The proportion of revertant and mutant phage in a growing population, as a function of mutation and growth rate

Gene, 1(1976) 27--32 27 @ Elsevier/North-HollandBiomedicalPress, Amsterdam- Printed in The Netherlands

APPENDIX

THE PROPORTION OF REVERTANT AND MUTANTPHAGE IN A GROWING POPULATION, AS A FUNCTIONOF MUTATION AND GROWTHRATE

EDUARD BATSCHELET*,ESTEBANDOMINGO** and CHARLESWEISSMANN** *Mathemafisehes Institut, Uni~rsit~t Z~rich, 8032 Zfirich and **lnstitut fdr Molekularbiologie I, Uniwrdtat Zi2rieh, 8093 Z~rich (Switzerland)

The accompanying paper (Domingo et al., 1976) describes experiments in which Q~ phage with an A -~ G mutation in the 40th nucleotide from the 3' terminus (mutant 40), either pure or mixed with an equal number of plaqueforming units of wild type Q0 phage, were propagated serially on E.coli. Infection was carried out at an overall multiplicity of 20, so as to favor intracellular c~mpetition of the phages. The lysate resulting from the first infectious cyc|e was used t o initiate a second one under the same conditions as before. This process was continued for 10 or more cycles and the ratio of mutant to wild type phage was determined by analyzing the 32P-|abeled viral RNA in regard to mutation 40. It was found that if the experiment was ~n~tiated with cloned mutant phage 3--8% of the phage was wild type after ti~xee infectiou~ cycles, and that the proportion of wild type increased rapidly with successive propagations. If the experiment was started with an equal mixture of mutant and wild type phage, the proportion of mutant dropped to about 20% after ~2, and to less than 0.5% after 10 transfers. These results could be explained by assuming that wild type phages (a) arise at a significant rate by reversion and (b) outgrow the mutant phage. We will formulate a model which enables us to describe the composition of the phage population as a function of reversion and growth rate in the population. Let N be the number of infectious cycles (or transfers), N = 0 , 1 , 2 , , MN ~ the amount of mutant phages, WN the amount of wild type phages at the end of the N-th cycle. Then

MN+ 1

ffi

kmMN --~mMN

+

#wWN ,

(1)

WN+ 1 -- KwWN -- #.WN + ~mMN" Herein, k m and k w denote the growth factors of MN and WN, respectively. The facto-fPw is the rate of forward mutation for a particular nucleotide, that is, mutation from the wild type to the mutant. Conversely, ~m is the rote of reversion ~ o m the mutant to the wild type. The mutation rates/~w and # m a r e believed to be several orders of magnitude smaller than k w and km. The equetions (1) form a system of linear, homogeneous difference equa-

28

tions with constant coefficients. An exact solution may be found by the method of trial functions: MN

-

AXN, WN

--

B~tN.

(2)

The characteristic equation is

[(k.-~m)--X

.. --0.

Um

(a)

( k w - ~ w ) --)-

For the ease of calculation we introduce the following notations: 6

(kw-~w)

=

+

(kin---pro),

= ( k - ~ w) -- ( k ~ m ) , R = (e=+ 4~w. m) ~ , S = R--e, T = R+eo

(4)

Notice that R = • which implies that S should be calculated from R2--e2

S=

4u

- --w#m. R+e T

(5)

With these notations in mind the roots of equation (3), that is, the eigenvslues ale

~,

1

1

~0+R), ~ --~0--R).

=

(6)

The coefficients of equations (2), that is, the eigenvectors turn out to be A~ = czS,

Bl = 2cl #m'

A=f--c2T,

B=ffi2c~ m,

(7)

with arbitrary coefficients cz and c=. Hence, the general solution of equatior.R

(1) is M N = A,~+A2X2N, wN

=

B,~ s, ÷B~,N

Now we are interested in the proportion of mutant phages. Let

MN MN+Ws .....,m,..

It follows from equa~ons (8) that

(8)

29

A. +A2 (),2 IX. )N PN=

(10) (A.+B.)+(A2 +B2 )(X2 IX. )N

We adjust this formula to the special initial condition Wo = 0 or po =1. From equations (7) and (8)we conclude that c. + c2 = 0 and that

8 + T(X2/X. )N PN

(11)

ffi

(S+2p m)+(T-'2p m)(X2 IX. )N As N -~ ~ , PN tends to

S/(S+2. m ) or. in view of equation (5). to 1 l+(T/2~w)

(12)

which is approximately equal to/~w/(kw--km) provided that "w and #m are small compared with k w and k m and that k w ~ k m. We shall first estimate the relative growth rates of mutant 40 and wild type phage from expe~ment 4 (Fig.3, Domingo et al., 1976) in which a mixture of about equal numbers of infectious wild type and mutant phage were propagated on £.coli for 10 infectious cycles. Assuming that Pm ~ Pw (= p) and that/~/k and , / k w are small (less than . m about 10 -2 ) equations (4). (5) and (6) simplify to 8

= k w + kin. e = k w - - k m . R = e. Tffi 2e.

S = 21~2le,

X,

= k w,

, Let q = pie =

X2 =

k m.

km /(1 --

kw

,,). kw

(13)

Then equation (11) reduces to

q2 + (km/kw)N

Ps-

(14) q(l+q)+(1--q)(k m/kw)N

Fig.1 shows a family of curves for , / k w ffi 3.5 • 10 -4 and different values of km/kw, while Fig,2 ~ o ~ a family of c ~ e s for km/kw = 0.6 and different v ~ u ~ i o f p/kw. It is Clear that a change in p/kw has no detectable effect on the slope of the m i d p ~ of the curves, which depends essentially on km/kw. For some N = No the first measured vPlue of experiment 4 was p = 0.39.

80 ,

0

2

4

6

8

10

12

14

"16

18

20

22

N

Fig.1. Co,~position of a phage population as 8 function of ~ number of ~ e r s and the relative browth rate of wild type and mutant. The curves for different relative Jrowtb rates of mutant and wild type phage, k n ~ from 0.1 to 0.9 were computed from equation (14), auuminl~ 8 relative mutation rate ~/kw = 3.5.10"4; the slopes of the midparts of the curves are largely independent of l, llcw~S~d depend mainly on kmlk w (el. Fiii.2). The data points of the experiments of Domingo et 81. (1976) have been fitted t o the theoretical curves. (o), Experiment 1; (a), experiment 2; (a), ezperiment 8; (s), ezperiment 4.

P# (%) 100

80 60 40

20

0

5'

4

6

8

10

12

14

16

18

~20

N

Fig.~. Composition of apk the relative mutation rate. 5.10,. to6 • lO")were

of mutant and wild ~

kL_

.re

31

As shown in Fig.l, the best fit for the data is obtained ior km/k w

0.25.

(15)

From equation (12) we may now obtain an estimate for #/k w. Using pN o + 9 <~ 0.005 as an approximation of p m we find #/k w ~ 0.004.

(16)

This is a maximal value because PNo + 9 was in fact not measurable. We shall next m~ lyze experiments 1--3 (Fig.2, Domingo et al., (1976)) which had been carried o u t as follows: Mutant phage were plated in high dilution on bacterial lawns. Single plaques thus contained the progeny derived from a single phage. A single plaque was cut out, the phage eluted and used to prepare 1 ml of virus stock with a titer of about 1011 infectious particles/m!. This stock was used to infect a bacterial culture in order to prepare the 32p. labeled phage required for analysis. This step is the first cycle in liquid culture but in fact prior to it the phage had undergone No cycles on the agar plate, resulting in an amplification of about 1011, the equivalent of about 40 doublings. If we equate infective cycles on a bacterial lawn with cycles in liquid culture, No would be between about 3 to 5. Assuming, No = 4 and using the data of experiments 1--3 to fit the curves of Fig.1 in regard to the slope of their midparts, we find a best fit with km/kw ffi 0.6 for expe~ments I and 2, and km/kw ffi 0.5 for experiment 3. From the curves for km/kw = 0.6 in Fig.2 we estimate #/kw to be about 3.5 • 10 .4 for experiments 1 and 2; the same value is found for experiment 3 (km/kw = 0.5; curves not shown). The value for #/kw depends on the assumptions made for No; assuming No = 3 or No = 5 the values for #/kw are about 2 • 10 "4 and 6 • 10 ~, respectively. There is a striking discrepancy between the values found for k m/kw in experiments 1--3 (0.5 to 0.6) on the one hand and experiment 4 (0.25) on the other. In the first three experiments the transfers were initiated with phage populations consisting predominantly of mutant phage and, despite infection at a multiplicity o f 20, most bacteria were infected by mutant particles alone until the proportion of wild type had risen to about 5%. Thus intracellular competition against wild type may not have been achieved in the initial transfers. ln experiment 4, competition was ensured by starting the serial transfers with an equal mixture of wild type and mutant particles. The data could thus be taken to mean that the larger difference in growth rate between mutant 40 and wild type phage is only expressed under conditions of intracellular competition and may even depend on the intracellular ratio of the two phage RNAs. In favor of this interpretation it was found that the number of particles released by cells infected with either wild type phage alone or mutant 40 alone does not differ substantially. If indeed km/kw is lower in the initial phase of experiments 1--3,/~/k w as calculated from these data may represent a lowest eS~imat~, . . . . . s u m ~ ~ it ~ possible t o obtain a rather reliable value for km/kw under competitive growth conditions, namely about 0.9.5; the value for the mutation

82

rote, #/kw, at position 40 f r o m the 3' e n d c a n only b e estimated to be between 2 . 1 0 " 4 (from experiments I--3) and 4 • 10 .3 (from experiment 4), with a most Hk~ly value of 3.5 • 10 "4. Wemay ask: What i s t h e probabfl/ty, ~, of a mutation occurflngat a particular site for each doubling of thephage? Then we h a v e t o solve theequatJon (2--v) n = 2n--~

(17)

where n is t h e n u m b e r of doublinas ~ infectious cycle. The;completion of an i n f e c t i o u s cycle under t h e c o n d i t i o u s 0 f our experiment (Le,, at a multiplicity of 20) resulted in the release of about 1300 infectious particles per bacterial cell; formally this corresponds to~6 t o l O doublings, depending on the number of infecting viral genomes participating as templates. Thus and

26 < k w < 21° 2 4 < km < 28

(18)

and n is between 6 and l O . F o r # / k w = 3.5- 10 ~ we get # = 0.022

for n = 6

# = 0.36

for n = 1 0 .

and It follows that

v -. 1.2 • I 0 ~

for n = 6

p ~-- 0.7 • 10 .4

for n = I0.

Thus, the probability of reversion at position 40 from the 3' end is about 10"4 per doubling. In summary, we have formulated a mathematical model which describes the composition of a phage popula~on consisting of wild type phage and a particular mutant a s a function of thek mutation and growth rotes, The resulting curves were fitted to experimental data obtained with the extracistronic ~ phage mutant 40 and its wild ~ counterp~. ~egro~ rate of the mutant relative to wild type, under competitive conditions, was 0.25, and the reversion rate of t h e m u t a n t was estimated to be a b o u t 10 "4p e r doubling. AC~O~E~~ENTS

This work was supported by a grant from the Schweizerische Nafionalfonds t o C.W. ( 3 . 4 7 5 , 0 . 7 5 ) . E.D. w a s a p a r t . t i m e F e l l o w o f E M B O ,

REFERENCES

D~.mingo, E., FlaveU,R.A. and W e ~ , C., In vitro s i t e - d i ~ mutsgenesis: generation ,,nd properties of an i~ectiou, ~ 1 (1976) e ~ , t r o n i c mutant otb,~rJophqe QB, ~ n e , ...... 3 - - 2 5 .

~

Communicated by W. Szybakki.

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