The death rate of lysogenic bacteria after ultraviolet induction follows gompertz's law

Mechanisms of Ageing and Development, 2 (1973) 125-139 ~'?) Elsevier Sequoia S.A., Lausanne - Printed in The Netherlands

T H E DEATH RATE OF LYSOGENIC BACTERIA A F T E R U L T R A V I O L E T I N D U C T I O N FOLLOWS G O M P E R T Z ' S LAW

JAMES S. M U R P H Y

The Rockefeller University, New York, N. Y. (U.S.A.) (Received November 10th, 1972)

SUMMARY

After ultraviolet induction of Bacillus megaterium 899a, both the rate of bacteriophage release and the rate of bacterial lysis increase with kinetics similar to those found in autocatalytic reactions. The virus-release curve may be measured over a span of five log units. Evidence is presented that this phenomenon is not mediated by information exchange among bacteria and, therefore, is not dictated by an extracellular autocatalytic reaction. The results show that the kinetics of death in induced cultures are similar to the kinetics of death found in higher organisms where the death rate increases exponentially with the age of the population (Gompertz's law). In other words, when a culture of lysogenic bacteria is induced, it acquires a defined lifespan and proceeds to die with kinetics obeying Gompertz's law. A possible mechanism is discussed in an attempt to explain the results at the molecular level.

INTRODUCTION

In the course of some experiments with ultraviolet light-induced Bacillus megaterium 899a we noticed that the bacterial lysis rate and the virus production rate appeared to increase exponentially with time. Further experiments showed that the important requirement for obtaining such results was to keep the bacteria in exponential growth phase during all manipulations. This paper presents evidence that the finding is not an experimental artifact and that under our conditions the bacterial death rate follows Gompertz's law 1. Golnpertz in 1825 discovered that the mortality rate of human populations over 30 years old rises exponentially with age according to the equation r = roe ~t where r0 is the initial mortality rate, a is the slope of the line and t is the age in years. This formula has been repeatedly shown to be approximately correct for humans and other higher animals '-). Strehler a has reviewed several theories which have been proposed to explain these results. The death rates of single celled organisms, particularly bacteria, have been studied during exposure to heat and disinfectants 4. The kinetics often approximate single-hit 5 or first-order reactions with the very different but equally simple equation

126 #t n0e -;It where no is the initial n u m b e r o f organisms, H the nunlber ,;it time t, ami 2, the death rate (a constant). In contrast, we have found that the death rate ,~I induced lysogenic bacteria is not a constant and the curves formed show no rcscmblance to single-hit kinetics. They may be better described as a p p r o x i m a t i n g inlinitchit kinetics*. In this reporl we present an experimental analysis o f the death rate of induced lysogenic bacteria and rule out several nlechanisms which might mask some o t h e r reaction and fortuitously produce kinetics following G o m p e r t z ' s la',~,. MATERIALS AND METHODS

Bacteria B, m¢gaterium 899a was the lysogenic organism used throughout. B. mq~,,aterium K M was used as a sensitive strain ['or all b a c t e r i o p h a g e assays. Culture methods described by M u r p h y and Philipson 7 were used throughout.

Determhtation o[ baeteria] coneentratio#z Bacterial counts were made by turbidity readings o f the broth cultures in a K l e t t - S u m m e r s o n colorimeter corrected with a calibration curve derived from direct h a e m o c y t o m e t e r cotints. U n d e r the optimal growth conditions used in this experiment the organisms under study a p p e a r in the h a e m o c y t o m e t e r to be mainly in pairs with some chains o f four. The decision o f whether a cell had divided or not was made on the basis o f the unequivocal a p p e a r a n c e o f a septum at 430 magnitication with light gentian violet staining. There was a g o o d correlation between the bacterium to chain ratios obtained by this method and those o b t a i n e d from e×amination of G r a m - s t a i n e d smears studied at 1000 .

l/irllS assay Bacteriophage assays were made by centrifuging, at 4 'C, 0. [ - to 0.5-ml samples diluted to 5 ml in cold p e p t o n e in conical 12-ml centrifuge tubes. After 15 rain at 1200 ,: g, aliquots o f the supernatants were removed immediately and either frozen or held at 4:'C to minimize the possibility o f further phage release by residual bacteria in the supernatant. At a later time, suitable dilutions o f the s u p e r n a t a n t were assayed for virus. These methods have a lower limit o f resolution o f between 10l and 10:' phage per ml. A n y true values below this were lost since they fall below the background level due, in all probability, to residual phage in the p r e l i m i n a r y washing steps and residual bacteria in the final supernatant. All phage titer values represent the mean o f at least two plates counted with suitable dilutions to give between 100 and 200 plaques per plate. Such values have a calculated s t a n d a r d deviation o f less than 7<';i.

* Atwood aand Norman's Formula 7 states that S - I (I e k~) '~ where S is the proportion of organisms surviving, D is the dose of radiation, k is a constant and #t the number of particlc~, within the organism that must be hit at least once. When n becomes large the equation ties much the same appearance as a Oornpertz curve of survivors~i.

127

Induction with ultraviolet radiation Ultraviolet induction was performed as follows: A 24-h slant culture of B. megaterium 899a was suspended in 5 ~ peptone and used as an inoculum for a 10-ml broth culture to give approximately 106 bacteria per ml. The broth culture was placed in a 30°C waterbath for 14 h before shaking was started. For the following 5 h, the culture was diluted from time to time with warm, aerated peptone broth to keep the bacterial concentration below 108 per ml. Just before the time selected for ultraviolet irradiation, the rapidly growing culture was centrifuged for 2 rain at 1400 >; g, the supernatant was discarded and the bacteria transferred to a tube containing 10 ml of aerated fresh medium at 30°C. This centrifugation procedure was repeated from two to four times, and the final sediment was suspended in 30 ml of warm aerated medium to a final concentration of approximately 5 • 107 bacteria per ml. The suspension was divided in half, one part for irradiation, the other for a nonirradiated control. Irradiation was performed in a 50-ml beaker containing a short teflon-covered bar magnet rotating at approximately 100 rev./min. The open beaker was placed directly under a G.E. germicidal lamp at a distance of 10 cm above the resting fluid level, and irradiated for 5 min, a dose found to cause the most rapid and complete induction. The above manipulations were performed with the greatest possible speed in order not to alter the growth rate of the culture. R ESU LTS

Basic observation Fig. 1 illustrates the basic observation of this report. It may be readily seen that throughout the entire time during which an increase in free bacteriophage could be measured the concentration of bacteriophage increased as a simple exponential function. Furthermore, repeated experiments (see also Figs 8 and 9) indicated that the bacteria, throughout the period of lysis measureable by turbidity, were lysing at an increasing rate, the inverse of the virus production curve. (The mathematical proof is contained in the Appendix.) This is completely different from what would be expected in single-hit or first-order reactions where the death rate is constant.

Test of probability function It might seem logical to assume that the lysis time, and thus the virus release time, could be expressed as a mean time with a standard deviation in minutes. If this is true, the virus titer would be described by a cumulative probability function*. Fig. 2A illustrates graphically that this is not the case. For simplicity, only one of several possible values for the standard deviation is considered. However, no standard deviation will predict the results over more than a short range of time.

*

I

F (x)

na/~___.,-oo,

1 t ..re - 2~- (x-H)'-' dx

where u -- mean, ~ -- standard deviation and x -- a discrete chance quantity.

128

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c o o

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6

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x

1 I

S

5 I

4i

0

30

6'0

120

90

150

Time (minutes) posl UV Fig. 1. The "'basic observation", Bacterial ( ) and free phage { ) concentrations al various times after ultraviolet ( U V ) induction. The a r r o w sho~,s t h a t tile culture had Jess than I0'; s u r q v o r s when tile final reading was made.

/,I /

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i / / //

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Fig. 2. A, left. A comparison o f experimental virus concentrations ( • • ) with those predicted by a cumulative probability curve ( .... ) assuming a standard deviation of 12 rain. On the lower right, bacterial concentrations ( . . . . . ) are compared with the inverse of a cumulative probability cur~c ( ). B, r(~,kt. S o m e of tile data from A plotted against a probability ordinate. Tile virus concentration ( • - - - • ) , a straight line normally, n o w plots as a curve, while the probability function f r o m A (12 rain standard deviation) plots as a straight line.

129 F o r the sake o f a clearer d e m o n s t r a t i o n o f the deviation o f p h a g e p r o d u c t i o n from a p r o b a b i l i t y function, the values for released phage f o u n d in this e x p e r i m e n t are plotted against time on a p r o b a b i l i t y o r d i n a t e * in Fig. 2B. P r o b a b i l i t y functions will plot as straight lines using this method. The deviation o f the experimental curve from a p r o b a b i l i t y function shows that no fit is possible. It is c o n c l u d e d that the d a t a c a n n o t be predicted by a cumulative p r o b a b i l i t y function. This result has been t b u n d repeatedly and consistently.

hltracellular virus The next step was to determine the relationship o f intracellular virus prod u c t i o n to the extracellular virus concentration. The main p u r p o s e was to distinguish between the following two possibilities: Virus m a y be p r o d u c e d intracellularly by all cells at a p p r o x i m a t e l y the same time but released after c o m p l e t i o n o f synthesis by an exponentially increasing lysis rate; or, the n u m b e r o f cells initiating phage p r o d u c t i o n may increase exponentially with time. A t intervals o f 20 min, samples o f an induced lysogenic culture were centrifuged for 5 min at 1200 × g (4°C). The s u p e r n a t a n t s were then r e m o v e d and the sediments were suspended in 0.2 M p h o s p h a t e buffer and placed in a Mickle disi n t e g r a t o r at 4°C for 15 rain. The resulting material was diluted and subsequently titered for bacteriophage. The results (Table I) showed that t h r o u g h o u t the e x p e r i m e n t the c o n c e n t r a t i o n of intracellular b a c t e r i o p h a g e increased exponentially, indicating that virus synthesis and cell lysis are coupled in an exponential process.

E.ff~ct of cells predestined to produce virus before induction The next hypothesis to be considered and ruled out is based on the a s s u m p t i o n that the straight-line p h a g e - p r o d u c t i o n curve is a fortuitous consequence o f the

TABLE 1 THE INTRACELLULAR BACTERIOPHAGE CONCENTRATION IN INDUCED AND NON-INDUCED CULTURES

Time ( r a i n )

Ultraviolet-induced Control

20 40 60 80* 95 115"*

3.4 " 104 1.2 " 105 1.0 • lOs 5.3 • 109 lysis began 8.0" 10TM

6.6 5.1 9.2 2.1 ---

• 10 ~ • 10.5 • 10.5 ' 10.5

* In another experiment, at 75 min, tile extracellular phage was 5.3 • l0 s while the intracellular was 5.1 • 108. ** Final titer of the lysate. * Probability graph paper has f (x) as an ordinate and/¢ _L cr as abscissa.

130

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Fig. 3. Experimentally determined bacteriophage concentrations of centrifuged, washed, unirradialcd control B. m<~,aterium 899a showing the biphasic nature of the curve. combination of two processes: the release of virus by bacteria which were alread\ synthesizing phage prior to radiation and ultraviolet-induced production and release of virus. To examine this hypothesis it is necessary to analyze phage production in unirradiated controls. Fig. 3 illustrates the results of one typical experiment and shows the biphasic nature of the curve with a slight peak at 100 rain which is probabl3 caused by the induction of a few cells from the manipulations at zero time. Analysis of the effects that this uninduced or normal phage production would have on the results in induction experiments has yielded the following general conclusions. First, since the initial measurements of virus yield in the controls arc invariably higher than those found in the irradiated cultures, it is concluded that the starting non-induced phage concentration is probably, decreased by irradiation. Second. during the major and most signiticant part of the exponential rise in bacteriophage after induction, a correction for non-induced production has no ell'eel on the results since percentagewise the value change is xerv small. Extracel/u/ar autocala/vlic reaction ruled oul

The shape of the experimental curves in the basic observation (Fig. 1) suggests the possibility o1" an extracellular autocatalytic reaction in which the lysis of one bacterium has somehow predetermined the time of lysis of others. However. Ihis has been shown not to be the case by the following experiments. Our argument is based on the assumption that any change in the shape or the phage-release curvc would be reflected in the bacterial lysis curve either by modifications in lysis lime. rate or both. Two aliquots of the same exponential culture, the first termed A and the second B, were irradiated 55 min apart. Fig. 4 summarizes the resuhs obtained with mixtures of these two (A and B) with appropriate controls. The resuhs indicate thai the two populations behave independently. Similar experiments have shown thai there is no interaction between induced and non-induced cultures. Furthermore, the lysis pattern is unaffected by changing the concentration of bacteriophage lysate oh-

I!

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131

//

/

~3

i

10 7

0

30

uv i i

60

I

90

120

150

L80

Time in minules

Fig. 4. Behavior of a mixture o f two populations o f B. megaterium 899a irradiated (UV) 55 min apart. Mixture (A and B) ( , --,~.~),);time zero (A) irradiated control ( > : - - - - x ) : time 55 (B) irradiated control ( 0 - - 0 ) .

tained from induced cultures or from a sensitive strain of bacteria (KM) after infection with virus. Centrifuging the induced bacteria and resuspending them in fresh medium likewise has no effect. These experiments rule out any explanation for the exponential increase in virus that depends for its mechanism on exchange of information between bacteria. This is true whether mediated by the medium, the virus, or by direct bacterium-to-bacterium contact. Therefore, it is concluded that the exponential rate of phage production is not a manifestation of an extracellular autocatalytic reaction.

The problem of adsorption It is well known that lysogenic bacteria, even though " i m m u n e " to their own phage, may still adsorb it 8,9. Thus, it was essential to determine the influence of adsorption on our findings. The first of this series of experiments was designed to test whether irradiation has any effect on the adsorption rate. The following experiments were completed before the ultraviolet-induced cells lysed and produced their own virus. Two cultures were prepared by the usual methods. One culture was ultraviolet irradiated, as usual, and the other culture was allowed to stand at room temperature for 5 rain. At zero time enough exogenous ultraviolet-induced bacteriophage was added to give a final titer of 108 per ml to each experimental tube and to a third tube containing medium alone. Repeated samples were taken and processed as usual. Results of Expt 1 in Table II show that the rates of adsorption are very rapid and insignificantly different in the two cultures. The next question was whether or not the adsorption rate was influenced by

132 TABLE 11 ADSORPTION RATES OF VARIOUS AMOUNTS OF PHAGE ON B. Mti(IA'FERILM Su% E.vpt No.

Type of culture

Time <{/ter

In/t/a/viru.~

inductk)n (nl/ll)

COllCellll'[lll'Otl

I

control induced

0 0

2.0 - I t) s 2.0 • I 0s

{).261 0.289

2

induced* induced induced induced

0 20 40 60

2.3 107 2.3" 107 1.0 • l0 s 5,6 • 10 ~'

0.265 0.242 0.195 0.070

3

induced induced induced induced

0 20 40 60

2.7 • I0~' 2.7 " I0~' 2.7 - l0 ~' 2.7 . I(Y'

0.053 0.064 0.062 0.05S

"

S/o/w**

* The entire experiment was performed in the same tube. ** Slope k In AP/t ,a.here P is the phage concentration and t is the lime in rain.

the concentration of bacteriophage in the system. The rate should be independent of phage concentration if the bacterial surface is essentially unsaturated. A culture was induced in the usual manner. At various times increasing n u m b e r s of bacteriophage obtained frorn an ultraviolet lysate were added to each tube. Samples were taken 20 rain after each addition, Results of Expt 2, Table 11, show that the system does not behave as an unsaturated one. Rather, the adsorption rate is markedly lowered as the phage concentration increases. Neither of the preceeding experiments provides any information a b o u t the adsorption rate just before lysis. The following experiment was designed to determine whether or not the adsorption characteristics of the bacteria change signilicantl3 between induction and lysis of the culture. A culture was induced in the usual m a n n e r and the bacteria equally distributed into four tubes. At 20-min intervals equal large n u m b e r s of phage were added to the tubes and samples subsequently taken in the usual manner. Expt 3 in Table I1 illustrates that no measurable change in adsorption rate of phage at high concentrations was found at various times between induction and lysis of the culture. E[l~,ct o f adsorption oli the virus concentration Theoretically, it should be possible to predict tile shape of the true phagerelease curve after incorporating a factor for the changing adsorption rate into the measured phage-release curve. It is of some value to approximate the solution of the equation experinaentally. After routine induction, a tube of organisms and a tube of medium alone were treated in precisely the same way. At 3-min intervals suitable n u m b e r s of virus particles in very small volumes were added to each tube to reproduce an approximate

133

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10 8

_l 106

IO:

;

1;

310

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dO

Time in m i n u t e s

Fig. 5. Experimental construction of the effect of decreasing adsorption rate as virus concentration increases. The virus was added in small increments to reproduce the usual virus production curve. Virus titers of a tube containing medium alone C :----~. ') are compared with titers of a tube containing induced lysogenic bacteria ( x x). All virus concentrations shown are far higher than the virus produced by the bacteria in the culture at the various times shown.

straight-line exponential increase in virus with a 10-fold increased time of 12 min. At 20-min intervals samples were taken shortly after virus additions and processed as usual. Fig. 5 illustrates the results. It can be seen that the control tube closely followed the curve expected from the virus addition schedule. The experimental curve is essentially straight and shows an increased slope due, of course, to rapid adsorption of phage at low concentrations and a gradually diminishing adsorption rate at higher concentrations because the bacterial surface has become saturated. The magnitude of the effect of adsorption in the overall curve is small and there is no direct evidence that a modification of the slope occurs in cultures of induced organisms.

Millipore "sandwich" experiments Although all the evidence found up to this point indicates that virus release from a population of induced lysogenic B. megaterium 899a increases at an exponential rate, the measurements have been made under one basic set of conditions. Therefore, another set of experiments was made under different and probably more rigid conditions. A technique was devised which should reduce the effect of interaction between bacteria and also reduce virus readsorption. A Millipore filter "sandwich" was made with two circles of Millipore filter paper in a Pyrex 250-ml Millipore filter assembly. A No. 27 3/4-inch hypodermic needle was inserted between the two filter papers and a seal of Dow Coming silicone stopcock grease applied to the ground glass as the filter was assembled. After routine induction a 1/100 dilution of the induced bacterial culture was made into filtered

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Fig. 6. "'Millipore sandwich" experimental resulls. 5 - 10:~ultraviolet-induced bacteria were added at lime zero. Samples were collected every 3 rain, titered and plotted as total phage per sample. Fig. 7. Cumulali~.c plot of the virus obtained from the "'Millipore sandwich".

(Millipore, HA) 5!',i peptone which had been w a r m e d and aerated prior to use. One ml o f this bacterial suspension was injected into the " s a n d w i c h " through the hypodermic needle. The needle was then carefully removed while m o m e n t a r i l y releasing the spring clamp. Immediately, a sufficient volume o f warm, aerated, filtered p e p t o n e was a d d e d to the funnel to provide the desired flow rate. Effluent From the funnel was collected in fractions at equal time intervals. Flow rate was kept constant by Frequently or continuously replacing lost volume in the filter funnel. The entire experiment was p e r f o r m e d at 29'~'C. F r a c t i o n s were collected in tubes s u r r o u n d e d by ice water. Fig. fl summarizes the total virus obtained from 5 • I0 ~ bacteria (initial count) in a " s a n d wiclY' o l ' t y p e HA Millipore filter p a p e r with a fluid head o f 60 ml. Fig. 7 shows the cumulative virus titer. It is clear that the previous conclusion o f this report is substantiated. Virus production, and p r e s u m a b l y bacterial lysis, increase exponentiallx with time in a p o p u l a t i o n o f ultraviolet-induced bacteria. (Also see A p p e n d i x . )

Requirements /br reproducibility Sherman and Albus H) showed that newly formed (log phase) bacteria were much more susceptible to mild toxic treatments that caused no mortality a m o n g cells from an older culture. A similar situation seems to exist in o u r experiments. If ultraviolet induction is performed on cells of B. megaterium that are not in log phase, ah-nost no lysis and virus p r o d u c t i o n occurs. Further, if cultures are used which arc shifting between log and lag phase, only partial lysis occurs with bacterial regrowth

135 beginning immediately. Only if the culture is carefully kept in log phase throughout will consistent results be obtained. In some experiments when a lag occurred after induction before the culture regained its original growth rate, the upper end of the cumulative bacterial lysis curve was somewhat displaced to the right. This deviation is almost surely due to inhomogeneity of the culture. One likely explanation is as follows. The measured curve may be thought of in this case as the sum of a series of parallel lines each representing a subpopulation that began to lyse at later and later times. As the earlier lysing subgroups disappear the resultant or overall curve is displaced toward the late lysing subgroups of the population. The overall effect is a slight shift to the right at the upper end of the curve. Such a phenomenon is seen in human population studies it. DISCUSSION Most studies of bacterial death rates have been concerned with the effects ot disinfectants, heat, or radiation l'-za. Many cases have been found which display first-order (or single-hit) curves, especially when spores are used. These results and conclusions have been criticized 16-1s. Some exceptional cases are reviewed by RahnlS. In contrast, lysogenic cells after induction, certain to die 2 h later, continue to multiply at nearly normal rates. They are different from cells in the process of being killed with heat or antiseptics because they have, in a real sense of the word, a defined lifespan of 2 h. After induction they have become more analogous to higher organisms which may show either type of death curve: a Gompertz function if they live out their normal lifespans, or an approximately constant death rate, after a delay, if they are subjected to acute thermal or chemical treatment 15. Sensitive bacterial cells infected with lytic viruses would not be expected to behave in the same way as induced lysogenic strains. After addition of virus to a sensitive bacterial culture, tile "lifespan" is generally quite short relative to the time necessary for adsorption and penetration. Therefore, the time when different cells begin viral synthesis varies considerably, resulting in a wide spread in lysis time. This effectively masks the kinetics of the underlying process. On the other hand, the experinaents reported here were performed in such a way that individual variation between cells was minimized, making detailed kinetic studies possible. The most likely interpretation of our results seems to be the following. The basic Gompertz death curve is combined with a factor for the growth of bacteria and a delay, equivalent to the latent period, representing tile time necessary for the bacteriophage growth cycle actually to kill the cell. What remains, if these two factors are removed, is a process lasting about 90 rain, initiated by a short exposure to ultraviolet light, which results in an exponentially increasing number of cells starting viral synthesis. In this model we suggest that the Gompertz curve is generated by exponentially increasing numbers of cells failing to repress the bacteriophage genome. Except for the fact that the process takes place largely after the ultraviolet treatment, the

136 kinetics bear some relation to those of the death o f ultraviolet-irradiated Neurovmr~e crassa as reported by Atwood and N o r m a n a. These authors assumc that with Ihcir

organisms killing is dependent on multiple hits on t u o or morc independent p r , , cesses, and they show both theoretical and experimental resuhs \~ilh cxponcn/kdl 5 increasing rates (see especially their Formula 16). To explain our results we assume that ultraviolet induction interrupts one or more biosynthetic pathways whose function is the repression or the bacteriophage genome 19. The unstable repressor molecules would then be thermally inactivated until they reached a concentration low enough to allow bacteriophage synthesis Io commence. If each repressor molecule inactivation was considered a hit and Iairh large numbers of repressors were involved, bacterial Ivsis curves resembling those found experimentally would be generated e". If, however, the resulting decline in resistance to lysis were found to be linear. the most likely explanation of our result would be Strehler and Mildvan's theor> ej in which stresses have a MaxwelI-Boltzmann distribution which, combined wilh ~t linear decline in vitality, produces an exponentially increasing death rate. The present experiments do not indicate that the mechanism underlying Ihc G o m p e r t z type of death rate in induced bacteria and in higher organisms is analogous except in the final manifestation. Difl'erent biochemical processes which result in an exponential increase in death rate are equally probable. In the case reported here lhc process is obviously at the subcellular and probably at the molecular level, whilc in higher organisms the hypothetical lethal cvents could be at either the molecular or the cellular level.

MATHEMATICAL APPENDIX Gon3pertz's law I states that the death rate, r, increases exponentially with age. r

(I)

: F ( ) e 6~1

where r, is the initial death rate, ~z is the G o m p e r t z slope, and I the age or the population. In a case where the individuals in the population multiply by binary lission at a constant growth rate k, the cmnulative number of individuals, &, at any timc i~ Bt

B.e ~'/

(2 I

In the present study, the bacteria are growing and lysing at rates k and r,c ''t respectively. The combined integral is calculated as follows. Bacterial growth

dB dr

kB

(3)

137 Bacterial death dB

(4)

-Broe ~t

--

dt

Combined growth and death dB --

kB

--

(5)

Broe at

dt

Integrating B

'B dB B

['t (k - - roe at) dt

(6)

0,

and (kt - ,~--~°e"+ ,-,~) (7)

Bt = Boe

The value of the integration constant ro/a makes the whole exponential term equal zero when t -- 0.

~×

k

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f:

bacleria

Total

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t

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310

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0L 6

Time

I

~

1

90

I 120

I

I 150

in minutes

Fig. 8. Bacterial growth curve ( > ~ - - - - x ) ,

and calculated bacterial death curve ( O . . . . 0 ) of B.

megaterium 899a after induction with ultraviolet light. Also indicated are the constructions (see text) for half the number of bacteria (Bd2), the estimated total number of bacteria (*), the time before bacteriophage release could be measured (r) and the bacteriophage concentration at time T divided by the burst size (Pr/b). The last is assumed to be equivalent to the cumulative bacterial death curve with slope k'. This curve is not precise above the mean lysis point.

138 r

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8

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60 T I~Tle ~

90

~20

r~/l~Ste c

Fig. 9. A comparison of the calculated bacterial curve from Formula 5 (O) \~iih the experimenlal values ( ) found after induction of B. m£~,alerium 899a.

The solution of Formula 7 depends on constructing a ctunulative bacterial death curve from the measured cumulative phage-release curve on the assumption that one average burst size of phage indicates one dead bacterium (Fig. 8). The tirst step is to constrtict a line parallel with and equal to one half of the bacterial growth curve. The intersection of this line with the bacterial lysis curve gives the time when one half of the total bacteria have lysed. This is also assumed to be the time when one half of the total virus has been produced. The final phage concentration divided b> the total bacteria (one half of the bacteria doubled) gives the burst size. The cumulati\.c phage-production curve is then divided by the burst size to find the bacterial death curve (k'). The unavoidable slight progression of virus release during sampling is corrected for when the bacterial death curve is synchronized with the bacterial growth curve at the half-way point mentioned above. The following is the form used for the solution of Formula 7. In Bt

In B , - -

[kt : PT/Brb

antilog(,L(l-

r) i InPT

In Br

In hi]

where (t -- r) •[: 1. The latent period during which virus could not be measured is indicated b\ r. Tile constant ,z for the rate of increase in the death rate is found by subtracting lhc bacterical growth constant (k) from the slope o f the virus production (or cumulative bacterial death) curve (k'). Pr is the phage concentration at time r, Br is the bacterial concentration at time -r, and b is the average burst size. Pr/Bvb is equivalent to r0/,z in Formula 7. It can be seen (Fig. 9) that the formula predicts the shape of the measurable lysis curve or, in other terms, the number of surviving bacteria at an} time with extraordinary precision.

139 ACKNOWLEDGEMENTS

I should like to acknowledge gratefully the help given me by Mr Harold A. Burger in finding the integral of Formula 4. My thanks go to Mrs Barbara Robey for her expert technical assistance.

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