Results from the Neoclassical Model

The analytical closed-form solution linked with the computational procedure described above was used by Vadasz and Vadasz to produce results that demonstrate the natural occurrence of an LIP and a lag phase. The impact of the parameters Mmax, rm and of the initial conditions xo, xo on the lag duration A, and on the LIP location in time, tLIP, was also investigated from these solutions.

The results of the solution in the time domain showing the effect of the maximum specific growth rate on the LIP as well as on the lag, obtained by using eqn [20] for different values of Mmax ranging from Mmax — 0.3 x 10- s~ and up to Mmax — 2 x 10~3s_1 are presented in Figure 6 corresponding to parameter values of rm — 10- (cell/ml)-1

2 x 10-3

1 .

1.3 x 10-3

10-3

0.7 x 1C

-3

^max

= 0.3 X 10"

3s -1

I 0.5 x 10-3

Figure 6 The effect of the maximum specific growth rate, ßmax, on LIP and lag. Analytical results in the time domain based on the solution [20] to eqn [9] for rm = 10"5 (cell/ml)"1, 6 = 1.51 x 108

cell ml cell ml subject to initial conditions of xo = 1.1458 x 105

xo = 7.5 x 10-3 cell (ml s)_1, and different values of mmax ranging from ^max = 0.3 x 10_3s_1 and up to ^max = 2 x 10_3s_1. Reproduced from Vadasz P and Vadasz AS (2005) Predictive modeling of microorganisms: LAG and LIP in monotonic growth. International Journal of Food Microbiology 102: 257-275, with permission from Elsevier.

x m max m x o o and S = 1.5 x 108cell ml"1. The accurate location of the lag and LIP is presented in the figure by the indicated points. As indeed expected, the smaller the maximum specific growth rate ^max the longer the lag duration A and the larger the value of tLIP.

Deriving a relationship showing how the LIP as well as the lag depend on the maximum specific growth rate ^max becomes appealing. Baranyi and Roberts indicate that the lag duration is inversely proportional to the maximum specific growth rate ^max. This implies that

where co is a constant, and after applying the log function on both sides of eqn [23] it can be presented in the form log A = log Co - log

showing that such an inversely proportional relationship should appear as a straight line on a log-log scale. Vadasz and Vadasz tested the neoclassical model solution to reveal whether it captures this fact, as well as the relationship between the LIP and the maximum specific growth rate. They used a wide variation of values of ^max over three orders of magnitude and evaluated the LIP time tLIP, producing the results presented on a log-log scale in Figure 7a for three different values of rm.

The results clearly indicate that the LIP time, tLIP, is accurately related as inversely proportional to the maximum specific growth rate ^max. The corresponding results for the lag duration using the present lag definition presented in Figure 7b for three different values of rm also show that the lag duration is inversely proportional to the maximum specific growth rate ^max. The latter is an approximation as observed by the very slight deviations of the dotted line (representing the model's results) from the straight line (continuous) in Figure 7b. The impact of the initial growth rate at inoculation on this inversely proportional relationship between the LIP time tLIP or lag duration A and maximum specific growth rate is presented in Figure 8, where it is evident that this relationship generally holds when zo ^ ^max, but when zo is closer to ^max it breaks down.

The effect of the viable biomass parameter rm on the solution was investigated by evaluating the solution for initial conditions of xo = 1.1458 x 10 cellml"1, Xo = 7.5 x 10"3 cell (ml s)"1, and different values of rm, ranging from rm = 5 x 10" (cell/ml)"1 and up to rm = 3 x 10" (cell/ml)"1.

These results are presented in Figure 9 corresponding to parameter values of ^max = 3 x 10" s"1 and S = 1.5 x 108cell ml"1. It is evident from Figure 9 that the lag duration A decreases as rm increases; however, this trend converges asymptotically to a constant nonzero value, which seems to depend on the initial conditions. In the case presented in Figure 9, this asymptotic value was slightly less than A = 5h.

The effect of the initial growth rate, X_o, on the solution is presented in Figure 10, corresponding to an initial cell concentration of xo = 1.1458 x 10 cell ml"1 and to the following parameter values rm = 10" (cell/ml)"1, Mmax = 3 x 10"4s"1 and S = 1.5 x 108cellml"1. The initial growth rate used varied from Xo = 7.5 x 10"4 cell (ml s)"1 to Xo = 50 cell (mls)"1. The anticipation that the lag duration is reduced as the initial growth rate increases is

o max

0.01

—o— rm = 2 x10-8 (cell/ml) - m = 10-6 (cell/ml) -1

0.0001 0.001 0.01 0.1 1 Maximum specific growth rate, ^max (s-1)

cd E

0.01

0.0001 0.001 0.01 0.1 1 Maximum specific growth rate, ^max (s-1)

0.01

0.0001 0.001 0.01 0.1 1 Maximum specific growth rate, ^max (s-1)

Figure 7 The relationship between the maximum specific growth rate, ^max, and LIP and lag based on the analytical results obtained from the solution [20] to eqn [9]. (a) The relationship between the maximum specific growth rate and LIP for three different values of rm. (b) The relationship between the maximum specific growth rate and lag for three different values of rm. Reproduced from Vadasz P and Vadasz AS (2005) Predictive modeling of microorganisms: LAG and LIP in monotonic growth. International Journal of Food Microbiology 102: 257-275, with permission from Elsevier.

0.01

0.01

■ n ■

1

• xo = 10-3 cell (ml s)-1 - xo = 10-1 cell (ml s)-1

□ . : > *

n xo = 1 cell (ml s)-1 - xo = 10 cell (ml s)-1

: □

in: i ■ * :

a : ■ : •

: ■ • :

i □ : " i *

1 B .

i i I ■

0.000 1 0.001 0.01 0.1 1 10 Maximum specific growth rate, ^max (s-1)

0.01

□ " • • '

• xo = 10-3 cell (ml s)-1

□ -1

• xo = 10-1 cell (ml s)-1

■ ■ : ° - : ■ n ■

□ xo = 1 cell (ml s)-1

_ 9

■ xo = 10 cell (ml s)-1

0.000 1 0.001 0.01 0.1 1 10 Maximum specific growth rate, ^max (s-1)

0.000 1 0.001 0.01 0.1 1 10 Maximum specific growth rate, ^max (s-1)

Figure 8 The relationship between the maximum specific growth rate, ^max, and LIP and lag based on the analytical results obtained from the solution [20] to eqn [9]. (a) The relationship between the maximum specific growth rate and LIP for four different values of the initial growth rate xo. (b) The relationship between the maximum specific growth rate and lag for four different values of the initial growth rate xo. Reproduced from Vadasz P and Vadasz AS (2005) Predictive modeling of microorganisms: LAG and LIP in monotonic growth. InternationalJournal of Food Microbiology 102: 257-275, with permission from Elsevier.

Figure 9 The effect of the viable biomass parameter rm on LIP and lag. Analytical results based on the solution [20] to eqn [9] for Mmax = 3 x 10-4s-1, 8 = 1.5 x 108 cell ml"1, subject to initial conditions of xo = 1.1458 x 105 cell ml-1, xo = 7.5 x 10-3 cell (ml s)-1, and for different values of rm, varying from rm = 5 x 10-6 (cell/ml)-1 and up to rm = 3 x 10-4 (cell/ml)-1. Reproduced from Vadasz P and Vadasz AS (2005) Predictive modeling of microorganisms: LAG and LIP in monotonic growth. International Journal of Food Microbiology 102: 257-275, with permission from Elsevier.

Figure 9 The effect of the viable biomass parameter rm on LIP and lag. Analytical results based on the solution [20] to eqn [9] for Mmax = 3 x 10-4s-1, 8 = 1.5 x 108 cell ml"1, subject to initial conditions of xo = 1.1458 x 105 cell ml-1, xo = 7.5 x 10-3 cell (ml s)-1, and for different values of rm, varying from rm = 5 x 10-6 (cell/ml)-1 and up to rm = 3 x 10-4 (cell/ml)-1. Reproduced from Vadasz P and Vadasz AS (2005) Predictive modeling of microorganisms: LAG and LIP in monotonic growth. International Journal of Food Microbiology 102: 257-275, with permission from Elsevier.

confirmed by these results. Furthermore, only five of the eight curves produce a lag. The curves corresponding to initial growth rates larger than or equal to 1 cell (mls)-1, that is, x_o>1 cell (mls)-1, have no lag, while the last curve corresponding to xo = 50 cell (ml s)- shows no LIP as well.

Figure 10 The effect of the initial growth rate xo on LIP and lag. Analytical results based on the solution [20] to eqn [9] for rm = 10"5 (cell/ml)"1, Mmax = 3 X 10"4s"1, S = 1.5 x 108 cell/ml, subject to the initial condition of xo = 1.1458 x 105 cell/ml, and for different values of xo, ranging from xo = 7.5 x 10"4 cell (ml s)"1 and up to xo = 50 cell (mls)"1. Reproduced from, Vadasz P and Vadasz AS (2005) Predictive modeling of microorganisms: LAG and LIP in monotonic growth. International Journal of Food Microbiology 102: 257-275, with permission from Elsevier.

Figure 10 The effect of the initial growth rate xo on LIP and lag. Analytical results based on the solution [20] to eqn [9] for rm = 10"5 (cell/ml)"1, Mmax = 3 X 10"4s"1, S = 1.5 x 108 cell/ml, subject to the initial condition of xo = 1.1458 x 105 cell/ml, and for different values of xo, ranging from xo = 7.5 x 10"4 cell (ml s)"1 and up to xo = 50 cell (mls)"1. Reproduced from, Vadasz P and Vadasz AS (2005) Predictive modeling of microorganisms: LAG and LIP in monotonic growth. International Journal of Food Microbiology 102: 257-275, with permission from Elsevier.

The effect of the initial cell concentration, xo, on the solution as evaluated by using the closed-form analytical solution [20] is presented on a phase diagram in terms of (X/x) versus x in Figure 11, corresponding to an initial growth rate of Xo = 1 cell (ml s)- and to the following parameter values rm = 5 x 10-4 (cell ml)-1, ^max = 3 x 10-4 j-1 and 6 = 1.5 x 108cellml-1. The initial cell

Figure 11 The effect of the initial cell concentration xo on LIP and lag on a phase diagram in terms of the specific growth rate x/x versus the cell concentration x. Analytical results based on the solution [20] to eqn [9] for rm = 5 x 10-4 (cell/ml)-1, ^max = 3 x 10-4s-1, 8 = 1.5 x 108 cell ml-1, subject to the initial growth rate condition of xo = 1 cell (mls)-1, and for different values of xo, ranging from xo = 1 x 107 cell ml-1 and up toxo = 6 x 107 cell ml-1. Reproduced from Vadasz P and Vadasz AS (2005) Predictive modeling of microorganisms: LAG and LIP in monotonic growth. International Journal of Food Microbiology 102: 257-275, with permission from Elsevier.

Figure 11 The effect of the initial cell concentration xo on LIP and lag on a phase diagram in terms of the specific growth rate x/x versus the cell concentration x. Analytical results based on the solution [20] to eqn [9] for rm = 5 x 10-4 (cell/ml)-1, ^max = 3 x 10-4s-1, 8 = 1.5 x 108 cell ml-1, subject to the initial growth rate condition of xo = 1 cell (mls)-1, and for different values of xo, ranging from xo = 1 x 107 cell ml-1 and up toxo = 6 x 107 cell ml-1. Reproduced from Vadasz P and Vadasz AS (2005) Predictive modeling of microorganisms: LAG and LIP in monotonic growth. International Journal of Food Microbiology 102: 257-275, with permission from Elsevier.

concentration used varied from xo = 1 x 107cellml-1 to xo = 6 x 10 cell ml-1. The lag and LIP location on this phase diagram as evaluated by the solution using eqns [21] and [22] is revealed by the black point markers.

The neoclassical definition of the lag duration is different from the classical definition primarily because the former is a direct consequence of the properties of the neoclassical model while the classical definition is an arbitrary geometric definition. The objective of this section is to show that our definition of lag is perfectly consistent with the classical one. The classical definition of lag is based upon the geometry of the growth curve presented in Figure 2. It is essentially the value of time at the intersection between the tangent line to the growth curve at (tLIP, yLIP) with the line y=yo, where by definition y = ln(x). The equation of the tangent line to the growth curve at (tLIP, yLIP) is obtained in the form y = yo + y_Lip(t- A) = yo + Mlip(^-A), or alternatively in terms of yLIP and tLIP in the form y=yLIP + y_Lip(t - tLip) = yLIP + MLIp(t - ¿Lip). From the first of these equations one can obtain the classical definition of lag in the form

By using now the definition of yULIP = yLIP = (X/x)LIP corresponding to the neoclassical model from eqn [13] we obtain the lag definition in the form

ftmax(rm8 2rmxLIP l) \

It may be shown analytically, although via a long derivation by using the closed-form solution of the neoclassical model, eqn [20], that the neoclassical lag definition [26] converges to the classical one [25] when rmS ^ 1 and xLIP ^ 6. These conditions are also consistent with ^max ~ ^LIP x (x/x)LIP. In order to check that our neoclassical definition of lag is consistent with the classical one even beyond the limits listed above we evaluated accurately the neoclassical lag duration according to the accurate eqn [22] and the accurate stationary points solutions [18]. The resulting values were compared to the classical definition of the lag evaluated by using eqn [25] and are presented in Figure 12. The wide range of variation of the lag was accomplished by varying the value of ^max over six orders of magnitude from ^max = 2 x 10-4s-1 up to Mmax = 1 x 10 s- . The lag threshold used in eqn [22] for its neoclassical definition was taken as about 1% above y3s = ln(x3s), which is equivalent to 17% above x3s, or b = 1.17. Figure 12 shows that the neoclassical definition of lag overlaps with its classical counterpart as their results lie on a straight line inclined at a 45° angle, identifying a complete overlap.

Lag

y

w

Ac|, Classical lag time (h)

20 40 60

Ac|, Classical lag time (h)

Figure 12 Comparison of the neoclassical values of the lag duration with the classical lag duration values obtained by a variation of ^max over six orders of magnitude from ^max = 2 x 10"4s"1 up to ^max = 1 x 102s"1. The value of b for evaluating the neoclassical lag was b = 1.17, that is, about 1% above y3s = ln(x3s), or 17% above the unstable stationary points x3s. The markers represent the evaluated results. Reproduced from Vadasz P and Vadasz AS (2005) Predictive modeling of microorganisms: LAG and LIP in monotonic growth. International Journal of Food Microbiology 102: 257-275, with permission from Elsevier.

subject to different concentrations of acacia condensed tannins (ACT) which affect substantially the lag phase. The latter represent microbial growth curves subject to different levels of stressed environmental conditions. The experimental data presented by O'Donovan and Brooker were digitized and used to test the model results. The trial-and-error approach for parameter estimation applied to each data set separately yielded four different curves for each of the data sets presented by them. Each curve is linked to three different parameter values and initial growth rates listed in the figure caption. The model results are presented in Figure 13 in comparison with the digitized data from O'Donovan and Brooker for each of the four different curves. It is evident from the results that a very good fit was obtained by using only three parameter values that control the monotonic version of the neoclassical model and without using a systematic solution to the inverse problem for parameter estimation. It is important to emphasize that all four data sets presented by O'Donovan and Brooker describe convex curves when plotted on the phase diagram in terms of X/x versus x, and therefore models such as Gompertz cannot capture them correctly because the Gompertz model produces concave curves on the phase diagram.

The second data set was based on the experimental data that Baranyi and Roberts used to test their model. The data which correspond to McClure et al. as well as Baranyi and Roberts' model results were digitized and used to test the neoclassical model. The results are

While the previous sections introduced results that show evidently that the neoclassical model captures all qualitative features that have been revealed experimentally, the most important test of this model is the direct comparison of its results with experimental data. To promote the objective of predictive modeling in microbiology, one should be in a position to indicate the values of the parameters once an experiment is undertaken. Unfortunately, the present status of predictive microbiology does not yet have the capability of indicating a priori the parameter values. The neoclassical model is therefore used to estimate the values of these parameters. A systematic way of doing so would require invoking the inverse problem for parameter estimation. The sensitivity of the parameters to data accuracy especially because of the need to provide measured data of growth rate and not only cell concentration, limit at this time the use of the latter systematic method for parameter estimation. Vadasz and Vadasz adopted a trial and error method for estimating the parameter values of rm, ^max, and S as well as the unknown value of the initial growth rate x_o. Two different experimental data sets were used for this purpose.

The first data set was selected from O'Donovan and Brooker showing the growth of Streptococcus gallolyticus

O (a) Experimental

10 15

Figure 13 Comparison of the neoclassical model analytical solution for monotonic growth based on Vadasz and Vadasz with the experimental data based on O'Donovan and Brooker (here redrawn from published data). The initial cell concentration is xo = 1.1458 x 105cfu ml"1 and the estimated parameter values are as presented by Vadasz and Vadasz. Reproduced from Vadasz P and Vadasz AS (2005) Predictive modeling of microorganisms: LAG and LIP in monotonic growth. International Journal of Food Microbiology 102: 257-275, with permission from Elsevier.

O (a) Experimental

10 15

Figure 13 Comparison of the neoclassical model analytical solution for monotonic growth based on Vadasz and Vadasz with the experimental data based on O'Donovan and Brooker (here redrawn from published data). The initial cell concentration is xo = 1.1458 x 105cfu ml"1 and the estimated parameter values are as presented by Vadasz and Vadasz. Reproduced from Vadasz P and Vadasz AS (2005) Predictive modeling of microorganisms: LAG and LIP in monotonic growth. International Journal of Food Microbiology 102: 257-275, with permission from Elsevier.

/ •

Model solution (Vadasz and Vadasz, 2002) Experimental data ( McClure et al., 1993)

200 300

Figure 14 Comparison of the neoclassical model analytical solution for monotonic growth based on Vadasz and Vadasz with the experimental data based on McClure etal. (here redrawn from published data). The estimated parameter values are Mmax = 1.6 x 10~5s_1 = 0.0576 h"1, <5 = 2.33 x 108 (cell ml)"1, rm = 5 x 10"4 (ml cell)"1. Reproduced from Vadasz P and Vadasz AS (2005) Predictive modeling of microorganisms: LAG and LIP in monotonic growth. International Journal of Food Microbiology 102: 257-275, with permission from Elsevier.

constant in Monod's substrate model, and Q(t) is the concentration of a 'critical substance' that acts as a 'bottleneck' in the growth process. The Newtonian time derivative notation x = dx/dt (or generally (:) = d()/dt) was introduced in eqn [27] for simplicity. We will swap between the Newtonian time derivative notation x: and the Leibnitz notation dx/dt for simplicity and clarity of presentation. Whenever the time derivative acts on a combined expression we prefer the Leibnitz notation, in most of the other circumstances Newtonian notation is simpler. The parameters ^max and 6 represent the maximum specific growth rate and the carrying capacity of the environment, respectively. In eqn [28] a different coefficient v was originally introduced by Baranyi and Roberts but eventually they recommended using v = ^max, based on biochemical arguments. This recommendation was followed in all subsequent applications. We will therefore adopt this relationship v = ^max too.

In food microbiology the use of the logarithm of the cell concentration is preferred and the following notation is then applied: y = ln(x) leading to z = y = (x/x)

Substituting these definitions into the Baranyi and Roberts model eqn [27] yields y —

presented in Figure 14 showing that the neoclassical model using the three generic model parameters (rm, ^max, and 6) fits very well the experimental data. The estimated value of ^max is 0.0576 h"1 compared to the value 0.058 estimated by Baranyi and Roberts.

In addition, the neoclassical model was shown (see Figure 8b) to capture the inversely proportional relationship between the lag duration and the maximum specific growth rate as part of the solution as well as producing a complete fit with the classical definition of the lag.

As q(t) is a function of time, the coefficient in front of the right-hand side of eqns [27] or [30], that is, the adjustment function q(t)/[1 + q(t)] is a function of time too. The model is therefore nonautonomous, a fact that conceals some of its salient properties. To reveal the latter, one aims at deriving an equivalent autonomous version of the model.

Oplan Termites

Oplan Termites

You Might Start Missing Your Termites After Kickin'em Out. After All, They Have Been Your Roommates For Quite A While. Enraged With How The Termites Have Eaten Up Your Antique Furniture? Can't Wait To Have Them Exterminated Completely From The Face Of The Earth? Fret Not. We Will Tell You How To Get Rid Of Them From Your House At Least. If Not From The Face The Earth.

Get My Free Ebook


Post a comment