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On the lag phase and initial decline of microbial growth curves
ARTICLE IN PRESS
Journal of Theoretical Biology 244 (2007) 511–517 www.elsevier.com/locate/yjtbi
On the lag phase and initial decline of microbial growth curves George T. Yates, Thomas Smotzer Department of Mathematics and Statistics, Youngstown State University, One University Plaza, Youngstown, OH 44555, USA Received 10 April 2006; received in revised form 7 July 2006; accepted 25 August 2006 Available online 1 September 2006
Abstract The lag phase is generally thought to be a period during which the cells adjust to a new environment before the onset of exponential growth. Characterizing the lag phase in microbial growth curves has importance in food sciences, environmental sciences, bioremediation and in understanding basic cellular processes. The goal of this work is to extend the analysis of cell growth curves and to better estimate the duration of the lag phase. A non-autonomous model is presented that includes actively duplicating cells and two subclasses of nonduplicating cells. The growth curves depend on the growth and death rate of these three subpopulations and on the initial proportion of each. A deterministic and a stochastic model are both developed and give the same results. A notable feature of the model is the decline of cells during the early stage of the growth curve, and the range of parameters when this decline occurs is identified. A limited growth model is also presented that accounts for the lag, exponential growth and stationary phase of microbial growth curves. r 2006 Elsevier Ltd. All rights reserved. Keywords: Lag phase; Microbial growth; Mathematical model; Predictive microbiology; Logistic growth
1. Introduction An important topic in predictive microbiology is the quantification of the lag phase during microbial growth and the adaptation of organisms to environmental toxins. The growth of cell cultures is usually divided into lag, exponential, stationary and death phases (Prescott et al., 2002). During the lag phase, cells undergo intracellular changes in an effort to adjust to a new environment, and little or no cell reproduction takes place. During the exponential phase, cells reproduce at a rate proportional to the number of cells leading to an exponential increase in the number of cells. The stationary phase follows when nutrients are limited or other environmental conditions restrict the number of cells that can be supported. Finally, cellular death and a declining population occur when the surroundings cannot maintain the population. This paper focuses on the lag phase and the transition to the exponential phase, and will consider the effects that the stationary phase may have on these initial phases. No attempt is made to model the final death phase in this Corresponding author. Tel.: +1 330 941 3782; fax: +1 330 941 3170.
E-mail address: [email protected] (G.T. Yates). 0022-5193/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.jtbi.2006.08.017
paper. Although the analysis presented here applies to most cellular growth, we will discuss examples of bacterial growth. Various models have been proposed for the initial growing phase of bacteria (Baranyi, 1998; Baranyi et al., 1993; Baranyi and Roberts, 1994; Hills and Wright, 1994; McKellar, 1997, 2001; see review by Swinnen et al., 2004). These and other research (Baty and Delignette-Muller, 2004) recognize the need for better understanding of the underlying mechanisms of the lag phase to improve predictive modeling. Both predictive and stochastic models that reflect the physiological state of the cells at inoculation are important to accurately model cellular growth (Baranyi, 1998; McKellar, 2001). We will develop a compartmental model to include (1) initially non-duplicating cells that adapt to become duplicating cells, (2) cells that are adapted for growth in a new environment from the outset, and (3) cells that fail to adapt to a new environment and die. This model requires the initial physiological state of the cells to be specified. A stochastic model based on individual cell behavior is developed that agrees with the compartmental model for large populations. An alternate way to define the lag phase is given when the population initially declines or the stationary phase develops before
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the exponential phase is clearly identifiable, and a limited growth model is presented.
where r2 r¼ . r1
2. Growth model
The total number of cells is given by
The total number of cells are divided into three subpopulations. The first, N 1 ðtÞ, is the number of initially non-duplicating cells that are undergoing internal cellular changes that will eventually enable them to become actively duplicating cells. The actively duplicating cells are denoted by N 2 ðtÞ. The third subpopulation, N 3 ðtÞ, represents cells that do not duplicate and cannot adapt to the new environment. These cells simply die. The constant r1 is the rate of conversion from non-duplicating to duplicating cells, r2 is the growth rate of the duplicating cells and r3 is the rate of death of the dying cells. In addition to an exponential growth term, the rate of change of N 2 ðtÞ has a source-like term equal to the negative rate of change of N 1 ðtÞ due to the conversion of non-duplicating cells to duplicating cells. A mathematical description of the number of cells of these three types is modeled by the following system of differential equations: dN 1 ¼ r1 N 1 , dt
(1)
dN 2 ¼ r2 N 2 þ r1 N 1 dt
(2)
dN 3 ¼ r3 N 3 dt
(3)
subject to the initial conditions N 1 ð0Þ ¼ abN 0 ; N 2 ð0Þ ¼ ð1 bÞN 0 N 3 ð0Þ ¼ bð1 aÞN 0 ,
and ð4Þ
where N 0 ¼ N 1 ð0Þ þ N 2 ð0Þ þ N 3 ð0Þ is the total initial number of all cells. The initial fraction of cells that are non-duplicating is given by b, and the initial fraction of the non-duplicating cells that convert into duplicating cells is given by a. Both a and b range from 0 to 1 and specify the initial configuration. Note that when a ¼ 0 or b ¼ 0, the system has uncoupled exponential solutions and these cases are not considered further. The first and third of these equations have simple exponential solutions: N 1 ðtÞ ¼ abN 0 er1 t , N 3 ðtÞ ¼ bð1 aÞN 0 e
(5) r3 t
.
(6)
The differential equation for N 2 ðtÞ then becomes, dN 2 ¼ r2 N 2 þ r1 abN 0 er1 t , dt which has the solution abN 0 r2 t abN 0 r1 t N 2 ðtÞ ¼ ð1 bÞN 0 þ e , e 1þr 1þr
(7)
(8)
(9)
NðtÞ ¼ N 1 ðtÞ þ N 2 ðtÞ þ N 3 ðtÞ,
(10)
and letting xðtÞ ¼
NðtÞ N0
(11)
we find ð1 bÞð1 þ rÞ þ ab r2 t abr r1 t xðtÞ ¼ þ bð1 aÞer3 t . e þ e 1þr 1þr (12) By examining yðtÞ ¼ ln xðtÞ
(13)
the exponential growth phase can be easily identified as the straight line ð1 bÞð1 þ rÞ þ ab ^ ¼ r2 t þ ln yðtÞ (14) 1þr which accurately models yðtÞ for large time. Finding the ^ ¼ yð0Þ ¼ 0, value of t ¼ l, when this line intersects yðlÞ gives a measure of the lag phase duration: 1 1þr l ¼ ln . (15) r2 ð1 bÞð1 þ rÞ þ ab This is exactly the same as the limit value of the population lag phase used by Kutalik et al. (2005), Hills and Wright (1994), Baranyi and Roberts (1994) and Baranyi (1998): 1 NðtÞ l ¼ lim t ln . (16) t!1 r2 N0 The lag phase given by Eq. (15) extends previous results by incorporating death and by generalizing the initial conditions to allow for cells that are adjusted to the new environment at the outset. These new features are quantified by the parameters a and b, respectively. For the special case when a ¼ 1 and b ¼ 1, that is, for N 1 ð0Þ ¼ N 0 , N 2 ð0Þ ¼ N 3 ð0Þ ¼ 0, rer1 t þ er2 t 1þr and l reduces to xðtÞ ¼
l0 ¼
1 lnð1 þ rÞ r2
(17)
(18)
and the results obtained by Hills and Wright (1994), Baranyi and Roberts (1994) and Baranyi (1998) are recovered. A notable feature of the current model is that a reduction of the population can precede the exponential growth. To explore this possibility we examine, y0 ðtÞ ¼
x0 ðtÞ . xðtÞ
(19)
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Since xðtÞ is always positive, we consider 8
r2 ½ð1 bÞð1 þ rÞ þ aber2 t r1 abrer1 t x ðtÞ ¼ 1þr r3 bð1 aÞer3 t , 0
6
ð20Þ y(t)
and y0 ð0Þo0 whenever x0 ð0Þo0 or r3 r½ð1 bÞð1 þ rÞ þ aboabr þ bð1 aÞð1 þ rÞ , r1
4
(21) 2
which reduces to the condition r0 4r, where r0 ¼
bð1 aÞr3 . ð1 bÞr1
(22)
To explore the dependence on the initial proportions of each subpopulation, that is, on the parameters a and b, we consider the case r3 ¼ r1 . This assumes that the rate of conversion of subpopulation N 1 from non-duplicating cells to duplicating cells is the same as the death rate of subpopulation N 3 . Such an assumption may have been reasonable from the outset since the dying cells are of minor interest and r1 could have been interpreted as a net rate of conversion and death. For the remainder of this section we will consider r3 ¼ r1 , and the general result (12) is retained for later investigation. When the population is initially decreasing a minimum of population occurs when x0 ðt0 Þ ¼ 0 where 1 bðr þ 1 aÞ ln (23) t0 ¼ r1 þ r2 r½ð1 bÞð1 þ rÞ þ ab and positive values of t0 are possible only when r0 4r. We find that whenever the combination of a and b give an r0 value greater than r, the population declines initially, obtains a minimum at t0 and increases thereafter. Conversely, when r0 or, the population is increasing for all time. Contours of constant r0 are shown in Fig. 1. For example, if r ¼ 1, as for the data shown in Fig. 2, and if a and b are above the r0 ¼ 1 contour in Fig. 1, there will be an initial dip in the growth curve. This is evident in Fig. 2
1.0 0.8 0.6
0 0
50
β
150 t (hours)
200
250
Fig. 2. Growth curve yðtÞ ¼ ln NðtÞ=N 0 versus time t. Data were obtained from ComBase data file B302_44 (www.combase.cc) for E. coli in culture medium at 20 C, pH 5.7 and 5.5% NaCl concentration. The solid curve plots Eq. (30) with parameter values r1 ¼ r2 ¼ 0:09, A ¼ 0:09, K ¼ 5000 obtained by a nonlinear least-square regression in programming language R and by an iterative process of experienced guesses.
and agrees with expectations, since decreasing the fraction of cells that convert from non-duplicating to duplicating (decreasing-a) or increasing the initial percent on nonduplicating cells (increasing-b) should result in a delay in the onset of growth and may even lead to a reduction in the overall cell count. All the contours in Fig. 1 approach a single point in the upper right corner where a ¼ 1 and b ¼ 1, which corresponds to the case when l ¼ l0 . Also note that r0 is not defined at this point or for any value of a when b ¼ 1. When the denominator inside the logarithm in (15) is less than 1, the lag phase l is increased above l0 . This means that if r0 4r, then l4l0 . In general, decreasing a or increasing b leads to an increase of the lag phase l. It can also be shown that for any fixed a, b and r2 , decreasing r1 causes l to increase. This analysis agrees with expectations based on the biological interpretation of the parameters. When the initial decline of cells is substantial, the value of l may not accurately indicate the onset of exponential growth in the growth curve. In these cases, we propose a modified lag phase l^ defined as the time when line (14) ^ ¼ yðt0 Þ or ^ lÞ takes on the minimum value yðt0 Þ. This gives yð 1 l^ ¼ t0 þ lnð1 þ rÞ r2
0.4
100
(24)
as an approximation for the lag phase that more accurately predicts the time when exponential growth commences. This expression is suggested only when r0 4r where t0 exists, and Eq. (15) should be used when r0 or.
0.2 0.0 0.0
0.2
0.4
0.6
0.8
1.0
α Fig. 1. Contour plot for r0 given by (22) with r3 ¼ r1 . Curves of constant r0 are plotted for r0 values ranging from 0.1 to 10 as indicated for each curve. The curves show how r0 varies with a and b. Whenever r0 is greater than r, l4l0 , and there exists a value t0 given by (23).
3. Stochastic model Buchanan et al. (1997) suggest that the basic physiological state of individual cells should form the basis of any growth model. Renshaw (1993) points out that when
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considering small populations stochastic models should be used. In the limit of large populations stochastic models based on individual cell physiology should give the same results as predictive models. In this section, we consider the initial state of every individual cell and use a Poisson birth process and pure death process to construct a stochastic model. Let the function x1 ðtÞ represent the number of bacteria undergoing internal cellular changes but not duplicating, x2 ðtÞ the number of bacteria actively duplicating and x3 ðtÞ denote the number of bacteria that are unable to undergo internal cellular changes and are thus dying. As in Renshaw (1993), suppose that the cell populations x1 and x2 grow according to the classical Poisson pure birth process with birth rate parameter r2 and that the cell population x3 dies according to a pure death process with death rate parameter r3 . The lag times for the ith cell in the initial N cells of x1 is denoted by ti . We assume that the lag times ti ði ¼ 1; 2; . . . ; NÞ are identically distributed independent random variables with expected value Eðti Þ ¼ t. The total number of bacteria as a function of time can be approximated by the sum of the exponential biphasic linear and linear functions xðtÞ ¼ x1 ðtÞ þ x2 ðtÞ þ x3 ðtÞ ¼
N X
er2 maxðtti ;0Þ
i¼1
þ
M X j¼1
er2 t þ
P X
er3 t ,
ð25Þ
k¼1
where N, M and P are the initial cell counts of x1 , x2 and x3 respectively. The logarithm of the population is again defined as yðtÞ ¼ ln xðtÞ. For t4 max ti , the formula can be rearranged to the following asymptotically linear form: yðtÞ ¼ lnðN þ M þ PÞ " # 1 N þMþP þ r2 t ln PN PM PP ðr þr Þt . r2 ti þ 2 3 r2 i¼1 e j¼1 1 þ k¼1 e
For N 0 large enough we have l¼
1 1 þ r2 t . ln r2 ab þ ð1 bÞð1 þ r2 tÞ
(29)
If the individual lag times follow the exponential distribution with t ¼ 1=r1 (Kutalik et al., 2005), then the lag time matches result (15) above. 4. Limited growth model In this section, we suppose the growth is limited by availability of nutrient or other growth limiting conditions. If the exponential growth phase is relatively short or has a time scale of the same order as the lag phase, the above model may need adjustments. We assume that the normalized population xðtÞ ¼ NðtÞ=N 0 is given by the sum of an exponential death and a logistic growth function instead of the sum of exponential death and exponential growth terms as obtained above. Thus, we suppose xðtÞ ¼ ð1 AÞer1 t þ
K . 1 þ ðK=A 1Þer2 t
(30)
The carrying capacity K is assumed greater than 1 throughout. The parameter A is a combined measure of the initial state b and the proportion of conversion a used earlier, and it is confined within the range 0oAo1. This model gives xð0Þ ¼ 1 and xðt ! 1Þ ¼ K, however, taking the limit indicated in (16) leads to an infinite value for l. An intermediate value of time greater than 0 and less than the time required for the constant stationary phase to manifest would allow us to construct an straight line approximation to the exponential growth phase occurring between 50 and 100 h for the example shown in Fig. 2. This entails finding the maximum slope of the graph yðtÞ ¼ ln xðtÞ, which occurs when y00 ðtÞ ¼ 0. To this end, we observe y0 ðtÞ ¼
x0 ðtÞ , xðtÞ
y00 ðtÞ ¼
xðtÞx00 ðtÞ ½x0 ðtÞ2 . x2 ðtÞ
ð26Þ For sufficiently large time, we define the population lag to be 1 N þM þP . lðN; M; PÞ ¼ ln PN r2 ti þ M r2 i¼1 e
(27)
Note that the variables expðr2 ti Þ for i ¼ 1; 2; . . . ; N are identically distributed. If the lag times ti follow an exponential distribution with mean value t, Baranyi (1998) showed that the expectation of expðr2 ti Þ is 1=ð1 þ r2 tÞ. So for large N we have that lðN; M; PÞ ¼
1 N þMþP . ln r2 N=ð1 þ r2 tÞ þ M
(28)
Using the notation of the growth model in the previous section, N ¼ abN 0 , M ¼ ð1 bÞN 0 and P ¼ ð1 aÞbN 0 .
(31)
Since xðtÞ40 for all time, y0 ðtÞ vanishes when x0 ðt0 Þ ¼ 0. Similar to the exponential growth model, there is a critical value of the parameters for which y0 ð0Þo0, which occurs whenever R0 ¼
Kð1 AÞ 4r. AðK AÞ
(32)
When R0 4r, there will be an initial dip in the growth curve and the minimum value of yðtÞ occurs at t0 . Finding t0 involves solving a quadratic equation when r ¼ 1, or a transcendental equation for general r, and values can be easily computed using Maple or other numerical software. See the Appendix for details.
ARTICLE IN PRESS G.T. Yates, T. Smotzer / Journal of Theoretical Biology 244 (2007) 511–517
The slope of the curve yðtÞ takes on a maximum value when y00 ðt Þ ¼ 0, which occurs when xðt Þx00 ðt Þ ¼ ½x0 ðt Þ2 .
515
2
(33) y(t)
0
Once t is obtained, we can find the line tangent to the curve yðtÞ at t as
-2
^ ¼ yðt Þ þ y0 ðt Þðt t Þ. yðtÞ ^ ‘Þ ¼ The value of t ¼ t‘ , when this line intersects yðt yð0Þ ¼ 0, gives the lag phase: yðt Þ , t‘ ¼ t 0 y ðt Þ
0
yðt Þ yðt0 Þ . y0 ðt Þ
(35)
As an example, we consider the data and fitted curve shown in Fig. 2 for Escherichia coli grown in a salt solution. For this example, r ¼ r2 =r1 ¼ 1 and from (32) R0 ¼ 10:11. Clearly, R0 4r, and the population initially decreases and reaches a minimum at " # 1 K A t0 ¼ ln pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi . r KAðK AÞ=ð1 AÞ A Details for this formula and the following computed results are given in the Appendix. Eq. (33) can be solved numerically to give t ¼ 54:18 and subsequently t‘ and t^‘ can be found from (34) and (35), respectively. The results are given in Table 1. A second example is shown in Fig. 3 for Listeria monocytogenes also grown in a salt solution. Here r ¼ r2 =r1 ¼ 1:15 and R0 ¼ 6:67 106 . Clearly, R0 4r and the population undergoes a pronounced initial decrease and reaches a minimum at t0 ¼ 362:2 before exponential growth occurs. The maximum slope during the exponential growth phase or LIP occurs at t ¼ 537:16 and the lag Table 1 Table of computed results for time of minimal cell counts t0 , maximum slope t and lag times t‘ and t^‘ for the growth curves in Figs. 2 and 3
E. coli (Fig. 2) Listeria monocytogenes (Fig. 3)
-6
(34)
which corresponds to the quantity l discussed earlier. Finding t entails solving a cubic equation when r ¼ 1, or a transcendental equation for general r, and Maple was used in computing specific values. Vadasz and Vadasz (2005) called t the logarithmic inflection point or LIP and use it similarly to find t‘ , however, they use a different model for xðtÞ. When yðt0 Þ is significantly below the initial value yð0Þ ¼ 0 and when exponential growth clearly develops before the initial cell count is recovered, it might be more appropriate ^ t^‘ Þ ¼ yðt0 Þ. This to define t^‘ as the time when the line yð suggests the alternative formula for the lag time: t^‘ ¼ t
-4
t0
t
t‘
t^‘
12.9 362
54.2 537
26.7 684
20.5 395
200
400
600 800 t (hours)
1000
1200
1400
Fig. 3. Growth curve yðtÞ ¼ ln NðtÞ=N 0 versus time t. Data were obtained from ComBase data file B259_148 (www.combase.cc) for Listeria monocytogenes in culture medium at 10 C, pH 7.0 and 10% NaCl concentration. The solid curve plots Eq. (30) with parameter values r1 ¼ 0:02, r2 ¼ 0:023, A ¼ 1:5 107 , K ¼ 16.
times are computed and given in Table 1. The parameters r1 ¼ 0:02, r2 ¼ 0:023, A ¼ 1:5 107 and K ¼ 16 were found using a weighted least-square regression in programming language R and by an iterative process of experienced guesses. Data near the minimum were more lightly weighted to obtain more realistic fitting of the initial death, exponential growth and stationary portions of the data. Although the minimum population yðt0 Þ is underestimated, the curve does provide a realistic estimate of the time t0 where the minimum occurs and the lag times t‘ and t^‘ . Other curve fitting techniques or model equations can be used to provide better overall fits to the data, however our aim is to understand the cellular mechanisms leading to the population decline and to quantify the lag phase. In Fig. 3, the inability of function (30) to follow the data near the minimum value is attributed to the assumption that the death rate r3 ¼ r1 in the limited growth model, and a more general model is needed to provide better agreement with the data. 5. Discussion and conclusion Three growth models were presented for microorganism growth curves. The compartmental growth model extends the model of Baranyi (1998) to include cellular death and includes more general initial conditions. We introduced the parameters a as a measure of the proportion of nonduplicating cells that become duplicating cells and b as the fraction of initial cells that are non-duplicating. This more general model accounts for the death of a fraction ð1 aÞ of the non-duplicating cells, and predicts the decline of total cell count during the initial adjustment period which is followed by exponential growth. The specific range of parameters were also identified when population decline occurs ðr0 4rÞ and when monotonic growth occurs ðr0 prÞ. The duration of the lag phase l was found to depend on a
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and b with the general result that l increases when a is decreased and b is increased. An alternative lag time l^ or t^‘ was suggested when substantial population decline occurs prior to exponential growth, and predicts a shorter lag time than l. An independent stochastic model was developed based on the physiological state of individual cells. This is a necessary consideration for small populations and the results agree with the compartmental model in the limit of large populations. Finally, a limited growth model was proposed when environmental factors lead to a stationary phase in the growth curve. Although simple formulas for the lag phase are possible only for special cases of the parameters numerical solutions are routine using standard computational methods. The solution behavior during initial phases and the dependence on parameters follow closely the compartmental growth model. Subsequent computation of the lag phase duration requires modification to identify the point of maximum rate of growth. Two lag phase times are identified, one using a more traditional approach starting when the exponential phase is extrapolated back to the initial population and a second by extrapolating to the minimal population point. Comparisons between the model and observed data are generally very good, and support the traditional explanation of the cellular activity during various phases of the growth curve. The current limited growth model does not capture the detailed behavior near the minimum population for the example presented in Fig. 3. The limited growth model assumes that the death rate and conversion rate are equal, and it is postulated that this is not the case for this example. Further research is needed to resolve this shortcoming. Acknowledgments
environment. The parameter A ranges from 0 to 1 and is a combined measure of the initial mix of duplicating and non-duplicating cells and of the proportion of nonduplicating cells that convert into duplicating cells. Analysis of the various phases of the growth curve is done by considering the graph of yðtÞ ¼ ln xðtÞ. Finding the local minimum and the maximum slope of the growth curve requires the first and second derivatives y0 ðtÞ ¼
x0 ðtÞ , xðtÞ
y00 ðtÞ ¼
xðtÞx00 ðtÞ ½x0 ðtÞ2 , x2 ðtÞ
(A.2)
where x0 ðtÞ ¼ r1 ð1 AÞer1 t þ
r2 KðK=A 1Þer2 t , ½1 þ ðK=A 1Þer2 t 2
x00 ðtÞ ¼ r21 ð1 AÞer1 t
r22 KðK=A 1Þer2 t ½1 ðK=A 1Þer2 t . ½1 þ ðK=A 1Þer2 t 3
ðA:3Þ
The times t0 and t are defined, respectively, by solving x0 ðt0 Þ ¼ 0,
(A.4)
xðt Þx00 ðt Þ ¼ ½x0 ðt Þ2 .
(A.5)
Before attempting to solve these equations for t0 and t , we observe that xð0Þ ¼ 1;
lim xðtÞ ¼ K,
t!1
r2 AðK AÞ ; K Note that x0 ð0Þo0 when x0 ð0Þ ¼ r1 ð1 AÞ þ
R0 ¼
lim x0 ðtÞ ¼ 0.
t!1
Kð1 AÞ 4r, AðK AÞ
(A.6)
Discussions with Jonathan Caguiat, Chester Cooper and Carl Johnston greatly helped keep our assumptions biologically reasonable. The summer projects of David Gohlke and David Martin stimulated this work. This work was supported by the National Science Foundation Grant DUE 0337558. The support of Youngstown State University in the form of a Research Professorship for G.T.Y. and T.S. is gratefully acknowledged.
where r ¼ r2 =r1 . When R0 4r, the population initially decreases before increasing to a final value of K41 as t ! 1. In this case, the growth curve has a minimum value at t0 . Finding t0 requires solving the equation
Appendix A
where
We assume that the normalized population xðtÞ ¼ NðtÞ=N 0 is given by the sum of an exponential death and a logistic growth function:
B0 ¼
xðtÞ ¼ ð1 AÞer1 t þ
K , 1 þ ðK=A 1Þer2 t
(A.1)
where N 0 is the initial population. The combined conversion/death rate r1 and the growth rate r2 are both taken positive and K is the carrying capacity for the specific
er1 t0 ¼
rB0 er2 t0 , ½1 þ ðK=A 1Þer2 t0 2
KðK AÞ . Að1 AÞ
Unless r1 is equal to r2 or an integer times r2 , this equation can only be solved numerically for t0 . Eq. (A.5) for t can be written as z1=r ð1 þ zÞ½ð1 þ rÞ2 z2 þ ð2 þ 2r r2 Þz þ 1 ¼
r2 zKðK=A 1Þ1=r , 1A
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where K 1 er2 t , z¼ A which must also be solved numerically for z in the general case, to obtain t . Taking r1 ¼ r2 or r ¼ r2 =r1 ¼ 1, these equations simplify somewhat, giving 2 K 1þ 1 er2 t0 ¼ B0 A which gives 1 K A t0 ¼ ln pffiffiffiffiffiffi . r2 A B0 A
(A.7)
A second solution is discarded since it is always negative. The solution for t reduces to solving 4z3 þ 7z2 þ 4z þ 1 B0 ¼ 0. We note that for K41 and 0oAo1, B0 must be greater than 1. It can be shown that this cubic equation always has one and only one positive real solution when B0 41. Thus, exactly one t always exists. References Baranyi, J., 1998. Comparison of stochastic and deterministic concepts of bacterial lag. J. Theor. Biol. 192, 403–408.
517
Baranyi, J., Roberts, T.A., 1994. A dynamic approach to predicting bacterial growth in food. Int. J. Food Microbiol. 23, 277–294. Baranyi, J., Roberts, T.A., McClure, P., 1993. A non-autonomous differential equation to model bacterial growth. Food Microbiol. 10, 43–59. Baty, F., Delignette-Muller, M.L., 2004. Estimating the bacterial lag time: which model, which precision? Int. J. Food Microbiol. 91, 261–277. Buchanan, R.L., Whiting, R.C., Damert, W.C., 1997. When is simple good enough: a comparison of the Gompertz, Baranyi and three-phase linear models for fitting bacterial growth curves. Food Microbiol. 14, 313–326. Hills, B.P., Wright, K.M., 1994. A new model for bacterial growth in heterogeneous systems. J. Theor. Biol. 168, 31–41. Kutalik, Z., Razaz, M., Baranyi, J., 2005. Connection between stochastic and deterministic modelling of microbial growth. J. Theor. Biol. 232, 285–299. McKellar, R.C., 1997. A heterogeneous population model for the analysis of bacterial growth kinetics. Int. J. Food Microbiol. 36, 179–186. McKellar, R.C., 2001. Development of a dynamic continuous–discrete–continuous model describing the lag phase of individual bacterial cells. J. Appl. Microbiol. 90, 407–413. Prescott, L.M., Harley, J.P., Klein, D.A., 2002. Microbiology, fifth ed. McGraw-Hill, New York. Renshaw, E., 1993. Modelling Biological Populations in Space and Time. Cambridge University Press, Cambridge. Swinnen, I.A.M., Bernaerts, K., Dens, E.J.J., Geeraerd, A.H., Van Impe, J.F., 2004. Predictive modelling of the microbial lag phase: a review. Int. J. Food Microbiol. 94, 137–159. Vadasz, P., Vadasz, A.S., 2005. Predictive modeling of microorganisms: LAG and LIP in monotonic growth. Int. J. Food Microbiol. 102, 257–275.
Journal of Theoretical Biology 244 (2007) 511–517 www.elsevier.com/locate/yjtbi
On the lag phase and initial decline of microbial growth curves George T. Yates, Thomas Smotzer Department of Mathematics and Statistics, Youngstown State University, One University Plaza, Youngstown, OH 44555, USA Received 10 April 2006; received in revised form 7 July 2006; accepted 25 August 2006 Available online 1 September 2006
Abstract The lag phase is generally thought to be a period during which the cells adjust to a new environment before the onset of exponential growth. Characterizing the lag phase in microbial growth curves has importance in food sciences, environmental sciences, bioremediation and in understanding basic cellular processes. The goal of this work is to extend the analysis of cell growth curves and to better estimate the duration of the lag phase. A non-autonomous model is presented that includes actively duplicating cells and two subclasses of nonduplicating cells. The growth curves depend on the growth and death rate of these three subpopulations and on the initial proportion of each. A deterministic and a stochastic model are both developed and give the same results. A notable feature of the model is the decline of cells during the early stage of the growth curve, and the range of parameters when this decline occurs is identified. A limited growth model is also presented that accounts for the lag, exponential growth and stationary phase of microbial growth curves. r 2006 Elsevier Ltd. All rights reserved. Keywords: Lag phase; Microbial growth; Mathematical model; Predictive microbiology; Logistic growth
1. Introduction An important topic in predictive microbiology is the quantification of the lag phase during microbial growth and the adaptation of organisms to environmental toxins. The growth of cell cultures is usually divided into lag, exponential, stationary and death phases (Prescott et al., 2002). During the lag phase, cells undergo intracellular changes in an effort to adjust to a new environment, and little or no cell reproduction takes place. During the exponential phase, cells reproduce at a rate proportional to the number of cells leading to an exponential increase in the number of cells. The stationary phase follows when nutrients are limited or other environmental conditions restrict the number of cells that can be supported. Finally, cellular death and a declining population occur when the surroundings cannot maintain the population. This paper focuses on the lag phase and the transition to the exponential phase, and will consider the effects that the stationary phase may have on these initial phases. No attempt is made to model the final death phase in this Corresponding author. Tel.: +1 330 941 3782; fax: +1 330 941 3170.
E-mail address: [email protected] (G.T. Yates). 0022-5193/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.jtbi.2006.08.017
paper. Although the analysis presented here applies to most cellular growth, we will discuss examples of bacterial growth. Various models have been proposed for the initial growing phase of bacteria (Baranyi, 1998; Baranyi et al., 1993; Baranyi and Roberts, 1994; Hills and Wright, 1994; McKellar, 1997, 2001; see review by Swinnen et al., 2004). These and other research (Baty and Delignette-Muller, 2004) recognize the need for better understanding of the underlying mechanisms of the lag phase to improve predictive modeling. Both predictive and stochastic models that reflect the physiological state of the cells at inoculation are important to accurately model cellular growth (Baranyi, 1998; McKellar, 2001). We will develop a compartmental model to include (1) initially non-duplicating cells that adapt to become duplicating cells, (2) cells that are adapted for growth in a new environment from the outset, and (3) cells that fail to adapt to a new environment and die. This model requires the initial physiological state of the cells to be specified. A stochastic model based on individual cell behavior is developed that agrees with the compartmental model for large populations. An alternate way to define the lag phase is given when the population initially declines or the stationary phase develops before
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the exponential phase is clearly identifiable, and a limited growth model is presented.
where r2 r¼ . r1
2. Growth model
The total number of cells is given by
The total number of cells are divided into three subpopulations. The first, N 1 ðtÞ, is the number of initially non-duplicating cells that are undergoing internal cellular changes that will eventually enable them to become actively duplicating cells. The actively duplicating cells are denoted by N 2 ðtÞ. The third subpopulation, N 3 ðtÞ, represents cells that do not duplicate and cannot adapt to the new environment. These cells simply die. The constant r1 is the rate of conversion from non-duplicating to duplicating cells, r2 is the growth rate of the duplicating cells and r3 is the rate of death of the dying cells. In addition to an exponential growth term, the rate of change of N 2 ðtÞ has a source-like term equal to the negative rate of change of N 1 ðtÞ due to the conversion of non-duplicating cells to duplicating cells. A mathematical description of the number of cells of these three types is modeled by the following system of differential equations: dN 1 ¼ r1 N 1 , dt
(1)
dN 2 ¼ r2 N 2 þ r1 N 1 dt
(2)
dN 3 ¼ r3 N 3 dt
(3)
subject to the initial conditions N 1 ð0Þ ¼ abN 0 ; N 2 ð0Þ ¼ ð1 bÞN 0 N 3 ð0Þ ¼ bð1 aÞN 0 ,
and ð4Þ
where N 0 ¼ N 1 ð0Þ þ N 2 ð0Þ þ N 3 ð0Þ is the total initial number of all cells. The initial fraction of cells that are non-duplicating is given by b, and the initial fraction of the non-duplicating cells that convert into duplicating cells is given by a. Both a and b range from 0 to 1 and specify the initial configuration. Note that when a ¼ 0 or b ¼ 0, the system has uncoupled exponential solutions and these cases are not considered further. The first and third of these equations have simple exponential solutions: N 1 ðtÞ ¼ abN 0 er1 t , N 3 ðtÞ ¼ bð1 aÞN 0 e
(5) r3 t
.
(6)
The differential equation for N 2 ðtÞ then becomes, dN 2 ¼ r2 N 2 þ r1 abN 0 er1 t , dt which has the solution abN 0 r2 t abN 0 r1 t N 2 ðtÞ ¼ ð1 bÞN 0 þ e , e 1þr 1þr
(7)
(8)
(9)
NðtÞ ¼ N 1 ðtÞ þ N 2 ðtÞ þ N 3 ðtÞ,
(10)
and letting xðtÞ ¼
NðtÞ N0
(11)
we find ð1 bÞð1 þ rÞ þ ab r2 t abr r1 t xðtÞ ¼ þ bð1 aÞer3 t . e þ e 1þr 1þr (12) By examining yðtÞ ¼ ln xðtÞ
(13)
the exponential growth phase can be easily identified as the straight line ð1 bÞð1 þ rÞ þ ab ^ ¼ r2 t þ ln yðtÞ (14) 1þr which accurately models yðtÞ for large time. Finding the ^ ¼ yð0Þ ¼ 0, value of t ¼ l, when this line intersects yðlÞ gives a measure of the lag phase duration: 1 1þr l ¼ ln . (15) r2 ð1 bÞð1 þ rÞ þ ab This is exactly the same as the limit value of the population lag phase used by Kutalik et al. (2005), Hills and Wright (1994), Baranyi and Roberts (1994) and Baranyi (1998): 1 NðtÞ l ¼ lim t ln . (16) t!1 r2 N0 The lag phase given by Eq. (15) extends previous results by incorporating death and by generalizing the initial conditions to allow for cells that are adjusted to the new environment at the outset. These new features are quantified by the parameters a and b, respectively. For the special case when a ¼ 1 and b ¼ 1, that is, for N 1 ð0Þ ¼ N 0 , N 2 ð0Þ ¼ N 3 ð0Þ ¼ 0, rer1 t þ er2 t 1þr and l reduces to xðtÞ ¼
l0 ¼
1 lnð1 þ rÞ r2
(17)
(18)
and the results obtained by Hills and Wright (1994), Baranyi and Roberts (1994) and Baranyi (1998) are recovered. A notable feature of the current model is that a reduction of the population can precede the exponential growth. To explore this possibility we examine, y0 ðtÞ ¼
x0 ðtÞ . xðtÞ
(19)
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Since xðtÞ is always positive, we consider 8
r2 ½ð1 bÞð1 þ rÞ þ aber2 t r1 abrer1 t x ðtÞ ¼ 1þr r3 bð1 aÞer3 t , 0
6
ð20Þ y(t)
and y0 ð0Þo0 whenever x0 ð0Þo0 or r3 r½ð1 bÞð1 þ rÞ þ aboabr þ bð1 aÞð1 þ rÞ , r1
4
(21) 2
which reduces to the condition r0 4r, where r0 ¼
bð1 aÞr3 . ð1 bÞr1
(22)
To explore the dependence on the initial proportions of each subpopulation, that is, on the parameters a and b, we consider the case r3 ¼ r1 . This assumes that the rate of conversion of subpopulation N 1 from non-duplicating cells to duplicating cells is the same as the death rate of subpopulation N 3 . Such an assumption may have been reasonable from the outset since the dying cells are of minor interest and r1 could have been interpreted as a net rate of conversion and death. For the remainder of this section we will consider r3 ¼ r1 , and the general result (12) is retained for later investigation. When the population is initially decreasing a minimum of population occurs when x0 ðt0 Þ ¼ 0 where 1 bðr þ 1 aÞ ln (23) t0 ¼ r1 þ r2 r½ð1 bÞð1 þ rÞ þ ab and positive values of t0 are possible only when r0 4r. We find that whenever the combination of a and b give an r0 value greater than r, the population declines initially, obtains a minimum at t0 and increases thereafter. Conversely, when r0 or, the population is increasing for all time. Contours of constant r0 are shown in Fig. 1. For example, if r ¼ 1, as for the data shown in Fig. 2, and if a and b are above the r0 ¼ 1 contour in Fig. 1, there will be an initial dip in the growth curve. This is evident in Fig. 2
1.0 0.8 0.6
0 0
50
β
150 t (hours)
200
250
Fig. 2. Growth curve yðtÞ ¼ ln NðtÞ=N 0 versus time t. Data were obtained from ComBase data file B302_44 (www.combase.cc) for E. coli in culture medium at 20 C, pH 5.7 and 5.5% NaCl concentration. The solid curve plots Eq. (30) with parameter values r1 ¼ r2 ¼ 0:09, A ¼ 0:09, K ¼ 5000 obtained by a nonlinear least-square regression in programming language R and by an iterative process of experienced guesses.
and agrees with expectations, since decreasing the fraction of cells that convert from non-duplicating to duplicating (decreasing-a) or increasing the initial percent on nonduplicating cells (increasing-b) should result in a delay in the onset of growth and may even lead to a reduction in the overall cell count. All the contours in Fig. 1 approach a single point in the upper right corner where a ¼ 1 and b ¼ 1, which corresponds to the case when l ¼ l0 . Also note that r0 is not defined at this point or for any value of a when b ¼ 1. When the denominator inside the logarithm in (15) is less than 1, the lag phase l is increased above l0 . This means that if r0 4r, then l4l0 . In general, decreasing a or increasing b leads to an increase of the lag phase l. It can also be shown that for any fixed a, b and r2 , decreasing r1 causes l to increase. This analysis agrees with expectations based on the biological interpretation of the parameters. When the initial decline of cells is substantial, the value of l may not accurately indicate the onset of exponential growth in the growth curve. In these cases, we propose a modified lag phase l^ defined as the time when line (14) ^ ¼ yðt0 Þ or ^ lÞ takes on the minimum value yðt0 Þ. This gives yð 1 l^ ¼ t0 þ lnð1 þ rÞ r2
0.4
100
(24)
as an approximation for the lag phase that more accurately predicts the time when exponential growth commences. This expression is suggested only when r0 4r where t0 exists, and Eq. (15) should be used when r0 or.
0.2 0.0 0.0
0.2
0.4
0.6
0.8
1.0
α Fig. 1. Contour plot for r0 given by (22) with r3 ¼ r1 . Curves of constant r0 are plotted for r0 values ranging from 0.1 to 10 as indicated for each curve. The curves show how r0 varies with a and b. Whenever r0 is greater than r, l4l0 , and there exists a value t0 given by (23).
3. Stochastic model Buchanan et al. (1997) suggest that the basic physiological state of individual cells should form the basis of any growth model. Renshaw (1993) points out that when
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considering small populations stochastic models should be used. In the limit of large populations stochastic models based on individual cell physiology should give the same results as predictive models. In this section, we consider the initial state of every individual cell and use a Poisson birth process and pure death process to construct a stochastic model. Let the function x1 ðtÞ represent the number of bacteria undergoing internal cellular changes but not duplicating, x2 ðtÞ the number of bacteria actively duplicating and x3 ðtÞ denote the number of bacteria that are unable to undergo internal cellular changes and are thus dying. As in Renshaw (1993), suppose that the cell populations x1 and x2 grow according to the classical Poisson pure birth process with birth rate parameter r2 and that the cell population x3 dies according to a pure death process with death rate parameter r3 . The lag times for the ith cell in the initial N cells of x1 is denoted by ti . We assume that the lag times ti ði ¼ 1; 2; . . . ; NÞ are identically distributed independent random variables with expected value Eðti Þ ¼ t. The total number of bacteria as a function of time can be approximated by the sum of the exponential biphasic linear and linear functions xðtÞ ¼ x1 ðtÞ þ x2 ðtÞ þ x3 ðtÞ ¼
N X
er2 maxðtti ;0Þ
i¼1
þ
M X j¼1
er2 t þ
P X
er3 t ,
ð25Þ
k¼1
where N, M and P are the initial cell counts of x1 , x2 and x3 respectively. The logarithm of the population is again defined as yðtÞ ¼ ln xðtÞ. For t4 max ti , the formula can be rearranged to the following asymptotically linear form: yðtÞ ¼ lnðN þ M þ PÞ " # 1 N þMþP þ r2 t ln PN PM PP ðr þr Þt . r2 ti þ 2 3 r2 i¼1 e j¼1 1 þ k¼1 e
For N 0 large enough we have l¼
1 1 þ r2 t . ln r2 ab þ ð1 bÞð1 þ r2 tÞ
(29)
If the individual lag times follow the exponential distribution with t ¼ 1=r1 (Kutalik et al., 2005), then the lag time matches result (15) above. 4. Limited growth model In this section, we suppose the growth is limited by availability of nutrient or other growth limiting conditions. If the exponential growth phase is relatively short or has a time scale of the same order as the lag phase, the above model may need adjustments. We assume that the normalized population xðtÞ ¼ NðtÞ=N 0 is given by the sum of an exponential death and a logistic growth function instead of the sum of exponential death and exponential growth terms as obtained above. Thus, we suppose xðtÞ ¼ ð1 AÞer1 t þ
K . 1 þ ðK=A 1Þer2 t
(30)
The carrying capacity K is assumed greater than 1 throughout. The parameter A is a combined measure of the initial state b and the proportion of conversion a used earlier, and it is confined within the range 0oAo1. This model gives xð0Þ ¼ 1 and xðt ! 1Þ ¼ K, however, taking the limit indicated in (16) leads to an infinite value for l. An intermediate value of time greater than 0 and less than the time required for the constant stationary phase to manifest would allow us to construct an straight line approximation to the exponential growth phase occurring between 50 and 100 h for the example shown in Fig. 2. This entails finding the maximum slope of the graph yðtÞ ¼ ln xðtÞ, which occurs when y00 ðtÞ ¼ 0. To this end, we observe y0 ðtÞ ¼
x0 ðtÞ , xðtÞ
y00 ðtÞ ¼
xðtÞx00 ðtÞ ½x0 ðtÞ2 . x2 ðtÞ
ð26Þ For sufficiently large time, we define the population lag to be 1 N þM þP . lðN; M; PÞ ¼ ln PN r2 ti þ M r2 i¼1 e
(27)
Note that the variables expðr2 ti Þ for i ¼ 1; 2; . . . ; N are identically distributed. If the lag times ti follow an exponential distribution with mean value t, Baranyi (1998) showed that the expectation of expðr2 ti Þ is 1=ð1 þ r2 tÞ. So for large N we have that lðN; M; PÞ ¼
1 N þMþP . ln r2 N=ð1 þ r2 tÞ þ M
(28)
Using the notation of the growth model in the previous section, N ¼ abN 0 , M ¼ ð1 bÞN 0 and P ¼ ð1 aÞbN 0 .
(31)
Since xðtÞ40 for all time, y0 ðtÞ vanishes when x0 ðt0 Þ ¼ 0. Similar to the exponential growth model, there is a critical value of the parameters for which y0 ð0Þo0, which occurs whenever R0 ¼
Kð1 AÞ 4r. AðK AÞ
(32)
When R0 4r, there will be an initial dip in the growth curve and the minimum value of yðtÞ occurs at t0 . Finding t0 involves solving a quadratic equation when r ¼ 1, or a transcendental equation for general r, and values can be easily computed using Maple or other numerical software. See the Appendix for details.
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The slope of the curve yðtÞ takes on a maximum value when y00 ðt Þ ¼ 0, which occurs when xðt Þx00 ðt Þ ¼ ½x0 ðt Þ2 .
515
2
(33) y(t)
0
Once t is obtained, we can find the line tangent to the curve yðtÞ at t as
-2
^ ¼ yðt Þ þ y0 ðt Þðt t Þ. yðtÞ ^ ‘Þ ¼ The value of t ¼ t‘ , when this line intersects yðt yð0Þ ¼ 0, gives the lag phase: yðt Þ , t‘ ¼ t 0 y ðt Þ
0
yðt Þ yðt0 Þ . y0 ðt Þ
(35)
As an example, we consider the data and fitted curve shown in Fig. 2 for Escherichia coli grown in a salt solution. For this example, r ¼ r2 =r1 ¼ 1 and from (32) R0 ¼ 10:11. Clearly, R0 4r, and the population initially decreases and reaches a minimum at " # 1 K A t0 ¼ ln pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi . r KAðK AÞ=ð1 AÞ A Details for this formula and the following computed results are given in the Appendix. Eq. (33) can be solved numerically to give t ¼ 54:18 and subsequently t‘ and t^‘ can be found from (34) and (35), respectively. The results are given in Table 1. A second example is shown in Fig. 3 for Listeria monocytogenes also grown in a salt solution. Here r ¼ r2 =r1 ¼ 1:15 and R0 ¼ 6:67 106 . Clearly, R0 4r and the population undergoes a pronounced initial decrease and reaches a minimum at t0 ¼ 362:2 before exponential growth occurs. The maximum slope during the exponential growth phase or LIP occurs at t ¼ 537:16 and the lag Table 1 Table of computed results for time of minimal cell counts t0 , maximum slope t and lag times t‘ and t^‘ for the growth curves in Figs. 2 and 3
E. coli (Fig. 2) Listeria monocytogenes (Fig. 3)
-6
(34)
which corresponds to the quantity l discussed earlier. Finding t entails solving a cubic equation when r ¼ 1, or a transcendental equation for general r, and Maple was used in computing specific values. Vadasz and Vadasz (2005) called t the logarithmic inflection point or LIP and use it similarly to find t‘ , however, they use a different model for xðtÞ. When yðt0 Þ is significantly below the initial value yð0Þ ¼ 0 and when exponential growth clearly develops before the initial cell count is recovered, it might be more appropriate ^ t^‘ Þ ¼ yðt0 Þ. This to define t^‘ as the time when the line yð suggests the alternative formula for the lag time: t^‘ ¼ t
-4
t0
t
t‘
t^‘
12.9 362
54.2 537
26.7 684
20.5 395
200
400
600 800 t (hours)
1000
1200
1400
Fig. 3. Growth curve yðtÞ ¼ ln NðtÞ=N 0 versus time t. Data were obtained from ComBase data file B259_148 (www.combase.cc) for Listeria monocytogenes in culture medium at 10 C, pH 7.0 and 10% NaCl concentration. The solid curve plots Eq. (30) with parameter values r1 ¼ 0:02, r2 ¼ 0:023, A ¼ 1:5 107 , K ¼ 16.
times are computed and given in Table 1. The parameters r1 ¼ 0:02, r2 ¼ 0:023, A ¼ 1:5 107 and K ¼ 16 were found using a weighted least-square regression in programming language R and by an iterative process of experienced guesses. Data near the minimum were more lightly weighted to obtain more realistic fitting of the initial death, exponential growth and stationary portions of the data. Although the minimum population yðt0 Þ is underestimated, the curve does provide a realistic estimate of the time t0 where the minimum occurs and the lag times t‘ and t^‘ . Other curve fitting techniques or model equations can be used to provide better overall fits to the data, however our aim is to understand the cellular mechanisms leading to the population decline and to quantify the lag phase. In Fig. 3, the inability of function (30) to follow the data near the minimum value is attributed to the assumption that the death rate r3 ¼ r1 in the limited growth model, and a more general model is needed to provide better agreement with the data. 5. Discussion and conclusion Three growth models were presented for microorganism growth curves. The compartmental growth model extends the model of Baranyi (1998) to include cellular death and includes more general initial conditions. We introduced the parameters a as a measure of the proportion of nonduplicating cells that become duplicating cells and b as the fraction of initial cells that are non-duplicating. This more general model accounts for the death of a fraction ð1 aÞ of the non-duplicating cells, and predicts the decline of total cell count during the initial adjustment period which is followed by exponential growth. The specific range of parameters were also identified when population decline occurs ðr0 4rÞ and when monotonic growth occurs ðr0 prÞ. The duration of the lag phase l was found to depend on a
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and b with the general result that l increases when a is decreased and b is increased. An alternative lag time l^ or t^‘ was suggested when substantial population decline occurs prior to exponential growth, and predicts a shorter lag time than l. An independent stochastic model was developed based on the physiological state of individual cells. This is a necessary consideration for small populations and the results agree with the compartmental model in the limit of large populations. Finally, a limited growth model was proposed when environmental factors lead to a stationary phase in the growth curve. Although simple formulas for the lag phase are possible only for special cases of the parameters numerical solutions are routine using standard computational methods. The solution behavior during initial phases and the dependence on parameters follow closely the compartmental growth model. Subsequent computation of the lag phase duration requires modification to identify the point of maximum rate of growth. Two lag phase times are identified, one using a more traditional approach starting when the exponential phase is extrapolated back to the initial population and a second by extrapolating to the minimal population point. Comparisons between the model and observed data are generally very good, and support the traditional explanation of the cellular activity during various phases of the growth curve. The current limited growth model does not capture the detailed behavior near the minimum population for the example presented in Fig. 3. The limited growth model assumes that the death rate and conversion rate are equal, and it is postulated that this is not the case for this example. Further research is needed to resolve this shortcoming. Acknowledgments
environment. The parameter A ranges from 0 to 1 and is a combined measure of the initial mix of duplicating and non-duplicating cells and of the proportion of nonduplicating cells that convert into duplicating cells. Analysis of the various phases of the growth curve is done by considering the graph of yðtÞ ¼ ln xðtÞ. Finding the local minimum and the maximum slope of the growth curve requires the first and second derivatives y0 ðtÞ ¼
x0 ðtÞ , xðtÞ
y00 ðtÞ ¼
xðtÞx00 ðtÞ ½x0 ðtÞ2 , x2 ðtÞ
(A.2)
where x0 ðtÞ ¼ r1 ð1 AÞer1 t þ
r2 KðK=A 1Þer2 t , ½1 þ ðK=A 1Þer2 t 2
x00 ðtÞ ¼ r21 ð1 AÞer1 t
r22 KðK=A 1Þer2 t ½1 ðK=A 1Þer2 t . ½1 þ ðK=A 1Þer2 t 3
ðA:3Þ
The times t0 and t are defined, respectively, by solving x0 ðt0 Þ ¼ 0,
(A.4)
xðt Þx00 ðt Þ ¼ ½x0 ðt Þ2 .
(A.5)
Before attempting to solve these equations for t0 and t , we observe that xð0Þ ¼ 1;
lim xðtÞ ¼ K,
t!1
r2 AðK AÞ ; K Note that x0 ð0Þo0 when x0 ð0Þ ¼ r1 ð1 AÞ þ
R0 ¼
lim x0 ðtÞ ¼ 0.
t!1
Kð1 AÞ 4r, AðK AÞ
(A.6)
Discussions with Jonathan Caguiat, Chester Cooper and Carl Johnston greatly helped keep our assumptions biologically reasonable. The summer projects of David Gohlke and David Martin stimulated this work. This work was supported by the National Science Foundation Grant DUE 0337558. The support of Youngstown State University in the form of a Research Professorship for G.T.Y. and T.S. is gratefully acknowledged.
where r ¼ r2 =r1 . When R0 4r, the population initially decreases before increasing to a final value of K41 as t ! 1. In this case, the growth curve has a minimum value at t0 . Finding t0 requires solving the equation
Appendix A
where
We assume that the normalized population xðtÞ ¼ NðtÞ=N 0 is given by the sum of an exponential death and a logistic growth function:
B0 ¼
xðtÞ ¼ ð1 AÞer1 t þ
K , 1 þ ðK=A 1Þer2 t
(A.1)
where N 0 is the initial population. The combined conversion/death rate r1 and the growth rate r2 are both taken positive and K is the carrying capacity for the specific
er1 t0 ¼
rB0 er2 t0 , ½1 þ ðK=A 1Þer2 t0 2
KðK AÞ . Að1 AÞ
Unless r1 is equal to r2 or an integer times r2 , this equation can only be solved numerically for t0 . Eq. (A.5) for t can be written as z1=r ð1 þ zÞ½ð1 þ rÞ2 z2 þ ð2 þ 2r r2 Þz þ 1 ¼
r2 zKðK=A 1Þ1=r , 1A
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where K 1 er2 t , z¼ A which must also be solved numerically for z in the general case, to obtain t . Taking r1 ¼ r2 or r ¼ r2 =r1 ¼ 1, these equations simplify somewhat, giving 2 K 1þ 1 er2 t0 ¼ B0 A which gives 1 K A t0 ¼ ln pffiffiffiffiffiffi . r2 A B0 A
(A.7)
A second solution is discarded since it is always negative. The solution for t reduces to solving 4z3 þ 7z2 þ 4z þ 1 B0 ¼ 0. We note that for K41 and 0oAo1, B0 must be greater than 1. It can be shown that this cubic equation always has one and only one positive real solution when B0 41. Thus, exactly one t always exists. References Baranyi, J., 1998. Comparison of stochastic and deterministic concepts of bacterial lag. J. Theor. Biol. 192, 403–408.
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