MUTATION RESEARCH
II1
G E N E T I C RECOMBINATION IN E S C H E R I C H I A C O L I
II. CALCULATION OF I N C O R P O R A T I O N F R E Q U E N C Y AND R E L A T I V E
MAP DISTANCE BY RECOMBINANT ANALYSIS
P. G. DE H A A N AND C. V E R H O E F
Laboratory of Microbiology, The State University, Utrecht (The Netherlands)
( R e c e i v e d S e p t e m b e r 9th, I965)
SUMMARY
In this paper a mathematical analysis based on the physical exchange of
genetic material is presented for a four-factor cross. The incorporation frequency
of donor markers and the relative map distances m a y be accurately estimated from
the frequencies of the eight recombinant classes. The results obtained in KI2 x KI2
and KI2 × B crosses are in good agreement with the theory.
INTRODUCTION
The genetic map of Escherichia coli is based, on information obtained from
transfer curves. The method is good for measuring the 4istance between markers
which are more than one minute apart; at lesser distances its accuracy is restricted.
Mapping by recombinant analysis may be achieved by selecting for a distal
donor marker and scoring for more proximal unselected markers. The cross-over
frequency of an unselected marker is a measure of its distance from the selected
marker. The cross-over frequencies observed in this kind of cross only represent the
probability that a cross-over occurs in zygotes which give rise to a given type ot
(selected) recombinant, but they give no information about the probability of the
same cross-over in all zygotes. The recombination frequencies or recombination
units are not additive and no mathematical test on the results is available. In this
paper a mathematical analysis based on the physical exchange of genetic material
will be presented for a four-point cross. The parameters of a cross, i,e. the incorporation frequency of donor markers and the relative map distances may be accurately
estimated from the observed numbers of recombinants
MATERIALS AND METHODS
The strains used in this paper and the method of crossing ~.re given in our
previous paper 5.
,Vlulation Res., 3 (~966) I I z - I 1 7
112
P.G.
D E HAAN, C. V E R H O E F
MATHEMATICAL ANALYSIS
In our previous paper 5 we have presented evidence that the linkage frequency
of an unselected proximal marker is:
fi =
~ - ~ ( I - - 2 ) e /,/,
(I)
The incorporation frequency (~) and the average number of breakage events (k) per
min transfer time were calculated from observed linkage frequencies and from
transfer times determined in separate blender experiments. The error in the calculation of the parameters c~ and k is unfortunately dependent on the precision in the
estimation of transfer times. A more precise m e t h o d for the estimation of the incorporation frequency and the relative m a p distance m a y be obtained b y the determination of frequencies of recombinant classes in a multi-factor cross.
The n u m b e r of recombinant classes in a cross of the type given in Fig. I with
n markers (one selected and n -- I unselected) is 2 n ~. The n u m b e r of unknowns is
n (c~and n -- I lengths of segments). The number of degrees of freedom (2"-1 -- I) must
be greater than the number of parameters (n) to be estimated in order to compare the
observed frequencies of the unselected markers with the calculated frequencies. A
four-point cross as given in Fig. I with seven degrees of freedom and four unknowns
leaves us thus three degrees of freedom for the mathematical test.
A+
•
Hfr
F-
1
B+
I
2
C+
I
3
D+
I
O
F i g . i. i ) i a g r a m of a f o u r - p o i n t cross. (), o r i g i n of t h e H f r c h r o l n o s o m e ; A, s e l e c t e d m a r k e r .
In this cross A+ reeombinants are selected and the markers B, C and D are
scored as unselected markers. We will make the same assumptions as in our previous
paper 5.
(I) The terminal ends of the Hfr fragment both initiate an obligatory breakage
event.
(2) F u r t h e r breakage events are r a n d o m l y distributed.
(3) The breakage events in the segments which are proximal to D and distal
to A have no influence on the incorporation of the markers.
(4) The incorporation frequency of a donor segment is independent of the
incorporation events of adjacent segments. E a c h Hfr segment between two adjacent
breakage events has thus a fixed probability for integration (incorporation frequency).
Let: ll, 12 and 13 correspond to the lengths of the segments I, 2 and 3 respectively
(Fig. I); k be the average n u m b e r of breakage events per unit length; ~ be the incorporation frequency of a donor segment.
The average number of breakage events in a segment with length l is kl. The
probability that no breakage event occurs in this segment is given b y the first term
(e -~z) of the Poisson distribution:
P(n) =
(kl)" e ~t
n!
Mutation Res., 3 (1966) ~ i r
ii 7
GENETIC RECOMBINATIONIN E. coll. II.
:13
Consequently the probability that (one or more) breakage events take place in this
segment is I -- e ~t.
For convenience we will further use the substitutions:
x1 =
I -- e-~ZI; I -- x 1 =
(2)
e-kZI etc.
We may now write down the expectations for the various breakage events and the
expectations for the incorporations necessary to obtain the eight recombinant classes.
Table I gives these expectations. It may be seen from this table that B+C+D + recombinants may be obtained as the result of eight different "breakage and reunion"
events. The total probability for this class is obtained by multiplication of the eight
TABLE
I
THE PROBABILITIES FOR THE BREAKAGE AND INCORPORATION EVENTS IN A FOUR-POINT CROSS
T h e t y p e of c r o s s is g i v e n i n Fig. i. All r e c o m b i n a t s a r e A+; + - + r e p r e s e n t s a B + C - D + r e c o m b i n a n t etc. T h e
first column gives the possible breakage events and the second column gives the corresponding probabilities.
I n t h e t h i r d c o l u m n t h e v a r i o u s i n c o r p o r a t i o n p r o b a b i l i t i e s for t h e e i g h t r e c o m b i n a n t c l a s s e s a r e g i v e n . T o
o b t a i n t h e t o t a l p r o b a b i l i t y for a g i v e n r e c o m b i n a n t , e a c h i n c o r p o r a t i o n p r o b a b i l i t y m u s t b e m u l t i p l i e d b y t h e
c o r r e s p o n d i n g p r o b a b i l i t y for t h e b r e a k a g e e v e n t s (see t e x t ) .
Breakage
events i n
segment
Probability
P r o b a b i l i t y o[ i n c o r p o r a t i o n J o t a r e c o m b i n a n t o[ t y p e :
+ + + + + _
+
+
_ + +
--
(I--,V1) ( I - - X 2 ) ( 1 - - / ' 3 )
I
X1
3
( I - - X I ) ( I - - X 2 ) X3
( I - - X 2 ) ( I --X3)
2
.rl
x2
I 3
X1
( I - - X 2 ) X3
~)
~2
(I--x3)
~3
~2(I --~)
~(I -- C()2
~(I --~) 2 ~2(I--~)
~3
~2(I --:()
~2(I --~) ~(i --~) 2
~3(I--~) ~(~--~)~ ~(I--~)~ ~(I
2 3
(: -rl) x2
xa
~3
I 2 3
Xl
23
~4
X2
_ +_
(X~
g(I
:
~ _ +
C(~(I ~)
~2(I --(X)
~)" ~(I--~) ~3(I--~) ~"(I--~)*
probabilities for the breakage events (column 2) with the corresponding incorporation frequencies (column 3). The summation of the eight terms is the probability for
a B+C+D+ recombinant. It may be seen from Table I that one recombinant class is
obtained as the result from eight different events; three recombinant classes as the
result of four events, etc. The expected numbers of recombinants in the eight classes
are obtained by multiplying the probabilities for the various recombinant classes by
the unknown number of zygotes (Z) which were plated. The expected and observed
numbers of recombinants obtained from Z zygotes are given in Table II.
The parameters may be estimated by means of the method of maximum likelihood (compare ref. I). In practice one wants to have a reasonable good set of approximate estimates with which to start the iterative scoring procedure. Estimations
based on the formulae:
(e ~ - g )
:-~-(b
(:
-
~)x,
(a+b+f+h)
+h) × (c + a + e + g )
=
d+e+f+h
M u t a t i o n R e s . , 3 (I966) I I I - i I
7
114
P. G. D E HAAN, C. V E R H O E F
c+g
(I - -
~)X e =
a?
b @ C -~ g
b
a+b
(I -- ~)x3
m a y be u s e d as initial e s t i m a t e s for the i t e r a t i v e calculation of the parameters. The
e x p e c t a t i o n s for the eight r e c o m b i n a n t classes m a y then be calculated with the aid
of the formulae which are given in Table II. The a p p r o p r i a t e test for goodness-of-fit
on calculated a n d observed d a t a is a Z ~ test with three degrees of freedom. The relat i v e m a p distance (kl) of a segment can be c a l c u l a t e d from the e s t i m a t e d p r o b a b i l i t y
(~) for one or more b r e a k a g e events in the segment with the aid of Eqn. 2.
"I'A 13I.t~ 11
I g X P E C T A T I O N S A N D O B S E R V A T I O N S F O R A F O U R . - P O I N T CROSS
T h e t y p e of z y g o t e s a r e g i v e n in Fig. :. All r e c o m b i n a n t s a r e A +. Z r e p r e s e n t s t h e n u m b e r of
z y g o t e s w h i c h h a v e r e c e i v e d t h e H f r f r a g m e n t o r i g i n - s e l e c t e d m a r k e r (A+). N o t e t h a t ~ Z is e q u a l
t o t h e n u m b e r of A + r e c o m b i n a n t s .
Recombi~m~lt class
J3
c
~>
Observed numbers
q
t
a
~z [,[ ([--~).r:l ['
t
t
t
....
b
c
~Z[:
~Z!I
~
--
....
--
i
Total
Expected numbers
(~ ~)A2] [t "(I--~)A'aj
(t -:<)xl
[r
(r--:<)x.~] (I - ~)-*'a
(~ ~).v,' ( t - - ~ ) * . , ( I
~x.~)
0{Z ([ --g)'Y1 (1 -" ~-Y2) 0[.v3
0{Z(I
0() ~10{X2 [ I
(I
~)1/3
:
g
~Z[t
(i
---
h
~Z ( 1
~).v I g.Y2 ( l
CdXl (L ~)a'~.r~
*~
~Z
g) ](3
I{ESULTS
The crosses p r e s e n t e d in this p a p e r were all p e r f o r m e d with the technique
described in our previous p a p e r 5. The f o r m a t i o n of m a t i n g pairs was restricted to a
period of 5 min a n d 5o min were allowed for chromosome transfer. The zygotes were
t h e n i n c u b a t e d u n d e r a e r a t i o n in b r o t h fl)r an a d d i t i o n a l period of 6o min. The
r e c o m b i n a n t s u'ere then p l a t e d on selective medium. The goodness-of-fit was
u n s a t i s f a c t o r y when the zygotes were p l a t e d w i t h o u t post-aeration. The e x p l a n a t i o n
is t h a t the eight t y p e s of r e c o m b i n a n t s h a v e different viabilities when zvf4otes are
p l a t e d on t h e selective plates. Leucine, in particular, suppresses the f o r m a t i o n of
r e c o m b i n a n t s . The effect of leucine d i s a p p e a r e d when the zygotes were i n c u b a t e d
for a period of 60 min in broth. The conclusion is t h u s t h a t the p o s t - a e r a t i o n period
of zygotes is essential to o b t a i n reliable results.
The complete results of two crosses are given in Tables I I I and IV. The p a r a m eters (4 a n d £) a n d the variances were e s t i m a t e d with the m e t h o d of m a x i m u m
likelihood with the aid of a computer. Table I I I gives the results of a H f r R4 × Kz2
adea.-thr-le~v-proA " cross. The incorporation frequency in this cross was 0.505 + o . o i 7
a n d this value is v e r y close to the value (0.5) found in the same cross b u t e s t i m a t e d
from linkage frequencies and distances in time u n i t s t The average n u m b e r of b r e a k a g e
3lulaliotz Res., B (I966) t I I - I 17
GENETIC
TABLE
RECOMBIN~ATION
iN
~E. coli. II.
III
OBSERVED AND EXPECTED NUMBERS OF
CROSS
The estimated
x2 ~
parameters
o.191 ± o . o t 5 ;
+
---+
m
+
--
+
--
Total
TABLE
ade,~+ RECOMBINANTS IN A H f r
R~
×
ade~ thr-leu-proA
KI2
in t h i s c r o s s a r e : d ~ 0 . 5 0 5 q= o . o 1 7 ; Xl = 0 . 5 4 4 ~ 0 - 0 3 0 ;
£~ = 0 . 6 8 7 ~ 0 . 0 2 6 ;
Geno(vpe of recombinant
l-hr leu pro
,
115
k~ 1 = 0.79;
Observednumber of
recombinants
k l 2 = o . 2 1 ; k [ a = 1.16; k~11+2+3= 2.16.
Expected number of
recombinants
664
337
66
242
125
23
38
17
660.6
339.8
68. 3
24o.o
127.6
26.0
36.3
13.4
1512
I512.o
232
=
P
=
1.59
0.75
IV
OBSERVED AND EXPECTED NUMBERS OF adelc+ RECOMBINANTS IN & H f r R 4 × B adejc-thr leu proBCROSS
The estimated
parameters
i n t h i s c r o s s a r e : ~ =- o . o 1 9 ± 0 . 0 0 3 ; 3~1 = 0 . 7 5 7 -L O.OLO;
; 2 = O.312 ~ O.O19; XZ = 0 ' 9 5 8 ~= O . O I 2 ;
k[
1
=
1 . 4 I ; k :2 = 0 " 3 8 ; k [ 3 = 3 " I 7 ;
Genotype of recombinant
thr leu proB
Observed number of
recombinants
Expected number of
recombinants
H+
t.
23
379
167
1615
23
2
4
14
24.1
373-5
172.1
I614.I
3o.1
0.6
3 .2
9-3
2227
2227.0
+
+
. .
_
_
_
+
-.
_
_
q
q
--
+
+
~
--
G
--
Total
k/l+Z+3 = 4 . 9 6 ,
Z22 = 5.65
o. I i > P > o . o
5
events over the segment adek~proA is 2.16, giving an average number of o.216 events
per rain transfer time. This value is in good agreement with that found in the
same cross with the method described earlier (o.211). The expected numbers in the
eight recombinant classes give a very satisfactory goodness-of-fit (Z32= 1.59).
Table IV gives the data from a Hfr R4 × ]3 ade~-thr-leu-proB- cross. The incorporation frequency in this cross is very low (o.o19). The average number of breakage
events over the segment ade~-proB is 4.96, giving an average number of 0.404 events
per rain transfer time. The estimated values for both parameters are again in
good agreement with the values calculated previously (O.Ol9 and 0.378 respectively).
The number of recombinants in two classes (-- + + and + -- + ) is very lowand
these classes were therefore taken together for the calculation of the goodnessof-fit. A quite satisfactory goodness-of-fit Z22 of 5.64 was obtained.
The relative map distances adek-thr, thr-leu and leu-proa in the KI2 × K I 2
cross are 36%, lO% and 54% of the segment adek-proA. The relative map distances
Mutation Res., 3 (1966) 1 1 1 - 1 1 7
116
P.G.
D E H A A N , C. V E R H O E F
O~
\\
\\
\\
\
\\
~thr
-1
pPoA ~
\
\
\
d
9
-2
~ppo8
-250
1
1
t
2
I
3
4
I
5
Average number of breokoge events
F i g . 2. P l o t o f l°log(/J - - ~) a g a i n s t r e l a t i v e m a p d i s t a n c e . F o r v a l u e s of r e l a t i v e m a p d i s t a n c e s (kl)
see T a b l e s l l [ a n d 1V.
,% H f r R~ x 1,~2 aden. thr l e u - p r o A ; .Z . . . . . . ~, H f r R~ × B
aden. thr lez* proB
in the K I 2 ><,B cross, based on a relative distance of the ade~-thr segment of 36°"/o,
are j~6°/,o, lO.3% a n d 82°o, for the segments ade~.-thr, thr lelt a n d leu,-pro~, respective13,. The relative m a p distances of the segment ade,,proR is thus 128% of the segm e n t ade~.@roA. The segment thr@ro~ is 44% larger t h a n the segment thr-proA. On
T a v~, l o r ' s m a p the difference is a b o u t ~o
O °'
/O "
Fig. 2 gives the plot of ~°log(/3 - - ~) against the average n u m b e r of b r e a k a g e
events (relative m a p distance 5) in the various segments. Parallel s t r a i g h t lines (with
a slope of 1 ° l o g e) were obtained.
DISCUSSION
I n this p a p e r the e x p e c t a t i o n s for the eight r e c o m b i n a n t classes in a four-point
cross have been developed. The results from two crosses given in Tables I I I and IV
show t h a t the o b s e r v e d and e x p e c t e d n u m b e r s in the eight r e c o m b i n a n t classes are
in good agreement. In a previous p a p e r 4 evidence was presented t h a t the m a t h e m a t i cal m o d e l of BAILEY ~ gave s a t i s f a c t o r y results. This m e t h o d , however, is based on
t h e a s s u m p t i o n t h a t odd n u m b e r s of cross-over events between two donor m a r k e r s
s e p a r a t e the markers. The evidence p r e s e n t e d in our previous p a p e r 5 shows t h a t this
a s s u m p t i o n is incorrect. The e x p l a n a t i o n for the s a t i s f a c t o r y results o b t a i n e d with
BAILEY'S model is t h a t the e x p e c t a t i o n s for the various r e c o m b i n a n t classes as presented in this p a p e r are basicallv identical with BAILEY'S e x p e c t a t i o n s when the
~VIutati(m Res., 3 (1966) l l I - I [ 7
GENETIC RECOMBINATION IN E . coli. II.
117
i n c o r p o r a t i o n frequency is e x a c t l y o.5. The p r o b a b i l i t y for a cross-over in the segment
b e t w e e n t h e m o s t p r o x i m a l m a r k e r a n d the origin in BAILEY'S model t h e n becomes
e x a c t l y 0.5. I t is t h u s e x p e c t e d t h a t crosses in which the i n c o r p o r a t i o n frequency is
a b o u t 0.5 will give s a t i s f a c t o r y results w i t h b o t h m e t h o d s , whereas crosses in which
tile i n c o r p o r a t i o n frequency is low will only give s a t i s f a c t o r y results with our m e t h o d .
This was indeed found in the K I 2 × B cross in which the incorporation frequency
was v e r y low (o.o19). A s a t i s f a c t o r y goodness-of-fit was o b t a i n e d with our m e t h o d
(Z2~ = 5.49) a g a i n s t a goodness-of-fit Z22 -~ 60 w i t h BAILEY'S m e t h o d .
T h e i n c o r p o r a t i o n frequency in the K I 2 × K I 2 cross is 0.5o5 ± o.o17. The
p r o b a b i l i t y for i n c o r p o r a t i o n of a donor m a r k e r is t h u s equal to the p r o b a b i l i t y for
i n c o r p o r a t i o n of an acceptor marker. "['his means t h a t the segments are r a n d o m l y
i n c o r p o r a t e d ; there is no preference for the i n c o r p o r a t i o n of donor or a c c e p t o r
segments.
I n most K I 2 × K I 2 crosses a value of a b o u t o.5 for the i n c o r p o r a t i o n frequency
was found (C. VERHOEF, unpublished) b u t in some crosses the incorporation frequency
is significantly lower t h a n o.5. The lowest value found in K I 2 × K I 2 cross was o.23
(ref. 5). No e x p l a n a t i o n for the low i n c o r p o r a t i o n frequency was found. The goodness-of-fit in K I 2 ;< t{I2 crosses is, however, less s a t i s f a c t o r y when the incorporation frequency is significantly lower t h a n 0.5. A possible e x p l a n a t i o n is t h a t the low
i n c o r p o r a t i o n frequency is caused b y v i a b i l i t y effects.
The low i n c o r p o r a t i o n f r e q u e n c y in the K I 2 x B cross m a y be e x p l a i n e d on
t h e basis of the genetic i n h o m o l o g y of the p a r e n t a l D N A strands. I t seems t h a t a
preference for the i n c o r p o r a t i o n of the a c c e p t o r D N A exists. The low i n c o r p o r a t i o n
frequency in this s y s t e m explains the low linkage observed in K I 2 x B crosses. The
g r e a t e r p r o b a b i l i t y for a b r e a k a g e event per unit length o b s e r v e d in the K I 2 X B
cross has a less p r o n o u n c e d influence on linkage t h a n the v e r y low i n c o r p o r a t i o n
frequency. The restriction for K , 2 D N A in E . coli B (refs. 2 a n d 3) is t h u s m a i n l y due
to the preferential incorporation of acceptor D N A into the r e c o m b i n a n t , p r o b a b l y
as the result of the preferential b r e a k d o w n of the donor segments after the b r e a k a g e
events.
The average n u m b e r (k) of b r e a k a g e events per unit length differs from cross
to cross, the e s t i m a t e d kl therefore represents the relative m a p distance. These
relative m a p distances are a d d i t i v e as long as interference is absent.
ACKNOWLEDGEMENT
Professor Dr. G. J. LEPPINK a n d Drs. A. TH. VAN DER BURGT are gratefully
a c k n o w l e d g e d for v a l u a b l e a i d in the m a t h e m a t i c a l t r e a t m e n t of our results.
REFERENCES
i BAILEY, N. T. J., Bacterial genetics, Chapter 8 in introduction to the Mathematical TheoJqv of
Genetic Linkage. Clarendon Press, Oxford, 196I p. 119-136.
2 BORER, H., Genetic control of restriction and modification in Escherichia coli. J. Bacgeriol.,
88 (t964) 1652-166o.
3 HOEKSTRA, \V. P. M. AND P. O. DE HAAN, The location of the restriction locus for ;~. I( in
Escherichia coli B. Mutation Res., 2 (196.5) 204-212.
4 STOUTHAMER, A. H., P. G. DE HAAN AND H. J. J. NIJKAMP, Mapping of purine markers in
Escherichia coli K12. Genet. Res., 6 (1965) 442 453.
5 VERHOEF,C. AND P. G. DE HAAN,Genetic recombination in Escherichia coll. I. Relation between
linkage of unselected markers and map distance. Mutation Res., 3 (1966) IOI-IIO.
3lutation Res., 3 (t 966) 11 i-1 t 7