Sirice each point has probability ~ ( ~ - of 1 ) occurring in *4 and probability 1 - u ( A ) of not occurring in A . XA4 has generating function
It is ttlso assuniecl that chromatid strands are involved in points of exchange a t random (i.e. no clrroniaticl interference). Map distance is commonly defined as 3 the expected number of exchanges in a given interval. Hence, for thij model, the map length of an interval I is given by
The 111ap length of the entire chromosome is clearly i p . Thr. ti-fold rrc.ombination fraction is defined for n disjoint (possibly adjacent) intervals as the I’robabili t y that a random gamete shows reconibination on all n intervals simultaneously. In a prerious paper (Lange Br Risch, 1977), it has been shown that in the absence of chroniatid interference. the wfold reconibination fr:tction for the disjoint intervals I l , ...,I,, is given by
Specifically. for a single interval I demarked by two loci, the recombination fraction y is giren by y ( I ) = 3Prob {XI> 0}, also noted by Jfather ( 1 938). Using the generating function in ( I ) , we calculate
yfl)
=
*(I - Prob {XI = 0))
=
+(l-G,(O)) jj(
1- Q( 1- ~ ( 1 ) ) ) .
Since from ( 2 ) ~ ( 1=) [2 x ( Z ) ] / p , we derive the mapping function
63
Alternative model of recombination
The n-fold recombination fraction may also be calculated using an inclusion-exclusion formula (Feller, 1968) and equation (3) :
i
y(ll,...,In) = (3)"Prob i=l
=
($)"(I -Prob
( U (A$, n
=
0)
i=l
n
=
(t)"(l-i=21 Prob (Nz,= 0} +
2
il
61
Priiitcd i n Great Britain
An alternative model of recombination and interference K. LANGE ATezu York State Psychiatric Institute, Department of Medical Genetics, 722 W . 16Sth Street, New York, N . Y . 10032 and Department of Biomathematics, University of California, Los Angeles, California 90024 BY N. RISCH
AND
INTRODUCTION
The correspondence between visually observed chiasmata and genetically observed recombinution has long been a subject of controversy. The correspondence is now generally accepted as being precise, particularly in light of additional recent cytological evidence (Tease, 1978). Cytological observations have shown that the number of chiasmata along a chromosome arm has R non-random (non-Poisson) distribution, while observations on genetic recombination hare revealed a non-random distribution of crossing-over points. These observations have led to the interpretation that already existent chiasmata somehow ‘interfere ’ with the forinat ion of subsequent ones, snd hence produce the non-randomness. The term ‘positive chiasma interference ’ has been used to denote an inhibitory effect, while ‘negative chiasma intcdkence ’ implies an enhancing effect. The non-randomness in the points of crossing-over has been quantified by the notion of coincidence. The coefficient of coincidence is defined on two disjoint segments as the ratio of the probability of double recombination in the two seginen ts to the product of the probabilities of recombination in the individual segments. The coefficient of coincidence is usually observed to be less than unity, which is taken as evit1rnc.e of ‘positive chiasma interference ’. l r i a recent paper, Sturt (1976) shows that it is unnecessary to assume an interference meclianism in order to account for both the cytological and genetic observations. The nonrandomness may be due to the existence of an obligatory chiasma. She proposes that an obligatory chiasma occurs randomly with respect t o map length, and subsequent chiasmata occur according to a Poisson process. In this paper, we present a more general model of chiasma formation, of which the Sturt proposal is a submodel. Our fundamental assumption is that the non-randomness in crossingover is in the numbers of chiasmata that occur, not in their locations. Coincidence is calculated in terms of the model, and its range of values as a function of the underlying parameters is examined. THE MODEL
Single-armed chromosome The points of exchange along a bivalent may be considered t o constitute a point process (Jagers, 1974). As stated in the introduction, we assume all non-randomness occurs in the distribution of numbers of points. Hence, we assume a general prior probability distribution for the total number of points qi = Probti points} with the associated probability generating function (0
Q(s) =
c qisi a==O
with mean ,u and variance cr2. 003-4800/79/0000-4308 $02.00 @ 1979 Universlty College London
62
N. RISCHAND K. LANCE
Once the number of points is given, the points are distributed independently along the chromosome arm according to some non-atomic probability measure w . If A is any subset of the arm, then w(A) gives the probability that a given point occurs within A . Note that if A is a finite union of n disjoint intervals I l , .. . ,I,,,then
We next define the random variable X4 as the nuiiiber of points that occur in the subset
A . Sirice each point has probability ~ ( ~ - of 1 ) occurring in *4 and probability 1 - u ( A ) of not occurring in A . XA4 has generating function
It is ttlso assuniecl that chromatid strands are involved in points of exchange a t random (i.e. no clrroniaticl interference). Map distance is commonly defined as 3 the expected number of exchanges in a given interval. Hence, for thij model, the map length of an interval I is given by
The 111ap length of the entire chromosome is clearly i p . Thr. ti-fold rrc.ombination fraction is defined for n disjoint (possibly adjacent) intervals as the I’robabili t y that a random gamete shows reconibination on all n intervals simultaneously. In a prerious paper (Lange Br Risch, 1977), it has been shown that in the absence of chroniatid interference. the wfold reconibination fr:tction for the disjoint intervals I l , ...,I,, is given by
Specifically. for a single interval I demarked by two loci, the recombination fraction y is giren by y ( I ) = 3Prob {XI> 0}, also noted by Jfather ( 1 938). Using the generating function in ( I ) , we calculate
yfl)
=
*(I - Prob {XI = 0))
=
+(l-G,(O)) jj(
1- Q( 1- ~ ( 1 ) ) ) .
Since from ( 2 ) ~ ( 1=) [2 x ( Z ) ] / p , we derive the mapping function
63
Alternative model of recombination
The n-fold recombination fraction may also be calculated using an inclusion-exclusion formula (Feller, 1968) and equation (3) :
i
y(ll,...,In) = (3)"Prob i=l
=
($)"(I -Prob
( U (A$, n
=
0)
i=l
n
=
(t)"(l-i=21 Prob (Nz,= 0} +
2
il
