Meiosis, recombination fractions
and genetic distance
Statistics 246, Spring 2004
Lecture 2A, January 22
Initially: pages 1-11.
Later: pages 12-18.
1
- the process which starts with
a diploid cell having one set of
maternal and one of paternal
The action of interest to us
chromosomes, and ends up
happens around here :
with four haploid
cells, each
of
•Chromosomes
replicate,
but stay
whichathas
a centromeres
single set of
joined
their
chromosomes,
•Bivalents
form these being
mosaics ofappear
the parental ones
•Chiasmata
•Bivalents separate by attachment
of centromeres to spindles.
Source:
http://www.accessexcellence.org
2
Four-strand bundle and exchanges
(one chromosome arm depicted)
sister
chromatids
sister
chromatids
2 parental chromosomes
Two exchanges
4-strand bundle (bivalent)
4 meiotic products
3
Chance aspects of meiosis
Number of exchanges along the 4-strand bundle
Positions of the exchanges
Strands involved in the exchanges
Spindle-centromere attachment at the 1st meiotic
division
Spindle-centromere attachment at the 2nd meiotic
division
Sampling of meiotic products
Deviations from randomness called interference.
4
A stochastic model for meiosis
A point process X for exchanges along the 4-strand
bundle
A model for determining strand involvement in
exchanges
A model for determining the outcomes of spindlecentromere attachments at both meiotic divisions
A sampling model for meiotic products
Random at all stages defines the no-interference
or Poisson model.
5
6
A model for strand involvement
The standard “random” assumption here is
No Chromatid Interference (NCI):
each non-sister pair of chromatids is equally likely
to be involved in each exchange, independently of
the strands involved in other exchanges.
NCI fits the available data pretty well, but there are
broader models.
7
The crossover process
on meiotic products
1 change
2 changes
1 change
no change
Changes of (grand)parental origin along meiotic products are
called crossovers. They form the crossover point process C
along the single chromosomes.
Under NCI, C is a Bernoulli thinning of X with p=0.5, that is,
each exchange has a probability of 1/2 of involving a given
chromatid, independently of the involvement of other
8
exchanges.
From exchanges to crossovers
Usually we can’t observe exchanges, but on suitably
marked chromosomes we can track crossovers.
Call a meiotic product recombinant across an interval
J, and write R(J), if the (grand)parental origins of its
endpoints differ, i.e. if an odd number of crossovers
have occurred along J. Assays exist for determining
whether this is so. We usually write pr(R(J))=r, and call
r the recombination fraction.
Recombination across the interval
No recombination
Recombination
No recombination
9
Counting recombinants R and non-recombinants NR
across the interval AB
4 NR
2R, 2NR
4NR
2R, 2NR
2R, 2NR
4R
10
Mather’s formula
Under NCI, if n>0, pr(R(J) | X(J) = n ) = 1/2.
Proof. Suppose that n>0. Consider a particular chromatid. It
has a probability of 1/2 of being involved in any given
exchange, and its involvement in any of the n separate
exchanges are independent events. Thus the chance that it is
involved in an odd number of exchanges is the sum over all
odd k of the binomial probabilities b(k; n, 1/2), which equals
1/2 (check).
Corollary (Mather): pr(R(J)) = 1/2  pr( X(J) > 0).
It follows that under NCI, the recombination fraction r = pr(R(J))
11
is monotone increasing in the size of J, and ≤ 1/2.
The Poisson model
Suppose that the exchange process X is a Poisson process,
i.e. that the numbers of exchanges in any pairwise disjoint
set of intervals are mutually independent Poisson random
variables. Denoting the mean number of exchanges in interval
J by (J), we can make a monotone change of the
chromosome length scale to convert this mean to |J|, where
|J| is the length of J. This foreshadows the important notion of
genetic or map distance, where rate = length.
Exercise: Prove that if X is a Poisson process, so is the
crossover process C.
12
Recombination and mapping
Sturtevant (1913) first used recombination fractions to
order (i.e. map) genes. Problem: the recombination
fraction does not define a metric.
Let’s consider 3 loci, denoted by 1, 2 and 3, and put
rij = pr(R(i--j)).
1
r12
2
r23
3
r13
In general, r13  r12 + r23
13
Triangle inequality
We will prove that under NCI, r13 ≤ r12 + r23 . To see this, define
p00 = pr(R(1--2)&R(2--3)), p01 = pr(R(1--2)&R(2--3))
p10 = pr(R(1--2)&R(2--3)), p11 = pr(R(1--2)&R(2--3)),
where the  denotes the complement (negation) of the event.
Now notice that
R(1--2)&R(2--3) + R(1--2)&R(2--3) = R(1--2),
R(1--2)&R(2--3) + R(1--2)&R(2--3) = R(2--3), and
R(1--2)&R(2--3) + R(1--2)&R(2--3) = R(1--3) (think about this one).
Thus we have
p10 + p11 = r12 , p01 + p11 = r23 , and p00 + p11 = 1-r13 .
Adding the three equations, and using the fact that the pij sum to 1 gives
r12 + r23 - r13 = 2p11 ≥ 0.
In general this inequality is strict. Under the Poisson model, p11 = r12r23 . 14
Map distance and mapping
Map distance: d12 = E{C(1--2)} = av # COs in 1--2
Unit: Morgan, or centiMorgan.
1
d12
2
d23
3
d13
d13 = d12 + d23
Genetic mapping or applied meiosis: a BIG business
• Placing genes and other markers along chromosomes;
•Ordering them in relation to one another;
15
•Assigning map distances to pairs, and then globally.
Haldane’s map function
Suppose that X is a Poisson process, and that the
map length of an interval J is d.
Then the mean number (J) of exchanges across
J is 2d, and by Mather, the recombination fraction
across J is
1
2d
r  (1 e ).
2
More generally, map functions relate recombination
16
fraction to genetic distance; r ~ d for r small.
The program from now on
With these preliminaries, we turn now to the
data and models in the literature which throw
light on the chance aspects of meiosis.
Mendel’s law of segregation: a result of
random sampling of meiotic products, with
allele (variant) pairs generally segregating in
precisely equal numbers.
As usual in biology, there are exceptions.
17
18
Random spindle-centromere attachment at 1st meiotic division
x
larger
smaller
In 300 meioses in an
grasshopper heterozygous
for an inequality in the size of
one of its chromosomes,
the smaller of the two
chromosomes moved with
the single X 146 times, while
the larger did so 154 times.
Carothers, 1913.
19
Tetrads
In some organisms - fungi, molds, yeasts - all
four products of an individual meiosis can be
recovered together in what is known as an
ascus. These are called tetrads. The four
ascospores can be typed individually.
In some cases - e.g. N. crassa, the red bread
mold - there has been one further mitotic
division, but the resulting octads are ordered.
20
21
Using ordered tetrads to study meiosis
Data from ordered tetrads tell us a lot about meiosis.
For example, we can see clear evidence of 1st and
2nd division segregation.
We first learned definitively that normal exchanges
occur at the 4-stand stage using data from N. crassa,
and we can also see that random spindle-centromere
attachment is the case for this organism.
Finally, aberrant segregations can occasionally be
observed in octads.
22
Meiosis in N.crassa
23
First-division segregation patterns
24
Second-division segregation patterns
25
Different 2nd division segregation patterns
Under random spindle-centromere attachment,
all four patterns should be equally frequent.26
Lindegren’s 1932 N. crassa data
27
2-strand double exchanges lead to FDS
There is a nice connexion
between the frequencies
of multiple exchanges
between a locus and its
centromere and the
frequency of 2nd division
segregations at that locus.
28
A simple calculation and result
Let Fk (resp. Sk ) denote the number of strandchoice configurations for k exchanges leading to
first (resp. second) division segregation at a
segregating locus. By simple counting we find
F0 =1 and So = 0, while for k>0,
Fk+1 = 2Sk , and Sk+1 = 4Fk + 2Sk .
Assuming NCI, the proportion sk of second-division
segregants among meioses having k exchanges
between our locus and the centromere is
29
2
1 k
sk  [1 ( ) ], k  0.
3
2
If the distribution of the # of exchanges is (xk),
then the frequency of SDSs is
1
3
s  x1  x 2  x 3  .....
2
4
If the distribution is Poisson (2d) then we find
2
3d
s  (1 e ).
3
This is a map-function: between the unobservable
map distance d and the observable SDS frequency30s.
Interference: the state of play
• Total number of exchanges on an arm rarely Poisson
• Positions of exchanges rarely Poisson in map
distance (i.e. crossover interference is the norm)
• Strand involvement generally random (i.e. chromatid
interference is rare)
• Spindle-centromere attachment generally random
(non-random attachments are quite rare)
•
The biological basis for crossover interference
is only slowly becoming revealed; stay tuned.
31
The Poisson model implies independence of
recombination across disjoint intervals
1
2
3
pr(R(1--2) & R(2--3)) = pr(R(1--2))  pr(R(2--3))
32
Morgan’s D. melanogaster data (1935)
I
sc
II
ec
cv
0: no recombination; 1: recombination
0
1
0
13670
824
1
1636
6*
*the number of double recombinants that we would
expect if recombination events across the two intervals
were independent is 85
Clearly there are many fewer double recombinants than the
independence model would predict.
This phenomenon is called crossover interference.. 33
A measure of crossover interference
1 2
3 4
The coincidence coefficient S4 for 1--2 & 3--4 is:
pr(R(1--2) & R(3--4))
pr(R(1--2))  pr(R(3--4))
=
pr(R(1--2) | R(3--4))
pr(R(1--2))
No crossover interference (for these intervals) if S4 = 1
Positive interference (inhibition) if S4 < 1. 34
An observation concerning
crossover interference
The coefficient S4 for short disjoint intervals, begins
at zero with zero cM separation for Drosophila and
Neurospora, and reaches unity at about 40 cM in
both organisms, despite the fact that the crossover
rate per kb is about ten times higher in N. crassa
than in D. melanogaster.
Thus interference somehow follows map distance
more than it does the DNA bp.
There are a number of other intriguing
observations like this concerning interference.
35
Stochastic models for exchanges
Count-location models
Renewal process models
Other special models, including a
polymerization model
36
Count-Location Models
Barrett et al (1954), Karlin & Liberman (1979) and Risch & Lange(1979)
These models recognize that interference influences
distribution of the number of exchanges, but fail to
recognize that the distance between them is
relevant to interference, which limits their usefulness.
N = # exchanges along the bivalent.
(1) Count distribution: qn = P(N = n)
(2) Location distribution: individual exchanges are
located independently along the four-strand bundle
according to some common distribution F.
Map distance over [a, b] is d = [F(b) – F(a)]/2, where
37
 = E(N).
The Chi-Square Model
Fisher et al (1947), Cobbs (1978), Stam (1979), Foss et al (1993), Zhao et al (1995)
Modeling exchanges along the 4-strand bundle as
events from a stationary renewal process whose
inter-event distribution is 2 with an even number of
degrees of freedom. The x events are randomly
distributed and every (m+1)st gives an exchange:
m=1 below.
Cx
X
Co
X
Cx
X
Co
X
Cx
X
C
C
C
C
C
Co
X
C
The chi-square model is denoted by Cx(Co)m.
38
m = 0 corresponds to the Poisson model.
Evidence in support of the
chi-squared model, I
The model fit the Drosophila data by
embodying two conspicuous features of
those data: the curve for S4 vs linkage
map distance had a toe of the right size
and reached a maximum a little short of
the mean distance between exchanges.
39
Coincidence here means S4 ;
the data are from 8 intervals
along the X chromosome of
D. melanogaster, 16,136
meioses, Morgan et al (1935)
40
McPeek et uno (1995)
Evidence in support of the
chi-squared model, II
The model predicts multilocus recombination
data in a variety of organisms pretty well,
typically much better than other models
The model fits human crossover location data
pretty well too, both in frequency and distribution
of location.
41
Model comparisons using Drosophila data
McPeek et uno (1995)
Recombination Pattern Observed
Data
10431
0000
771
1000
1579
0100
1221
0010
1994
0001
4
1100
7
1010
4
0110
46
1001
3
0101
25
0011
1
1110
1
1101
1
1011
1
0111
1
1111
Chi-square
Expected
Poisson
11014
597
1247
931
1664
68
50
105
90
188
141
40
40
40
40
40
773
Expected
gamma
10497
739
1538
1214
1980
2
12
4
73
68
10
0
0
0
0
0
51
Expected
count-location
10434
778
1556
1185
2036
16
12
23
20
42
32
2
2
2
2
2
67
42
Human
43
Broman &Weber, 2000
Biological interpretation of the
chi-squared or Cx(Co)m model
The biological interpretation of the chi-squared model
given in Foss, Lande, Stahl, and Steinberg 1993, is
embodied in the notation Cx(Co)m : the C events are
crossover initiation events, and these resolve into
either reciprocal exchange events Cx, or gene
conversions Co, in a fairly regular way: crossovers are
separated by an organism-specific number m of
conversions.
In some organisms the relative frequency of crossover
associated and non-crossover associated conversion
events can be observed.
Question: who’s counting?
44
Fitting the Chi-square Model to Various Organisms
Gamete data:
D. melanogaster:
Mouse:
m=4
m=6
Tetrad data:
N. crassa:
S. cerevisiae:
S. pombe:
m=2
m = 0 - 3 (mostly 1)
m=0
Pedigree data:
Human (CEPH):
m=4
The chi-square model has been extremely successful in
fitting data from a wide variety of organisms rather well.45
Failure of the Cx(Co)m model with yeast
The biological interpretation of the chi-squared model
embodied in the notation Cx(Co)m is that crossovers
are separated by an organism-specific number of
potential conversion events without associated
crossovers.
It predicts that close double crossovers should be
enriched with conversion events that themselves are
not associated with crossovers.
With yeast, this prediction can be tested with suitably
marked chromosomes.
It was so tested in Foss and Stahl, 1995 and failed.
46
Very brief summary of some current
research on recombination
It appears that many organisms have two meiotic
recombination pathways, one of which lacks
interference. There the protein MSH4 binds to
recombinational intermediates and directs their
resolution as Cx’s, while in its absence these resolve
as Co’s. The intermediates seem to be brought into
clusters, called late recombination nodules, and MSH4
binds to one member per cluster, e.g. the middle one.
This resolves as a crossover while the others resolve
as noncrossovers, leading to the “counting model”.
47
Challenges in the statistical study of meiosis
Understanding the underlying biology
Combinatorics: enumerating patterns
Devising models for the observed phenomena
Analysing single spore and tetrad data,
especially multilocus data
Analysing crossover data
48
Acknowledgements
Mary Sara McPeek, Chicago
Hongyu Zhao, Yale
Karl Broman, Johns Hopkins
Franklin Stahl, Oregon
49
References
www.netspace.org/MendelWeb
HLK Whitehouse: Towards an Understanding of the
Mechanism of Heredity, 3rd ed. 1973
Kenneth Lange: Mathematical and statistical
methods for genetic analysis, Springer 1997
Elizabeth A Thompson Statistical inference from
genetic data on pedigrees, CBMS, IMS, 2000.
50
51
Testing and generalizing NCI
NCI implies inequality constraints on (multilocus)
recombination probabilities which can be tested
against statistical alternatives.
We also have biological alternatives: models for
strand choice going beyond NCI.
The best known is due to Weinstein (1938) which
postulates a Markov model for the pairs of nonsister chromatids being involved in successive
exchanges; the cost is just two extra parameters.
There is not much evidence that it is needed.52
At the inaugural meeting of the
Biometrics Society, Woods Hole, 1947
Ronald A Fisher to Joshua Lederberg:
“Young man: it is not a two-strand model,
it is a one-strand model!”.
53