J. theor. Biol. (1979) 76, 387-402
An Autocatalytic J. G.
Model for Bacterial Spore Germination
CLOUSTON,
C. P. GILBERT AND P. MISKELLY
Australian Atomic Energy Commission, Lucas Heights, Sutherland, N.S. W., Australia 2232 Received 27 June 1977, and in revised,form 20 June 1978) The sigmoidal response observed in kinetic studies of germinating bacterial spores, using nephelometric techniques and phase contrast microscope photometry, is analysed by postulating an autocatalytic process. A scheme A to B to C, where B catalyses the degradation of A, is applied to describe the response accurately by three rate constants. The autocatalytic model is compared with previous proposals for quantifying germination studies and for interpreting the effect of different experimental conditions on initiation of germination. Its effectiveness is demonstrated by quantifying published results for a germinating single spore determined by phase contrast microscopy, and nephelometric results for spore suspensions, including the effect of temperature on the rate of initiation of germination. Although developed for quantitative analysis of spore germination, the model is applicable to other autocatalytic phenomena. To assist the experimentalist, a simple accurate method for deriving the three constants specifying the sigmoidal characteristic is described.
1. Introduction
Numerous kinetic studies of germination have been made by observing some change in a chemical or morphological property of dormant bacterial spores. Various investigators have usedloss of heat and radiation resistance,uptake of dilute stains, loss of refractile appearance under phase contrast microscopy, releaseof calcium and dipicolinic acid, or changesin turbidity of suspensionsas criteria for initiation of germination (Rode & Foster, 1960). Despite several approaches to the problem, no satisfactory interpretation in terms of reaction rate processeshas been developed. Germination rates have been reported as a percentage decreasein optical density either during a single or over several constant time intervals (Hachisuka et al., 1955; Woese & Morowitz, 1958; Rode & Foster, 19621, and as the slope of the linear portion of a semilogarithmic plot of the percentage of ungerminated spores with time (O’Connor & Halvorson, 387
0022-5193/79/040387+16$02.0010
(‘ 1979 Academic
Press Inc. (London)
Ltd.
388
J
G.
CLOUSTON
ET
AL
1961). An empirical expression describing the response, when a change in optical density is used to measure the rate, was developed and applied by McCormick (1964, 1965) to relate values of the parameters to changes in environment conditions. Vary & Halvorson (1965) have shown that the frequency distribution of spores in a germinating suspension is adequately represented by a distribution function (Weibull, 1951) which is not very different from that devised by McCormick (1964). A disadvantage of such an approach is that insight into possible biochemical processes is limited because the parameters cannot be readily related to reaction schemes (Koch. 1966, 1969). The germination of a single spore can be studied by phase contrast microscopy and the spore is seen to remain bright for some time before a rapid change to phase dark occurs. These events have been called “microlag” and “microgermination” respectively (Vary & Halvorson. 1965). Microscopic photometry of single spores suggests that the microgermination stage is biphasic (Hashimoto et al., 1969a); both the microlag period and the duration of the first and second phases of microgermination can be influenced by environmental conditions (Hashimoto et al., 19696). A similar effect is observed when the change in optical density of a bacterial spore suspension is used to measure the extent of germination (Powell, 1950.1951). An initial lag or induction period followed by a relatively rapid decrease in optical density produces a response which is invariably sigmoidal. In this paper we show that curves for germinating spores when either a change in optical density and/or refractility is used to measure the rate of germination, can be generated by a model representing an autocatalytic process. The kinetic data of McCormick (1965), Vary & Halvorson (1965) and Hashimoto ct a/. (1969~) are re-examined. 2. Theoretical
Aspects
The prime requirements for the model were that it should correlate with available data, have a minimum number of parameters and represent a process acceptable as a plausible mechanism for germination. Consider the simple scheme kl k, A$BtiCC, (II h I k 2 where the intermediate state B does not necessarily involve a measurable change in the physical or chemical property used to follow the reaction. A lag will occur before C is observed and the magnitude of the lag and germination periods will depend on the relative values of the rate constants.
AUTOCATALYTIC
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This approach was first tried with data from the literature (McCormick, 1965) and least-squares fitting for several different, increasingly complex, first order models, and had varying success. It was assumed that n consecutive irreversible intracellular reactions with rate constants k, (n = 1,2,3, . . .) controlled the rate at which a single spore transformed from the phase bright (state A) to the phase dark (state C) through the intermediate state B. Such models did not satisfactorily accommodate the wide variations in lag and germination periods observed by different workers (Hashimoto et al., 1969a,b ; Vary & Halvorson, 1965) although reasonable correlation with the central portion of the sigmoid could be obtained. A better fit resulted when the process A to B was replaced by a diffusion mechanism, but the results did not warrant detailed evaluation. Attempts to reproduce the initial delay by postulating a threshold barrier or a transportation lag (pure time delay) were found to have only limited application. However, autocatalytic processes such as the following reaction schemes, in which the products B or C catalyse the degradation of A or B gave encouraging correlation with data where k-i 4 k, and ke2 + k, :
(2)
(3)
(4)
3. Data Characterization All data are sigmoidal curves of the form shown in Fig. 1. To characterize them, it is adequate to define the three points shown and record trO, t,, and rsO, the times at which the curve reaches 10, 50 and 80% of its final value. This is more accurate than it might at first appear since it is implied that the curve height is zero at t,, and that the slope of the curve is zero at t, and t,. Thus, we can normalize by defining T,, = t,&,, and T,, = tso/t,, to describe all sigmoidal curves with t 5O= 1, that is, to describe all possible shapes provided only that they pass through 50% at tSO = 1. Figure 2 shows four extreme sigmoids normalized in this way, and a fifth representing the type of response generally observed with spore suspensions. Normalization
390
J. G. CLOUSTON
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means that only two parameters are necessary to describe the shape of the curve, the third merely indicating the time scale. Since each curve can be characterized by two parameters, the possible range of experimental transients is best illustrated by representing each complete curve by a single point on a T-plane. as in Fig. 3 which shows points corresponding to the five curves of Fig. 2. Point 2 approximates to a simple first order reaction, i.e. exponential. Clearly 0 < T,, < 1 and so Fig. 3 covers the complete range of T,,. Similarly, 1 < &, but, as there is no obvious maximum value, the T-plane is inevitably open at the top. To be of value to the experimentalist, a model for quantifying the kinetics of germination must be capable of generating transients representing a very wide area of the T-plane. Although it should be simple, the model must contain at least three independent parameters. As discussed. these can be normalized to NO by a process akin to time scaling to provide a range of results over the T-plane. If there were only one such normalized parameter (McCormick, 1964) all possible sigmoids produced by the model could be described by a single line in the T-plane, but this is clearly inadequate. The dashed line in Fig. 3 characterizes all curves available from McCormick’s
FIG. I. The sigmoid. are used to characterize
The times at which the curve reaches the response for each experiment.
IO?,, 50”, and WY; of its final value
AUTOCATALYTIC
MODEL-SPORE
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391
model and, although it does pass through areas occupied by experimental results, it can represent only a few curves satisfactorily. 4. The Kinetic Model
Analogue computer evaluation showed that reaction scheme (2) is superior to (3) and (4) because it covers a wider range of the T-plane. Suppose the germination of a single spore is being studied using phase contrast microscopy and suppose also A is the concentration of its refractiie component and that B is the concentration of the substance which reacts with A to decrease its concentration, then when k-r + k, and k-z 4 k, the
I:‘-; __-__---____c-___-----------80
IW
,3
I/// I Q 50 ~-_-_-^___ -___---___ E ctij 40 a
30 20 l-zf-
-
/
0
o-5
I.0 Normalized
-
1.5
1
2-O
1
2.5
time
FIG. 2. Four extreme sigmoids (l-4) and one more representative curve (5). The values of T,, and T,, are (1) 0.05, 9.9; (2) 0.15, 2.33; (3) 0.7, 9.9; (4) 0.97, 1,OS; and (5) O-2, 2-O.
I. G. CLOUSTON ET AI. 392 scheme for a single spore can be represented by:
-;;=
k,A+kaA.B-k,B,
rsi
dC >;- = k,B,
(6)
where A=A,andB=C=Oatt=O; A+B+C = A, for all t; and A, B and C are all real and positive for all t. Taking A,, = 1, the results may be directly expressed as a fraction or percentage of the initial refractility or other property under study. It is stressed that the autocatalytic scheme (2) is supposed to model a possible intracellular process causing loss of the refractile appearance when a single spore germinates. When a nephelometric method is used to study a heterogeneous spore population the resultant sigmoid rate characteristic is
FIG. 3. The sigmoid curves of Fig. 2 represented as points I -5 in the T-plane. The dashed line characterizes all curves available from the expression developed by McCormick (1965). The model of section 4 can generate curves characterized by any point in the unshaded area.
AUTOCATALYTIC
393 of all sigmoid characteristics for the spores
MODEL-SPORE
GERMINATION
that resulting from a summation in the population. Normalizing equations (5) and (6) by defining K, = k,/k8, K2 = k,/k, and k, = 1 means that all response shapes can be investigated in terms of K, and K, only. Since B is zero initially, the k, path for scheme (2) produces no effect. The initial rise in B is due solely to the first order reaction with rate constant k,. The curve for C is consequently horizontal at the origin. As B accumulates, autocatalytic action stimulates its production, and rate of increase in C is determined by the magnitude of k2. The model has shown itself capable of generating transients over the whole unshaded area of Fig. 3, which is bounded approximately by points representing the four extreme shapes of Fig.2. The range of parameter values is 10 > K, > 1O-25 and 10 > K, > 0.35. In some regions of Fig. 3, however, solutions to equations (5) and (6) are not easy to obtain. For instance, as K, becomes small (to generate curves with a long initial delay or microlag), the model becomes very sensitive to small errors in B and to changes in K2 of the order of one part in 106. 5. Using the Model
The prime objective is to evaluate k,, k, and k, from experimental values for t,,, t,, and tso. If an analytical solution to equations (5) and (6) existed, reaction rates could be found directly from experimental results; this was the approach adopted by McCormick (1964,1965). No such solution has been found and, while one may exist, it is unlikely that it will be sufficiently compact either to give insight into the biological processes or to simplify the above evaluation. A generalized solution was attempted using a hybrid computer employing a self-optimizing program to control high speed analogue solutions. Although evaluation of the reaction rates was complete in 10-20 seconds, the program was abandoned when it could not be made reliable in the more critical regions of the T-plane. The approach finally adopted involved digital solution of the equations over a wide range of parameters. Accurate results could be obtained provided care was taken to select the integrating method and its error bounds. In this respect, the techniques of Milne (1972) and Gear (1971) were adequate. Contours of constant K,, K, and T, were plotted, where T, is the computed time for 50% reaction. The contours cover a large area of the Tplane and are of the form illustrated in Fig. 4. The preparation of the contour graph (Clouston, Gilbert & Miskelly, 1975) was costly in computer time, but it can be used by experimentalists to derive kl, k2 and k, simply and quickly.
394
I. G.
CLOUSTON
ET
.4L
To use the chart for evaluating reaction constants the times for 10, 50 and 80% reaction are measured. This presumes a knowledge of the value corresponding to lOOY, reaction, but the derived rate constants are not particularly sensitive to errors in estimating completion of the reaction. The normalized values T,, = t,,/t50 and Ts, = tBO/tSO are calculated and the required point is located on the chart (Fig. 4). Values of K 1, K2 and T, are interpolated from the contours. The rate constants are determined from the relationships k, = Tc/rso, k, = K, x k, and k, = K2 x k,. These values describe the experimental response when substituted into equations (5) and (6).
6. Application
of the Model
The accuracy of the model for quantifying sigmoidal responses and its value for the experimentalist was investigated by applying it to a variety of published data. The individual spores in a non-synchronous spore population have a broad asymmetrical distribution of microlag and microgermination times
FIG. 4. A sketch of the chart of ‘J-plane contours obtained from solutions to (5) and (6). The rate constants K,, contours relate experimental values of I,,, tjO and tgO to the reaction KZ and k,.
AUTOCATALYTIC
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395
(Vary & Halvorson, 1965); consequently, nephelometric techniques for observing the rate of germination produce curves, which represent the resultant response of the population. When nephelometry is used, the parameters k,, k2 and k, may be derived to quantify the behaviour of the population for specific conditions. As an example, we can consider a study by Vary & Halvorson (1965) which compared a normalized curve of the change in optical density of 625 nm of a germinating suspension with the total number of phase bright spores which had not commenced germination. If it is assumed that the initial number of spores in the population is N, and that ii, is the initial mean concentration of the refractile component per spore, then Noti0 = N,ii,+N,ti+N,~, where a is the mean concentration of the refractile component per spore some time after germination is initiated. Letting N, represent the number of spores which have not been initiated, and N, those which have been partially initiated, then N,c is the mean concentration of the non-refractile component. Letting A = N,&,, B=NJ and C=N,E then A+B corresponds to the uncommitted and transitional phase bright cases while C represents those spores for which the degradation reaction is complete. The model gives very close agreement for the values of the rate constants reported in Fig. 5. The agreement of the curve for A with the phase bright data is not good; if it were, this would imply that B could be identified precisely with spores in the transitional state, which is unlikely, quite apart from the necessarily subjective nature of the observations. An analysis of the data of McCormick (1965) for the effect of temperature on germination rate is an interesting application. McCormick studied the germination of Bacillus cereus strain T spores when initiated using L-alanine as a trigger by recording the decrease in optical density at 625 nm. It was shown that the time course of germination could be described by an empirical relationship. OD,=OD,[l-(l-a)exp(-kt-‘)I,
(7)
where OD, is the optical density reading at a time t after the reaction is triggered and ODo is the initial optical density. The constant czis the ratio of the final to the initial optical density. Both the k and c parameters were influenced by temperature but only k was sensitive to preliminary heat activation and the concentration of the trigger. Values for kl, k, and k,, which accurately generate McCormick’s curves for the effect of temperature on the time course of germination, are plotted in Fig. 6.
I. G.
396
CLOUSTON
ET
Al.
Since the magnitude of kI influences the lag period, the autocatalytic model suggests that induction is not very sensitive to increasing temperature for temperatures less than 25°C. The reduction in the lag period above 25°C is associated with a rapid increase in k,. Increasing values for k, and X-, which reflect the transformation from phase bright to phase dark pass through optima at about 34°C. The net effect is that. even though the lag period is reduced, the fraction of spores which transfer from phase bright to phase dark decreases as the temperature is increased and, for a temperature the process is inhibited. The temperature sufficiently above optimum, optimum exhibited by the system might be attributed to the effect of increasing temperature on an enzyme reaction(s) controlling the second phase of the germination process (Johnson, Eyring & Polissar. 1954). The effect of temperature on the microlag phase which results in a linear relationship for k, may arise from the displacement of an equilibrium
0
I
IO0
200
306
Time
0
400
(s ) J
o-5
I.0 Normalized
l-5
2.0
time
FIG. 5. Comparison of a curve generated by the autocatalytic model for k, = 7.6 x IO- ’ s ‘. k, = 7.6 x to-* s- ’ and k, = 2.7 x lO-‘s-’ with the accumulative number of B. cereus strain T spores transforming from phase bright to phase dark as reported by Vary and Halvorson in Sussman & Halvorson (1966).
AUTOCATALYTIC
MODEL-SPORE
GERMINATION
391 between the enzyme(s) and an intercellular inhibitor in a direction which reduces the concentration of the enzyme inhibitor complex (Ogston, 1956). The model has potential for quantitatively distinguishing the effect of different variables on the microlag and microgermination phases. Although specifically developed for representing the sigmoidal characteristics displayed by nephelometric studies of concentrated suspensions of germinating bacterial spores, the chart (Fig. 4) may readily be adapted to represent biphasic responses such as those obtained by Hashimoto et al. (1969a) when the germination of single spores was recorded by a microscope photometer. The autocatalytic model can be applied to quantify such behaviour by assuming that the microlag and microgermination phases are a combination of either two parallel or consecutive processes. While curve A could be approximated as a single sigmoidal process, curves B and C suggest that two events are involved in the loss of the refractile appearance of a
i03/lAbsolute
temperature)
FIG. 6. Rate constants k,, k2 and k, derived by applying the autocatalytic model to the data of McCormick (1965) for spores of B. cereus T, heat activated for 4 h and triggered by 0.10 M Lalanine plotted for the temperature range 11.2”C to 40°C.
J. G.
398
CLOUSTON
ET
..lL
bacterial spore during germination. The agreement for curve B (Fig. 7) between the predicted response and the experimental results shows that values for k,, k, and k, can be derived for quantifying a biphasic response even for protracted microlags. The constants specify the net effect of the reaction(s) causing loss of refractility for the particular experimental conditions. In this example loss of refractility was assumed to be the sum of two components, both commencing at zero time, one contributing 60”,, and the other 407; to the total refractility. An alternative and perhaps more appropriate approach is to postulate that the biphasic characteristic arises from consecutive autocatalytic mechanisms. the later reaction being triggered when the first reaches some critical stage. This analysis is simpler to apply to a type C curve for example and more satisfactory when the amplitude of the trigger point is either arbitrarily defined or determined experimentally. Provided the effect of the variation of a parameter on the relative value of the rate constants as illustrated by Fig. 6
0 end A Expenmenioi
,&
0
---0nd ---m-d- - Autocotolyhc t,
’
4
blphanc model
restrxees results
I
I
I
5
6
7
A
8
9
IO
II
12
4
FIG. 7. Microgermination curves of B. wwus T spores recording the effect of heat treatment on the kinetics of single spore germination. The curves A, B and C refer to results pubhshed as Fig. 1 by Hashimoto et al. (1972). Curve B has been analysed by assuming consecutive processes. Constants
for points 0 : k, = 8.3~ lo-”
Constants
for points a : k, = 1.67 x lo-”
s-‘. s-‘.
k,=
1.67~10.‘s
kg=1~67x10-Zs-‘.
‘.
k, = 6.67 x 10 ’ s- ’ k,=5,83xlO-“s
1,
AUTOCATALYTIC
MODEL-SPORE
399
GERMINATION
is of interest this approach would adequately summarize data for comparative evaluation of different spore species. Finally, to exemplify a response involving a protracted microlag followed by a rapid microgermination (Fig. 8) rate constants have been obtained for the data of Hashimoto et al. (1969~). The single phase response for a spore treated with formic acid is compared with the curve generated. by the autocatalytic model. These authors postulate that the initial phase of the germination curve of a single spore is associated with rapid hydration of the M spore. 7. Models and Mechanisms The cycle, vegetative cell-sporulation-spore-germination-vegetative cell is one in which the dormant spore state is axiomatically an enzyme inhibited system. A suggestion that a specific enzyme(s) controlled the germination
, 0
3
6
9
12 I5 Time (5 )
I
1
I
0
O-5
l-0 Normalued
18
21
I
I .5
24
I
2-o
hme
FIG. 8. The experimental response for a single spore recorded by Hashimoto Fig. 6) is compared with the response calculated using the autocatalytic k, = 3.95 x lo-” s-‘, ka = 3.95 SC’ and kz = 1.93 SC’.
ef al. (1969a, model for
400
J. G.
CLOUSTON
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AL
mechanism (McCormick, 1964; O’Connor & Halvorson, 1961) led Woese, Vary & Halvorson (1968) to propose a kinetic model based on the concept of a germination substance which must attain a critical concentration before a spore can germinate. A step-response time distribution for germination of a heterogeneous population was derived by assuming a set of sub-populations with a characteristic number of germination enzymes per spore distributed according to Poisson statistics. A good fit (Fig. 9) to an experimental result was found for fi = 9 except for a small fraction of spores which germinated earlier than predicted. The disadvantages of this simple enzyme model, and of an expanded version in which the germination substance is labile, are that neither can be applied either to quantify the response of spores to
I.0 i
0.3 c
Experimental results Enzyme model
---
Autocotalytlc
l
0.2/
O.l’i’ I, ,/9 s’ 0
s’
100
, 200
1 300
model
I 400
I
/ I
500
FIG. 9. The data used by Woese ef aI. (1968) to exemplify an enzyme model which postulate5 that germinating enzymes are distributed within individual spores according to a Poisson distribution is reproduced. The step function response of this enzyme model is compared with the continuous response generated by the autocatalytic model for k, = 3.96 x 1O-5 s-l ; k, = 6.6 x lo-* s-l and k, = 2.3 x lo-’ SC’.
AUTOCATALYTIC
MODEL-SPORE
GERMINATION
401
different environmental conditions or to compare different species in similar environments. To overcome both these difficulties and the empiricism of McCormick’s approach, and to provide a continuous rather than a step response, Warren (1969) adapted the enzyme postulate to modify the McCormick expression. It was supposed that the germination substance was synthesized by an enzyme distributed so that the fraction of spores Y with n or more molecules of enzyme was given by Y = exp (-n/7). The germination expression became In Y-’ = Bk,‘(exp
(yt)- 1)
(8)
in which the parameters are functions of a hypothetical mechanism. Because the rate of the initiating reaction per molecule of enzyme is proportional to k, and accelerates with time, Warren’s suggestion implies an auto-accelerating process. The quantitative model we propose has the advantage that it embraces concepts implicit in the enzyme model (Woese et al., 1968) and the McCormick empirical relationship as modified by Warren. Experimental germination data (Fig. 2 in Woese et al., 1968) were used to derive values of k,. kz and k, according to the autocatalytic model. Figure 9 compares the data with the curves generated by the autocatalytic model and the step function model. The lag and germination regimes are accurately reproduced by the autocatalytic model to within 1% for more than 80% of the response and within 10% for the complete reaction. From a kinetic viewpoint, the initial portion of the curve is most important and, although possible, no attempt was therefore made to obtain a more precise fit for the tail. Rather than postulate a germinating enzyme(s) and a specific germination substance, an alternative concept would be to suppose that a specific inhibiting mechanism is functioning. The concept of a specific intracellular metabolic inhibitor or structure is a plausible hypothesis for interpreting the distribution of lag and germination times in a heterogeneous population, The distribution simply reflects a variation in the inhibitor concentration per spore. Dormancy might then be viewed as a metastable state controlled by an intracellular inhibitor-enzyme complex such that any stress dissociating the complex favours metabolic activity and vice versa. Referring to the single spore experiments exemplified by Figs 5 and 6, the transition phase bright to phase dark can be explained in terms of the kinetics of reversible enzyme inhibition (Ogston, 1956). If this approach is coupled with the postulates of Johnson, Eyring & Polissar (1954), the effect of temperature on the constants k,, k, and kr (Fig. 6) can be qualitatively explained. The net rate of germination is determined by an activated enzyme15
cJ. G. CLOUSTON EiAl 402 substrate complex whose concentration is intluenced by two reversible equilibria, one being that which controls the relative concentrations of an enzyme in an active as opposed to an inactive configuration, and the other being that which controls the degree of intracellular inhibition.
REFERENCES CLOUSTON, J. G.. GILBERT, C. P. & Mts~bt~.y. P. (1975). Australian Atomic Energy ~‘onlmission unpublished report. GEAR. C. W. (1971). In R6912 (M. J. Hopper. rd.). Harwell. England. 4rotnic Eric-rgq Rexarch Establishment. HACHISUKA, Y., ASANO. N.. KAIU. N.. OUJIMA. M.. KITAOKI. M. Kc K[I\o. I (1955). .I
Buc,teriol. 69, 399. HASHIMOTO. HASHIMOTO, HASHIMOTO. & L. L. Microbiology. JOHNSON. F. London: KOCH. A. L. KOCH, A. L. MILNE. W.
T.. FRIEBIN. T.. FRIEEIN. TV FRIEBIN. Campbell.
W. R. W. R. W. R. eds),
& CON’TI. S. F. ( 1969~). ./. BUC ir~o/. 98, 101 1. & CONTI. S. F. ( 19696). J. Ru~eri& 100, 1385. & CONTI. S. F. (1972). In .Spore,r (H. 0. Halvorson, D.C.. U.S.A., American Vol. V. Washington.
K. Hanson Society for
H.. EYRING. H. & POLISSAK. J. J. ( 1954). Thr ktne/rt Btr.~.~ of M&I&U B;o/oy~. J. Wiley & Sons. (1966). J. rheor. Biol. 12, 276. (1969). J. rheur. Biol. 23, 251. E. In Modelling Program C:‘rm Munuul .Sj stem, 3611 ~onritwow J;).
An Autocatalytic J. G.
Model for Bacterial Spore Germination
CLOUSTON,
C. P. GILBERT AND P. MISKELLY
Australian Atomic Energy Commission, Lucas Heights, Sutherland, N.S. W., Australia 2232 Received 27 June 1977, and in revised,form 20 June 1978) The sigmoidal response observed in kinetic studies of germinating bacterial spores, using nephelometric techniques and phase contrast microscope photometry, is analysed by postulating an autocatalytic process. A scheme A to B to C, where B catalyses the degradation of A, is applied to describe the response accurately by three rate constants. The autocatalytic model is compared with previous proposals for quantifying germination studies and for interpreting the effect of different experimental conditions on initiation of germination. Its effectiveness is demonstrated by quantifying published results for a germinating single spore determined by phase contrast microscopy, and nephelometric results for spore suspensions, including the effect of temperature on the rate of initiation of germination. Although developed for quantitative analysis of spore germination, the model is applicable to other autocatalytic phenomena. To assist the experimentalist, a simple accurate method for deriving the three constants specifying the sigmoidal characteristic is described.
1. Introduction
Numerous kinetic studies of germination have been made by observing some change in a chemical or morphological property of dormant bacterial spores. Various investigators have usedloss of heat and radiation resistance,uptake of dilute stains, loss of refractile appearance under phase contrast microscopy, releaseof calcium and dipicolinic acid, or changesin turbidity of suspensionsas criteria for initiation of germination (Rode & Foster, 1960). Despite several approaches to the problem, no satisfactory interpretation in terms of reaction rate processeshas been developed. Germination rates have been reported as a percentage decreasein optical density either during a single or over several constant time intervals (Hachisuka et al., 1955; Woese & Morowitz, 1958; Rode & Foster, 19621, and as the slope of the linear portion of a semilogarithmic plot of the percentage of ungerminated spores with time (O’Connor & Halvorson, 387
0022-5193/79/040387+16$02.0010
(‘ 1979 Academic
Press Inc. (London)
Ltd.
388
J
G.
CLOUSTON
ET
AL
1961). An empirical expression describing the response, when a change in optical density is used to measure the rate, was developed and applied by McCormick (1964, 1965) to relate values of the parameters to changes in environment conditions. Vary & Halvorson (1965) have shown that the frequency distribution of spores in a germinating suspension is adequately represented by a distribution function (Weibull, 1951) which is not very different from that devised by McCormick (1964). A disadvantage of such an approach is that insight into possible biochemical processes is limited because the parameters cannot be readily related to reaction schemes (Koch. 1966, 1969). The germination of a single spore can be studied by phase contrast microscopy and the spore is seen to remain bright for some time before a rapid change to phase dark occurs. These events have been called “microlag” and “microgermination” respectively (Vary & Halvorson. 1965). Microscopic photometry of single spores suggests that the microgermination stage is biphasic (Hashimoto et al., 1969a); both the microlag period and the duration of the first and second phases of microgermination can be influenced by environmental conditions (Hashimoto et al., 19696). A similar effect is observed when the change in optical density of a bacterial spore suspension is used to measure the extent of germination (Powell, 1950.1951). An initial lag or induction period followed by a relatively rapid decrease in optical density produces a response which is invariably sigmoidal. In this paper we show that curves for germinating spores when either a change in optical density and/or refractility is used to measure the rate of germination, can be generated by a model representing an autocatalytic process. The kinetic data of McCormick (1965), Vary & Halvorson (1965) and Hashimoto ct a/. (1969~) are re-examined. 2. Theoretical
Aspects
The prime requirements for the model were that it should correlate with available data, have a minimum number of parameters and represent a process acceptable as a plausible mechanism for germination. Consider the simple scheme kl k, A$BtiCC, (II h I k 2 where the intermediate state B does not necessarily involve a measurable change in the physical or chemical property used to follow the reaction. A lag will occur before C is observed and the magnitude of the lag and germination periods will depend on the relative values of the rate constants.
AUTOCATALYTIC
MODEL-SPORE
GERMINATION
389
This approach was first tried with data from the literature (McCormick, 1965) and least-squares fitting for several different, increasingly complex, first order models, and had varying success. It was assumed that n consecutive irreversible intracellular reactions with rate constants k, (n = 1,2,3, . . .) controlled the rate at which a single spore transformed from the phase bright (state A) to the phase dark (state C) through the intermediate state B. Such models did not satisfactorily accommodate the wide variations in lag and germination periods observed by different workers (Hashimoto et al., 1969a,b ; Vary & Halvorson, 1965) although reasonable correlation with the central portion of the sigmoid could be obtained. A better fit resulted when the process A to B was replaced by a diffusion mechanism, but the results did not warrant detailed evaluation. Attempts to reproduce the initial delay by postulating a threshold barrier or a transportation lag (pure time delay) were found to have only limited application. However, autocatalytic processes such as the following reaction schemes, in which the products B or C catalyse the degradation of A or B gave encouraging correlation with data where k-i 4 k, and ke2 + k, :
(2)
(3)
(4)
3. Data Characterization All data are sigmoidal curves of the form shown in Fig. 1. To characterize them, it is adequate to define the three points shown and record trO, t,, and rsO, the times at which the curve reaches 10, 50 and 80% of its final value. This is more accurate than it might at first appear since it is implied that the curve height is zero at t,, and that the slope of the curve is zero at t, and t,. Thus, we can normalize by defining T,, = t,&,, and T,, = tso/t,, to describe all sigmoidal curves with t 5O= 1, that is, to describe all possible shapes provided only that they pass through 50% at tSO = 1. Figure 2 shows four extreme sigmoids normalized in this way, and a fifth representing the type of response generally observed with spore suspensions. Normalization
390
J. G. CLOUSTON
ET
AL
means that only two parameters are necessary to describe the shape of the curve, the third merely indicating the time scale. Since each curve can be characterized by two parameters, the possible range of experimental transients is best illustrated by representing each complete curve by a single point on a T-plane. as in Fig. 3 which shows points corresponding to the five curves of Fig. 2. Point 2 approximates to a simple first order reaction, i.e. exponential. Clearly 0 < T,, < 1 and so Fig. 3 covers the complete range of T,,. Similarly, 1 < &, but, as there is no obvious maximum value, the T-plane is inevitably open at the top. To be of value to the experimentalist, a model for quantifying the kinetics of germination must be capable of generating transients representing a very wide area of the T-plane. Although it should be simple, the model must contain at least three independent parameters. As discussed. these can be normalized to NO by a process akin to time scaling to provide a range of results over the T-plane. If there were only one such normalized parameter (McCormick, 1964) all possible sigmoids produced by the model could be described by a single line in the T-plane, but this is clearly inadequate. The dashed line in Fig. 3 characterizes all curves available from McCormick’s
FIG. I. The sigmoid. are used to characterize
The times at which the curve reaches the response for each experiment.
IO?,, 50”, and WY; of its final value
AUTOCATALYTIC
MODEL-SPORE
GERMINATION
391
model and, although it does pass through areas occupied by experimental results, it can represent only a few curves satisfactorily. 4. The Kinetic Model
Analogue computer evaluation showed that reaction scheme (2) is superior to (3) and (4) because it covers a wider range of the T-plane. Suppose the germination of a single spore is being studied using phase contrast microscopy and suppose also A is the concentration of its refractiie component and that B is the concentration of the substance which reacts with A to decrease its concentration, then when k-r + k, and k-z 4 k, the
I:‘-; __-__---____c-___-----------80
IW
,3
I/// I Q 50 ~-_-_-^___ -___---___ E ctij 40 a
30 20 l-zf-
-
/
0
o-5
I.0 Normalized
-
1.5
1
2-O
1
2.5
time
FIG. 2. Four extreme sigmoids (l-4) and one more representative curve (5). The values of T,, and T,, are (1) 0.05, 9.9; (2) 0.15, 2.33; (3) 0.7, 9.9; (4) 0.97, 1,OS; and (5) O-2, 2-O.
I. G. CLOUSTON ET AI. 392 scheme for a single spore can be represented by:
-;;=
k,A+kaA.B-k,B,
rsi
dC >;- = k,B,
(6)
where A=A,andB=C=Oatt=O; A+B+C = A, for all t; and A, B and C are all real and positive for all t. Taking A,, = 1, the results may be directly expressed as a fraction or percentage of the initial refractility or other property under study. It is stressed that the autocatalytic scheme (2) is supposed to model a possible intracellular process causing loss of the refractile appearance when a single spore germinates. When a nephelometric method is used to study a heterogeneous spore population the resultant sigmoid rate characteristic is
FIG. 3. The sigmoid curves of Fig. 2 represented as points I -5 in the T-plane. The dashed line characterizes all curves available from the expression developed by McCormick (1965). The model of section 4 can generate curves characterized by any point in the unshaded area.
AUTOCATALYTIC
393 of all sigmoid characteristics for the spores
MODEL-SPORE
GERMINATION
that resulting from a summation in the population. Normalizing equations (5) and (6) by defining K, = k,/k8, K2 = k,/k, and k, = 1 means that all response shapes can be investigated in terms of K, and K, only. Since B is zero initially, the k, path for scheme (2) produces no effect. The initial rise in B is due solely to the first order reaction with rate constant k,. The curve for C is consequently horizontal at the origin. As B accumulates, autocatalytic action stimulates its production, and rate of increase in C is determined by the magnitude of k2. The model has shown itself capable of generating transients over the whole unshaded area of Fig. 3, which is bounded approximately by points representing the four extreme shapes of Fig.2. The range of parameter values is 10 > K, > 1O-25 and 10 > K, > 0.35. In some regions of Fig. 3, however, solutions to equations (5) and (6) are not easy to obtain. For instance, as K, becomes small (to generate curves with a long initial delay or microlag), the model becomes very sensitive to small errors in B and to changes in K2 of the order of one part in 106. 5. Using the Model
The prime objective is to evaluate k,, k, and k, from experimental values for t,,, t,, and tso. If an analytical solution to equations (5) and (6) existed, reaction rates could be found directly from experimental results; this was the approach adopted by McCormick (1964,1965). No such solution has been found and, while one may exist, it is unlikely that it will be sufficiently compact either to give insight into the biological processes or to simplify the above evaluation. A generalized solution was attempted using a hybrid computer employing a self-optimizing program to control high speed analogue solutions. Although evaluation of the reaction rates was complete in 10-20 seconds, the program was abandoned when it could not be made reliable in the more critical regions of the T-plane. The approach finally adopted involved digital solution of the equations over a wide range of parameters. Accurate results could be obtained provided care was taken to select the integrating method and its error bounds. In this respect, the techniques of Milne (1972) and Gear (1971) were adequate. Contours of constant K,, K, and T, were plotted, where T, is the computed time for 50% reaction. The contours cover a large area of the Tplane and are of the form illustrated in Fig. 4. The preparation of the contour graph (Clouston, Gilbert & Miskelly, 1975) was costly in computer time, but it can be used by experimentalists to derive kl, k2 and k, simply and quickly.
394
I. G.
CLOUSTON
ET
.4L
To use the chart for evaluating reaction constants the times for 10, 50 and 80% reaction are measured. This presumes a knowledge of the value corresponding to lOOY, reaction, but the derived rate constants are not particularly sensitive to errors in estimating completion of the reaction. The normalized values T,, = t,,/t50 and Ts, = tBO/tSO are calculated and the required point is located on the chart (Fig. 4). Values of K 1, K2 and T, are interpolated from the contours. The rate constants are determined from the relationships k, = Tc/rso, k, = K, x k, and k, = K2 x k,. These values describe the experimental response when substituted into equations (5) and (6).
6. Application
of the Model
The accuracy of the model for quantifying sigmoidal responses and its value for the experimentalist was investigated by applying it to a variety of published data. The individual spores in a non-synchronous spore population have a broad asymmetrical distribution of microlag and microgermination times
FIG. 4. A sketch of the chart of ‘J-plane contours obtained from solutions to (5) and (6). The rate constants K,, contours relate experimental values of I,,, tjO and tgO to the reaction KZ and k,.
AUTOCATALYTIC
MODEL-SPORE
GERMINATION
395
(Vary & Halvorson, 1965); consequently, nephelometric techniques for observing the rate of germination produce curves, which represent the resultant response of the population. When nephelometry is used, the parameters k,, k2 and k, may be derived to quantify the behaviour of the population for specific conditions. As an example, we can consider a study by Vary & Halvorson (1965) which compared a normalized curve of the change in optical density of 625 nm of a germinating suspension with the total number of phase bright spores which had not commenced germination. If it is assumed that the initial number of spores in the population is N, and that ii, is the initial mean concentration of the refractile component per spore, then Noti0 = N,ii,+N,ti+N,~, where a is the mean concentration of the refractile component per spore some time after germination is initiated. Letting N, represent the number of spores which have not been initiated, and N, those which have been partially initiated, then N,c is the mean concentration of the non-refractile component. Letting A = N,&,, B=NJ and C=N,E then A+B corresponds to the uncommitted and transitional phase bright cases while C represents those spores for which the degradation reaction is complete. The model gives very close agreement for the values of the rate constants reported in Fig. 5. The agreement of the curve for A with the phase bright data is not good; if it were, this would imply that B could be identified precisely with spores in the transitional state, which is unlikely, quite apart from the necessarily subjective nature of the observations. An analysis of the data of McCormick (1965) for the effect of temperature on germination rate is an interesting application. McCormick studied the germination of Bacillus cereus strain T spores when initiated using L-alanine as a trigger by recording the decrease in optical density at 625 nm. It was shown that the time course of germination could be described by an empirical relationship. OD,=OD,[l-(l-a)exp(-kt-‘)I,
(7)
where OD, is the optical density reading at a time t after the reaction is triggered and ODo is the initial optical density. The constant czis the ratio of the final to the initial optical density. Both the k and c parameters were influenced by temperature but only k was sensitive to preliminary heat activation and the concentration of the trigger. Values for kl, k, and k,, which accurately generate McCormick’s curves for the effect of temperature on the time course of germination, are plotted in Fig. 6.
I. G.
396
CLOUSTON
ET
Al.
Since the magnitude of kI influences the lag period, the autocatalytic model suggests that induction is not very sensitive to increasing temperature for temperatures less than 25°C. The reduction in the lag period above 25°C is associated with a rapid increase in k,. Increasing values for k, and X-, which reflect the transformation from phase bright to phase dark pass through optima at about 34°C. The net effect is that. even though the lag period is reduced, the fraction of spores which transfer from phase bright to phase dark decreases as the temperature is increased and, for a temperature the process is inhibited. The temperature sufficiently above optimum, optimum exhibited by the system might be attributed to the effect of increasing temperature on an enzyme reaction(s) controlling the second phase of the germination process (Johnson, Eyring & Polissar. 1954). The effect of temperature on the microlag phase which results in a linear relationship for k, may arise from the displacement of an equilibrium
0
I
IO0
200
306
Time
0
400
(s ) J
o-5
I.0 Normalized
l-5
2.0
time
FIG. 5. Comparison of a curve generated by the autocatalytic model for k, = 7.6 x IO- ’ s ‘. k, = 7.6 x to-* s- ’ and k, = 2.7 x lO-‘s-’ with the accumulative number of B. cereus strain T spores transforming from phase bright to phase dark as reported by Vary and Halvorson in Sussman & Halvorson (1966).
AUTOCATALYTIC
MODEL-SPORE
GERMINATION
391 between the enzyme(s) and an intercellular inhibitor in a direction which reduces the concentration of the enzyme inhibitor complex (Ogston, 1956). The model has potential for quantitatively distinguishing the effect of different variables on the microlag and microgermination phases. Although specifically developed for representing the sigmoidal characteristics displayed by nephelometric studies of concentrated suspensions of germinating bacterial spores, the chart (Fig. 4) may readily be adapted to represent biphasic responses such as those obtained by Hashimoto et al. (1969a) when the germination of single spores was recorded by a microscope photometer. The autocatalytic model can be applied to quantify such behaviour by assuming that the microlag and microgermination phases are a combination of either two parallel or consecutive processes. While curve A could be approximated as a single sigmoidal process, curves B and C suggest that two events are involved in the loss of the refractile appearance of a
i03/lAbsolute
temperature)
FIG. 6. Rate constants k,, k2 and k, derived by applying the autocatalytic model to the data of McCormick (1965) for spores of B. cereus T, heat activated for 4 h and triggered by 0.10 M Lalanine plotted for the temperature range 11.2”C to 40°C.
J. G.
398
CLOUSTON
ET
..lL
bacterial spore during germination. The agreement for curve B (Fig. 7) between the predicted response and the experimental results shows that values for k,, k, and k, can be derived for quantifying a biphasic response even for protracted microlags. The constants specify the net effect of the reaction(s) causing loss of refractility for the particular experimental conditions. In this example loss of refractility was assumed to be the sum of two components, both commencing at zero time, one contributing 60”,, and the other 407; to the total refractility. An alternative and perhaps more appropriate approach is to postulate that the biphasic characteristic arises from consecutive autocatalytic mechanisms. the later reaction being triggered when the first reaches some critical stage. This analysis is simpler to apply to a type C curve for example and more satisfactory when the amplitude of the trigger point is either arbitrarily defined or determined experimentally. Provided the effect of the variation of a parameter on the relative value of the rate constants as illustrated by Fig. 6
0 end A Expenmenioi
,&
0
---0nd ---m-d- - Autocotolyhc t,
’
4
blphanc model
restrxees results
I
I
I
5
6
7
A
8
9
IO
II
12
4
FIG. 7. Microgermination curves of B. wwus T spores recording the effect of heat treatment on the kinetics of single spore germination. The curves A, B and C refer to results pubhshed as Fig. 1 by Hashimoto et al. (1972). Curve B has been analysed by assuming consecutive processes. Constants
for points 0 : k, = 8.3~ lo-”
Constants
for points a : k, = 1.67 x lo-”
s-‘. s-‘.
k,=
1.67~10.‘s
kg=1~67x10-Zs-‘.
‘.
k, = 6.67 x 10 ’ s- ’ k,=5,83xlO-“s
1,
AUTOCATALYTIC
MODEL-SPORE
399
GERMINATION
is of interest this approach would adequately summarize data for comparative evaluation of different spore species. Finally, to exemplify a response involving a protracted microlag followed by a rapid microgermination (Fig. 8) rate constants have been obtained for the data of Hashimoto et al. (1969~). The single phase response for a spore treated with formic acid is compared with the curve generated. by the autocatalytic model. These authors postulate that the initial phase of the germination curve of a single spore is associated with rapid hydration of the M spore. 7. Models and Mechanisms The cycle, vegetative cell-sporulation-spore-germination-vegetative cell is one in which the dormant spore state is axiomatically an enzyme inhibited system. A suggestion that a specific enzyme(s) controlled the germination
, 0
3
6
9
12 I5 Time (5 )
I
1
I
0
O-5
l-0 Normalued
18
21
I
I .5
24
I
2-o
hme
FIG. 8. The experimental response for a single spore recorded by Hashimoto Fig. 6) is compared with the response calculated using the autocatalytic k, = 3.95 x lo-” s-‘, ka = 3.95 SC’ and kz = 1.93 SC’.
ef al. (1969a, model for
400
J. G.
CLOUSTON
ET
AL
mechanism (McCormick, 1964; O’Connor & Halvorson, 1961) led Woese, Vary & Halvorson (1968) to propose a kinetic model based on the concept of a germination substance which must attain a critical concentration before a spore can germinate. A step-response time distribution for germination of a heterogeneous population was derived by assuming a set of sub-populations with a characteristic number of germination enzymes per spore distributed according to Poisson statistics. A good fit (Fig. 9) to an experimental result was found for fi = 9 except for a small fraction of spores which germinated earlier than predicted. The disadvantages of this simple enzyme model, and of an expanded version in which the germination substance is labile, are that neither can be applied either to quantify the response of spores to
I.0 i
0.3 c
Experimental results Enzyme model
---
Autocotalytlc
l
0.2/
O.l’i’ I, ,/9 s’ 0
s’
100
, 200
1 300
model
I 400
I
/ I
500
FIG. 9. The data used by Woese ef aI. (1968) to exemplify an enzyme model which postulate5 that germinating enzymes are distributed within individual spores according to a Poisson distribution is reproduced. The step function response of this enzyme model is compared with the continuous response generated by the autocatalytic model for k, = 3.96 x 1O-5 s-l ; k, = 6.6 x lo-* s-l and k, = 2.3 x lo-’ SC’.
AUTOCATALYTIC
MODEL-SPORE
GERMINATION
401
different environmental conditions or to compare different species in similar environments. To overcome both these difficulties and the empiricism of McCormick’s approach, and to provide a continuous rather than a step response, Warren (1969) adapted the enzyme postulate to modify the McCormick expression. It was supposed that the germination substance was synthesized by an enzyme distributed so that the fraction of spores Y with n or more molecules of enzyme was given by Y = exp (-n/7). The germination expression became In Y-’ = Bk,‘(exp
(yt)- 1)
(8)
in which the parameters are functions of a hypothetical mechanism. Because the rate of the initiating reaction per molecule of enzyme is proportional to k, and accelerates with time, Warren’s suggestion implies an auto-accelerating process. The quantitative model we propose has the advantage that it embraces concepts implicit in the enzyme model (Woese et al., 1968) and the McCormick empirical relationship as modified by Warren. Experimental germination data (Fig. 2 in Woese et al., 1968) were used to derive values of k,. kz and k, according to the autocatalytic model. Figure 9 compares the data with the curves generated by the autocatalytic model and the step function model. The lag and germination regimes are accurately reproduced by the autocatalytic model to within 1% for more than 80% of the response and within 10% for the complete reaction. From a kinetic viewpoint, the initial portion of the curve is most important and, although possible, no attempt was therefore made to obtain a more precise fit for the tail. Rather than postulate a germinating enzyme(s) and a specific germination substance, an alternative concept would be to suppose that a specific inhibiting mechanism is functioning. The concept of a specific intracellular metabolic inhibitor or structure is a plausible hypothesis for interpreting the distribution of lag and germination times in a heterogeneous population, The distribution simply reflects a variation in the inhibitor concentration per spore. Dormancy might then be viewed as a metastable state controlled by an intracellular inhibitor-enzyme complex such that any stress dissociating the complex favours metabolic activity and vice versa. Referring to the single spore experiments exemplified by Figs 5 and 6, the transition phase bright to phase dark can be explained in terms of the kinetics of reversible enzyme inhibition (Ogston, 1956). If this approach is coupled with the postulates of Johnson, Eyring & Polissar (1954), the effect of temperature on the constants k,, k, and kr (Fig. 6) can be qualitatively explained. The net rate of germination is determined by an activated enzyme15
cJ. G. CLOUSTON EiAl 402 substrate complex whose concentration is intluenced by two reversible equilibria, one being that which controls the relative concentrations of an enzyme in an active as opposed to an inactive configuration, and the other being that which controls the degree of intracellular inhibition.
REFERENCES CLOUSTON, J. G.. GILBERT, C. P. & Mts~bt~.y. P. (1975). Australian Atomic Energy ~‘onlmission unpublished report. GEAR. C. W. (1971). In R6912 (M. J. Hopper. rd.). Harwell. England. 4rotnic Eric-rgq Rexarch Establishment. HACHISUKA, Y., ASANO. N.. KAIU. N.. OUJIMA. M.. KITAOKI. M. Kc K[I\o. I (1955). .I
Buc,teriol. 69, 399. HASHIMOTO. HASHIMOTO, HASHIMOTO. & L. L. Microbiology. JOHNSON. F. London: KOCH. A. L. KOCH, A. L. MILNE. W.
T.. FRIEBIN. T.. FRIEEIN. TV FRIEBIN. Campbell.
W. R. W. R. W. R. eds),
& CON’TI. S. F. ( 1969~). ./. BUC ir~o/. 98, 101 1. & CONTI. S. F. ( 19696). J. Ru~eri& 100, 1385. & CONTI. S. F. (1972). In .Spore,r (H. 0. Halvorson, D.C.. U.S.A., American Vol. V. Washington.
K. Hanson Society for
H.. EYRING. H. & POLISSAK. J. J. ( 1954). Thr ktne/rt Btr.~.~ of M&I&U B;o/oy~. J. Wiley & Sons. (1966). J. rheor. Biol. 12, 276. (1969). J. rheur. Biol. 23, 251. E. In Modelling Program C:‘rm Munuul .Sj stem, 3611 ~onritwow J;).
