Kinetic study of growth throughout the lag phase and the exponential phase of Escherichia coli

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1 FEMS Microbiology Ecology 45 (1987) Published by Elsevier 291 FECO SUMMARY Kinetic study of growth throughout the lag phase and the exponential phase of Escherichia coli Masami Mochizuki and Tsutomu Hattori Institute for Agricultural Research, Tohoku Umuersity, Senda~ 98, Japan Received 2 January 1987 Revision received 27 May 1987 Accepted 13 June 1987 Key words: Growth equation; First order reaction kinetics; Microcultural study; Lag time; Rate of growth initiation of parent cells An equation describing bacterial growth throughout lag and exponential phases was derived based on the assumption that parent cells inoculated into a fresh medium initiate cell growth according to the first order reaction kinetics (the FOR model) and that all cells once cell growth is initiated continue cell growth at a constant rate. The length of a lag time was given by three parameters: the retardation time, t,, the rate of growth initiation of parent cells, X, and the specific growth rate, ~1. Escherichia coli inoculated into a fresh medium was shown to grow in accordance with theoretical prediction. When non-growing cells taken from different-aged cultures were used as inocula, changes in the length of the lag time depended solely on the h value. 2. INTRODUCTION During cultivation of bacterial cells in a liquid medium, a time elapsing between inoculation and Correspondence to: M. Mochizuki, Institute for Agricultural Research, Tohoku University, Sendai 98, Japan. the establishment of exponential growth is known as the lag phase. Although it has not been generally accepted, Buchanan [l] proposed a unique model of growth kinetics during the lag phase. He assumed that the lag phase represents the time required for all the viable bacteria planted to germinate, and the germination of the individuals follows a mathematical distribution. According to his idea, as soon as an organism begins dividing, the rate of increase becomes at once constant and thus the rate of increase of a culture at any instant is proportional to the dividing cell number ((Y) at that time. Recently, Hattori [2] proposed a kinetic model which described bacterial colony formation according to the first order reaction kinetics (the FOR model); the colony number n at time t is given by n= i n,[l-exp{-x(t-t,)}] (tht,) (t < t,> in which nm is the expected number of colonies when time goes to infinity, A is the rate of colony appearance per unit time, and t, is the retardation time. The h value was shown to decrease with (1) /87/$ Federation of European Microbiological Societies

2 292 inoculum culture age. The FOR model was also shown to reflect the process of growth initiation of parent cells in microculture [3]. In this paper, we derive an equation describing bacterial growth throughout the lag and exponential phases by using equation (1) as the probability function of Buchanan s model. From the equation, the length of a lag time is derived as a function of three parameters: t,, A, and the specific growth rate ~1. Secondly, we confirm the validity of the equation using E. coli as a test organism. 3. MATERIALS AND METHODS 3.1. Bacterial strain and growth medium The bacterial strain used in this study was E. coli IAM116, supplied by the Institute of Applied Microbiology, Tokyo University, Japan. The growth medium was a 1-fold diluted nutrient broth (DNB), which was filtered through.2 pm pore size cellulose nitrate filter to eliminate suspended particles [3]. The solid medium for the microculture and the plate count experiment contained 1.% Noble agar. 1. ml of an E. cofi culture grown for 24 h at 27 C in the DNB medium was inoculated into 5 ml of DNB medium in a looo-ml Erlenmeyer flask. The culture was stirred and maintained at 27 C. For microculture chamber and liquid medium growth experiments, samples were taken after.3, 1, 4, 7 and 1 days incubation. In each culture, populations were in the exponential and stationary phase after.3 and 1 and in the decline phases after 4, 7, and 1 days incubation, respectively. Cells taken from the.3-day culture were referred to as growing and those from the 1, 4, 7 and IO-day culture as non-growing cells Microculture Samples were grown on agar block microcultures at 27 C as described previously [3]. Following incubation, the number of parent cells and microcolonies was counted by phase contrast microscopy using an Olympus Biological Microscope (BHS) with a ~1 oil immersion objective. The term i-cell microcolony was used to describe a small cell aggregate consisting of i or more cells. The term t,(i) was the retardation time in the FOR model for the formation of i-cell microcolonies Growth of E. coli cells in liquid medium 1. ml aliquots of a.3-day culture diluted lo-fold with prewarmed DNB medium and of the 1, 4, 7 and lo-day culture diluted 1-fold with prewarmed DNB medium were transferred to test tubes containing 9. ml of prewarmed DNB medium and the cultures were incubated without shaking at 27 C. Following incubation,.2-ml samples were withdrawn from each test tube and the number of viable cells determined by the pour-plate method using 5 replicate plates incubated for 2 days at 27 C. 4. RESULTS AND DISCUSSION 4. I. The equation describing microcolony formation of bacterial cells Cells taken from a population in the exponential growth phase are referred to as growing and those in the stationary or decline phase as nongrowing cells. When growing cells are inoculated into a fresh medium, growth will ideally proceed in an exponential manner without lag. When nongrowing cells are inoculated, however, there is invariably an appreciable lag before exponential growth starts. We will first consider the kinetics of microcolony formation by assuming that parent cells spread onto an agar medium will initiate cell growth according to the FOR model and that all cells, once cell growth is initiated, form 2-cell microcolonies after the time lapse until division of parent cells. Here, let n be the number of clones which have formed 2-cell microcolonies at time t. From the FOR model, we have n= n,[l-exp{-x(t-t,(2))}] (tht,(2)) 1 (t < t,(2)) in which n, is the expected number of clones formed 2-cell microcolonies when time goes to infinity, X is the rate of growth initiation of (2)

3 293 parent cells, and r,(2) is the retardation time, that is the time interval between inoculation and the division of the first parent cell. In a previous report [3], X was evaluated by the slope of the linear relationship between ln(n, - n) on t: ln(n, -n) -X(t-t,(2))+ln n, (tgt,(2)) = i Inn, (t < C(2)) 4.2. Derivation of an equation describing bacterial growth Let N and cx be the total number of cells and the dividing cell number, respectively, at time t. Suppose the number of parent cells at inoculation be n,, which is independent of the incubation time t and the total cell number N at that time. When growing cells are inoculated into a fresh liquid medium, the rate of the increase in cell numbers (d N/dt ) at any time (t) is proportional to the dividing cell number (CY) already present, which is equal to the total cell number N [4]. Then we have the differential equation dn/dt = /JN (4) where p is the growth rate constant. A particular solution of integration of equation (4) is N=n, exppt (5) Thus, the number of cells increases exponentially without any lag. When non-growing cells are inoculated into a fresh liquid medium, the rate of the increase in cell numbers (dn/dr) at any time (t) is considered to be proportional to the dividing cell number (a), as postulated by Buchanan [l]. Since parent cells inoculated into a fresh liquid medium should initiate cell growth in accordance with equation (2) and, once having initiated, should continue dividing at a constant rate, the number of non-growing cells at time t is n, - n and thus the dividing cell number CY at that time is taken as {N-(ni- n )}. Then we have a differential equation dn/dt = p{n-(ni- n)> (t h t,(2)) (t< G(2)) (3) (6) Substituting equation (2) into equation (6) gives the equation dn/dt = (t 2 t,(2)) (t< G(2)) A particular solution of integration of equation (7) yields 1 [cl exp{ -A@ - t,(2))) +texp{~l(t-tt,(2))}lnl/(~+x) N= (t 2 t,(2)) (8) nl (t < t,(2)) Therefore, the number of cells increases in accordance with equation (8) throughout the lag and exponential phase. Next, we derived a lag time from equation (8). When non-growing cells are inoculated into a fresh liquid medium, the number of cells becomes N, after time t, (t, > t,(2)). From equation (8), we have N, = [P ew{ -Act, - t,(2))) +hexp{cl(ta-fr(2))}lnl/(cl+a) (9) Suppose t, is great enough for p exp{ -h(t, - t,(2))} to be negligibly small in equation (9) and we have t,=t,(2)+(1/c1)ln[{(~+-t))/x}{n,/n,}] (1) When growing cells are inoculated into a fresh liquid medium, the number of cells becomes N, after time t:; from equation (5) we have t1= (I/P) ln( K/n, > (II) As proposed by Lodge and Hinshelwood [S], the lag time of a culture is defined as the difference between the observed time when the culture reaches a certain density which is chosen within the exponential phase and the time when another culture with the same size of inoculum reaches the same density without any lag. Using the equation (7)

4 294 of bacterial growth, the lag time L is given by L = t, - t;: (12) Incorporating equation (1) and (11) in equation (12) yields 1 A L=r,(2)+(1/~1)ln{(~~+h)/X} (13) Therefore, we can estimate the length of the lag time (L) from values of its 3 parameters Growth kinetics of E. coli during the first few generations Differently aged cultures of E. coli were grown on agar medium and microcolony formation was followed. Microcolony formation showed a good fit with the FOR model using a l-day-old inoculum (Fig. l), although some discrepancies were observed in the early stages. Such discrepancies are probably due to variation in the generation times [6]. Similar results were also obtained for other non-growing populations; for the exponentially growing population, the theoretical curve served as a rough approximation since the individual delay to the first cell division of the growing population was due to their age distribution [3,7]. The X value, which was estimated from the slope of the linear relationship in Fig. 2, was maximal c Time Fig. 1. Fit of microcolony formation with the FOR model. E. co/i aged 1 day was grown on agar medium at 27 C. The solid curves were computed using data depicted in closed symbols.,, 2-cell; a, A, 4-cell;,, S-cell microcolonies. (h).. c 1 I c Time( h ) Fig. 2. Relationship between the logarithm of rz, - n and incubation time t on agar medium. (A) Parent cells were aged.3 days (); (B) parent cells were aged 1 day (). 4 days (A), 7 days (A) and 1 days (Cl). The solid lines are the expected curves estimated by the non-linear least squares method. The slopes of the curves in the l-day, 4-day, 7-day and IO-day culture correspond to the rate of growth initiation of parent cells X = 2.2 h-, 1.16 h-,.84 hk and.33 h-, respectively. when parent cells were growing and decreased with the increase of culture age. As shown in Fig. 3, the generation time calculated for E. coli was almost the same (61 min) whatever the age of the inoculum was. Estimated values for each parameter are described in Table 1. The time lapse between inoculation and division of parent cells is given by the parameter t,(2), the length of which for non-growing cells of E. cofi was 164 min and was 2.7 times larger than the generation time, although the t,(2) value for nongrowing cells of Agromonas sp. was the same as the generation time [3]. It is pointed out that a chloramphenicol-sensitive process prior to the initiation of DNA replication of E. coli B/r was a

5 295 h AZ V.I- L 5 I- 3, / 1 - / /,D / I I I Microcolony size i Fig. 3. Relationship between t,(i) and the logarithm of microcolony size i. Parent cells were aged.3 days () 1 day (o), 4 days (A), 7 days (A) and 1 days (D), respectively. Table 1 Parameter values in the FOR model for microcolony formation of parent cells Culture age a Microcolony size b X 1 c (days) (i) (h-t) ;h) / g > > a E. co/i was arown in the DNB medium at 27O C and a sample was taken after.3, 1, 4, 7 and 1 days incubation. Cells of each sample were spread onto the agar medium and microcolony formation was followed. The value of the parameters was estimated by the non-linear least squares method. b A small cell aggregate consisting of the indicated number of cells. The number of parent cells observed at inoculation in the.3-day, l-day, 4-day, I-day and lo-day culture were, respectively, 19, 82, 92, 93 and 13. physiological event necessary for the initiation of chromosome replication [8], which was a landmark event controlling the initiation of cell growth, and the time lapse until the division of parent cells was 8 min at 37 o C [8]. In the present study, cells of E. coli were incubated at 27 C and thus the time lapse can be assumed to double owing to temperature decrease; we may have a time lapse of 16 min between the growth initiation and the division of parent cells. Therefore, the process of growth initiation of bacterial cells inoculated onto a fresh medium may have started at the beginning of cultivation, as first postulated from plate count data [2] and later reestablished with Agromonas sp. using a microculture chamber and phase contrast optics [3]. These results indicate with non-growing cells: (1) that parent cells initiated cell growth according 5 Time (h) Fig. 4. Differently aged cultures of E. coli were inoculated into the DNB medium and the cultures were incubated at 27 o C. The relative viable number of cells aged.3 days () and 1 days (III, ) are the averages of 5 plates. The solid lines are the theoretical growth curves, h, =l, h =.33 h-, r,(2) = 2.74 h, and c =.77 h-. The length of the observed lag time and the observed growth rate constant (n) were calculated from the intercept and the slope, respectively, of the linear relationship (broken line) estimated by the least squares method using data of closed symbols. 1

6 296 to the FOR model, (2) that the rate of growth initiation fell with culture age of the parent cells, and (3) that all cells, once cell growth was initiated, continued cell growth at a constant rate Growth curve of E. coli in the fresh liquid medium Differently aged cultures of E. coli were inoculated into DNB medium and, following incubation, numbers of cells were determined by the plate count method. Two sets of experimental data are shown in Fig. 4. When a sample taken from the.3-day culture was inoculated into DNB medium, parent cells that had grown exponentially in the DNB medium continued exponential growth. A generation time of 55 min (cl =.77 h-t) was estimated by linear regression analysis as = ; L observed lag time (hour) Fig. 5. Effect of values of 3 parameters, X (). r,(2) (A), and n (m) on the length of the observed lag time. The values of X and t, (2) are described in Table 1. See legends of Fig. 4 for the estimation of the length of the observed lag time and the observed growth rate constant. The solid lines are the theoretical curves. f,(2) = 2.74 h and n =.77 h-. shown in Fig. 4. When a sample taken from the lo-day culture was inoculated into DNB medium, exponential growth started after a lag. The theoretical growth curve estimated by equation (8) confirmed a good fit with the observed number of cells. Similar results were also obtained with other non-growing populations. The present study using E. coli cells showed that, in the culture with an inoculum of growing cells, the number of cells increased in an exponential manner without any lag in accordance with equation (5) whereas in the culture with an inoculum of non-growing cells, the number started to increase exponentially after a lag which correlated well with equation (8). The two equations, (5) and (8), are likely applicable to make precise predictions of the bacterial growth throughout the lag and exponential phases, when the values of parameters are determined. From equation (13) the length of the lag time L is given by 3 parameters: the retardation time, t,(2), the rate of growth initiation of parent cells, X, and the specific growth rate, ~1. As shown in Fig. 5, changes in the length of the observed lag time depended solely on the value of the parameter X. REFERENCES PI PI Buchanan, R.E. (1918) Life phases in a bacterial culture. J. Infect. Dis. 23, Hattori, T. (1985) Kinetics of colony formation of bacteria: An approach to the basis of the plate count method. Rep. Inst. Agr. Res. Tohoku Univ. 34, l Mochizuki, M. and Hattori, T. (1986) Kinetics of microcolony formation of a soil oligotrophic bacterium, Agromonas sp. FEMS Microbial. Ecol. 38, [41 Monod, J. (1949) The growth of bacterial cultures. Ann. Rev. Microbial. 3, [51 Lodge, R.M. and Hinshelwood, C.N. (1943) Physicochemical aspects of bacterial growth, Part IX (The lag phase of Bact. Lmtis aerogenes). J. Chem. Sot., (61 Schaechter, M., Williamson, J.P., Hood, Jr., and Koch, A.L. (1962) Growth, cell and nuclear divisions in some bacteria. J. Gen. Microbial. 29, [7] Powell, E.O. (1956) Growth rate and generation time of bacteria, with special reference to continuous culture. J. Gen. Microbial. 15, [8] Messer, W. (1972) Initiation of deoxyribonucleic acid replication in Escherichia coli B/r: Chronology of events and transcriptional control of initiation. J. Bacterial. 112, 7-12.