Download Statistical Analysis of the Patterns of Spore Formation

Journal of General Microbiology (1980), 121, 113-1 16. Printed in Great Britain
113
Statistical Analysis of the Patterns of Spore Formation
in Bacillus subtilis
By G . D U N N 7
Microbiology Unit, Department of Biochemistry, University of Oxford,
Oxford OX1 3QU
(Received 20 May 1980)
~
A description of the patterns of sporulation in chains of Bacillus subtilis cells has been
obtained by fitting a statistical model to data obtained from a resuspended culture. The
underlying assumptions of this model are : (a) resuspension causes an immediate response
in the growing bacteria which is necessary for subsequent sporulation ; (b) the probability
of this response occurring is the same for all the bacteria present at the time of resuspension;
(c) after resuspension some of the bacteria pass through a single round of cell division to
produce a pair of sporulating bacteria, while the rest divide twice to produce four; and
(a) the two sub-populations described in (c) differ in their ability to complete sporulation.
INTRODUCTION
There is evidence that the initiation of sporulation in Bacillus subtilis occurs at a specific
point in the cell cycle (Mandelstam et al., 1971 ; Mandelstam & Higgs, 1974; Dunn et al.,
1978). Once spore formation has started, the process passes through several morphologically
distinct states (Ryter, 1965). At stages 0 and I the sporangium is practically indistinguishable
from a vegetative cell. Stage I1 is the formation of the prespore septum, and stage 111is the
subsequent engulfment of the prespore. At stages IV and V cortex and coat materials,
respectively, are laid down, and by stage V the spore appears bright when observed by
phase-contrast microscopy. Later stages will not be discussed in the present paper, and it
will be assumed that sporulation has been successfully completed if a phase-bright spore is
visible under a light microscope.
This paper provides a description of the patterns of sporulation (that is, whether cells
contain spores or not, irrespective of their position) in chains of four cells (quads) of B.
subtilis that have been resuspended from a rich medium to a poor one to induce sporulation.
It is meant to be a companion paper to that of Dunn (1980) which described the patterns of
spore positions within these quads. As in the previous investigation, the aim of the work
was to provide a statistical model that would summarize the observed patterns of spore
formation. The behaviour of sister cells in sporulating cultures of B. subtilis has been investigated by Dawes et al. (1971) who found that, in general, either both cells within a pair
contained spores, or neither did. They suggested that sporulation is induced in a growing
cell that subsequently divides to produce two synchronously developing sporangia. Here it
is shown that the situation is considerably more complex than this, but that the conclusions
of Dawes et al. (197 1) essentially still hold.
METHODS
Bacillus subtilis 168 (trpC2) was grown on a hydrolysed casein and salts medium (Sterlini & Mandelstam,
1969). Cultures were shaken at 37 "C and harvested while in exponential growth (0.25 mg dry wt ml-l), and
sporulation was induced by transferring the cells to a resuspension medium containing glutamate and
t Present address: General Practice Research Unit, Institute of Psychiatry, De Crespigny Park, Denmark
Hill, Camberwell, London SE5 6AF.
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114
G. DUNN
Table 1. Probabilities of observing the ten classes of quad predicted by model 2*
The proportion of quads derived from a single cell present at the time of resuspension is p, so that
the proportion derived from pairs is (1-p). The probability q is that of induction of the fist
sporulation operons in a cell at the time of resuspension. The parameter r is the probability of
completion in an individual cell of a quad derived from a single cell at the time of resuspension.
The parameter s is the probability of completion in an individual cell of a quad derived from a cell
pair at the time of resuspension.
Class
Probability
ssss
SSSE
SSES
SSEE
SESE
SEES
ESSE
SEEE
ESEE
EEEE
+
pqr 4 (1 -p)q 2s 4
2pqr3(1- U) 2(1 -p)qzs3(1- S)
2pqr3(1-r ) 2(1 -p)q2s3(1- s)
2pqr2(1-r)$+2(1 -p)qzs2(1 - ~ ) ~ + 2 (-p)q(l
1
-q)s2
2pqry1- r ) 2(1 -p)q2s2(1- s)2
pqr 2( 1 - r ) (1 -p)q2s2( 1 - s) 2
pqr2(1-r)2+(l-p)q2s2(1-s)2
2pqr(l-r)3+2(1 -p)q%(l -s)3+2(1 -p)q(l -q)s(l -s)
2pqr(l-r)3+2(1-p)q2s(l-s)3+2(1-p)q(l-q)s(l-s)
-4) + (1 --PI4 2(1- s> 2(1 -pMl -4x1 P4(1 - r )
+
+
+
+
+
* The corresponding probabilities for model 1 are obtained by constraining the values of
equal.
Y
and s to be
inorganic ions, also at 37 "C(Sterlini & Mandelstam, 1969). All media were supplementedwith L-tryptophan
(20 pg ml-l).
Samples were removed from sporulating cultures 6 to 8 h after resuspension and stored in a solution of
formaldehyde (lo%, v/v). The patterns of spore formation in chains of cells were then examined using
phase-contrast microscopy. Sporangia that contained phase-bright spores were scored as ' S ', those that did
not as 'E'. When scoring pairs of cells there are three distinguishableclasses: SS, SE and EE. When scoring
quads there are ten: SSSS, SSSE, SSES, SSEE, SESE, SEES, ESSE, SEEE, ESEE and EEEE.
RESULTS A N D DISCUSSION
A culture of B. subtilis was harvested during exponential growth on the casein hydrolysatecontaining medium and resuspended in the glutamate and salts medium to induce sporulation. After about 6 h a sample was removed from the culture and the patterns of sporulation
in quads were recorded (quads accounted for about 40 % of the cells in this culture, the rest
being mostly in the form of pairs). In a sample of 1794 the counts for the ten classes of
quad were as follows: SSSS 1119, SSSE 92, SSES 93, SSEE 102, SESE 9, SEES 8, ESSE 8,
SEEE 28, ESEE 17, EEEE 318. The proportion of cells containing phase-bright spores was
0.81 (with standard error of 0-01).
Consider a possible statistical model to explain the observed pattern of counts in the
above culture. First, it will be assumed that there is a response, necessary for subsequent
sporulation, at, or close to, the time of resuspension in the poor medium. Secondly, it will
be assumed that the probability of a cell making this response is q. Since resuspension is
followed by a threefold increase in cell numbers (Dunn et al., 1976) it would be reasonable
to suggest that about half of the cells initially transferred grow and divide once to produce
two cells while the rest pass through two generations to produce four bacteria. Now, if we
consider the resulting quad population, we can say that a proportion ( p ) will have been
derived from single cells present at the time of resuspension, and the rest (1 - p ) from cell
pairs. Given that there is about a threefold increase in cell numbers, and if quads are typical
of the culture as a whole, the value of p ought to be about 0.67. Finally, allowance is made
for the proportion of the final population that subsequentlycomplete sporulation, assigning
the probability, r, to completion. These assumptions are the basis of model 1.
Now, consider a more complicated situation in which model 1 is modified to allow
heterogeneity in the proportion of bacteria completing sporulation in a resuspended culture,
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Patterns of sporulation in B. subtilis
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Table 2. Comparison of observed and expected counts for the ten types of quad
Class
ssss
Observed
1119
92
93
102
9
SSSE
SSES
SSEE
SESE
SEES
8
ESSE
8
SEEE
28
ESEE
17
EEEE
318
Total
1794
$Pearson’s chi square
Degrees of freedom
Expected
(model 1)*
1077
119
119
132
7
3
3
7
7
318
Expected
(model 2)t
1119
93
93
102
12
6
6
23
23
318
105.795
4.680
6
5
* Rounded to nearest integer. The maximum likelihood estimates for this model were: p = 0-770 (standard error 0.023),q = 0.784 (s.E. 0.009)and Y = 0.947 (s.E. 0.003).
t Rounded to nearest integer. The maximum likelihood estimates for this model were: p = 0.775 (S.E
0.018), q = 0.788 (s.E. 0.009), r = 0.979 (s.E. 0.004) and s = 0-815 (s.E. 0.020).
$ Calculated using expected values that were not rounded to the nearest integer.
Cells of a quad derived from one cell at the time of resuspension might be more, or less,
likely to complete sporulation than those of a quad derived from two. In the modified
model, model 2, let the probability of completion in the former bacteria be r and the
corresponding probability for the latter be s. The probabilities of observing the ten classes
of quad, assuming model 2 (and if r is equal to s, model l), are given in Table 1.
Using the values already given for these classes the maximum likelihood estimates of the
parameters of the two models were determined iteratively using a modification of the
Newton Raphson method (see Bailey, 1961, Appendix 1). The expected frequencies of the
ten classes of quad, found using these estimates, are compared with the observed frequencies
in Table 2. Using Pearson’s chi-square statistic to assess the goodness of fit of the two
models, it can be seen that model 1 is inadequate, but that model 2 fits the data very well.
Clearly a simpler model than model 1 would not have explained the data, and a model more
complex than model 2 would have been an unnecessary complication. It is concluded,
therefore, that model 2 gives a reliable picture of the process of cell division and sporulation
in resuspended cultures of B. subtilis.
An assumption underlying both models is that resuspension causes an immediate response
(that is, occurring within a few minutes) in the growing bacteria that is essential for subsequent sporulation. Clearly termination of the final round of DNA replication is a necessary prerequisite for spore formation (Mandelstam et al., 1971 ; Dunn et al., 1978), and for
some of the bacteria this final round is started after resuspension to induce starvation (Dunn
et al., 1978). The minimum period of DNA synthesis required in the poor medium is about
35 min, suggesting that bacteria that are over 20 min into a round of DNA replication (a
complete round takes about 55 min) have to start a further round in order to sporulate.
This and other evidence has been used to support the idea that the initiation of sporulation
occurs at a critical point 15to 20 min into a round of chromosome replication (Mandelstam
& Higgs, 1974). Now, if there is an immediate response to resuspension, that is necessary
for sporulation, this response cannot be equated with ‘initiation’ (as used by Mandelstam &
Higgs, 1974). However, recent experiments (J. Mandelstam & S. Clarke, personal communication) have indicated that the first sporulation ‘operons ’ (containing, for example, the
genes coding for serine protease and ribonuclease) are induced at the time of resuspension,
and that their induction does not require concomitant DNA synthesis. The probability q,
therefore, is assumed to be that of inducing these first sporulation operons. It will also be
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116
G. DUNN
assumed that if these operons have been induced in the growing bacterium, ‘initiation’ (as
used by Mandelstam & Higgs, 1974, and Dunn et al., 1978) is certain to occur. The latter
can be thought of as the event that commits the bacterium to form a septum near to one of
its ends, rather than in the centre of the cell. These assumptions are consistent with the
conclusions of Dawes et al. (1971), provided that one interprets their observations in terms
of the present model. Model 2 differs from model 1 in predicting that the probability of
sporulation for cells that pass through a single round of cell division after resuspension is
less than that for cells that pass through two rounds. The values determined for these
probabilities were 0.82 and 0.93, respectively.
This work was supported by the Science Research Council. The results were analysed
while the author was a student of statistics in the Department of Biomathematics, University
of Oxford. I wish to acknowledge the receipt of a T.O.P.S. Award from the Training
Services Division of the Manpower Services Commission. I also wish to thank Professor J.
Mandelstam, Dr F. H. C . Marriott and Dr B. Dancer for their advice and criticism.
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