## Evaluation of the Lag Duration

The next objective is to look for a relationship between the lag and maximum specific growth rate. The classical definition of lag is presented in Figure 2 by the value of t at the intersection of the tangent to the growth curve at the LIP (line 'b') with the straight line parallel to the t-axis starting atyo (line 'a'), that is, Acl in Figure 2. The solution expressed by eqn [41] leads asymptotically toward the tangent line to the growth curve at the LIP (line 'b'). The equation of this line is obtained by setting t in the exponential term e_mmaxt leading to y = yo + ln

The straight line 'c' in Figure 2 is represented by part of eqn [47], that is, y = yo + Mmax t. As the logarithmic term is negative, that is, ln(zo/Mmax) < 0, because zo< Mmax, the actual line 'b' tangent to the growth curve at the LIP point, represented by eqn [47], is shifted to the right with respect to 'c' as described in Figure 2. The intersection (t = Acl) of line 'b', eqn [47], with line 'a', is obtained by substituting y=yo into [47] leading to

Mmax

Substituting [48] into [47] yields the solution for line 'b' in the form

which is the classical representation of the lag. Equation [48] is identical to a corresponding equation given by Baranyi and Roberts when we use the relationship between qo and xo, xo that was derived here and presented in eqn [34] and the condition xo ^ 6. However, now following this substitution the relationship between the lag duration, Acl, and the maximum specific growth rate, max z o

^max, is slightly changed. Vadasz and Vadasz showed that when zo ^ ^max, eqn [48] produces the familiar inversely proportional relationship result Aci — C/^max, where C is a constant (i.e., independent of ^max). However, as soon as the value of ^max becomes close to zo, the logarithmic term introduces a substantial effect as demonstrated graphically in Figure 16. The figure represents eqn [48] on a log-log scale, that is, log(Ad) — log[ln(^max/zo)] -log(^max). As observed from Figure 16, the previous analysis results are being confirmed, that is, as long as zo ^ ^max, a linear approximation revealing the inversely proportional relationship between the lag duration, Aci, and the maximum specific growth rate, ^max, is valid; however, for values of ^max that are closer to zo the linear approximation (i.e., the original inversely proportional relationship) breaks down.

A new, more natural (rather than artificial), and biologically meaningful formulation of lag duration is now appealing as a result of revealing the existence of a set of unstable stationary points presented in eqn [45] that depend on the initial conditions. The suggested new lag definition is introduced because the classical definition is just an arbitrary geometrical convenience without any biological significance. The new proposed definition has a profound (and accurate) biological meaning and significance in terms of the initial cell concentration and initial growth rate, that is, their values being in the neighborhood of one of the unstable stationary points. By referring to the unstable stationary points defined by eqn [45] in the form xs — xo(1 — zo/^max) we redefine the lag duration Ard as the amount of time that elapses until the cell

-m- |
Zo |
= 5 x 10-7 |

—0— |
Zo |
= 5 x 10-6 |

—•— |
Zo |
= 5 x 10-5 |

Zo |
= 5 x 10-4 | |

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

Figure 16 The relationship between the lag and the maximum specific growth rate, ^max, on a log-log scale. An inversely proportional relationship is revealed as long as zoxyo < ^max. The units of zo are [s_1j. Reproduced from Vadasz P and Vadasz AS (2006) Biological implications from an autonomous version of Baranyi & Roberts growth model. International Journal of Food Microbiology 114: 357-365, with permission from Elsevier.

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

Figure 16 The relationship between the lag and the maximum specific growth rate, ^max, on a log-log scale. An inversely proportional relationship is revealed as long as zoxyo < ^max. The units of zo are [s_1j. Reproduced from Vadasz P and Vadasz AS (2006) Biological implications from an autonomous version of Baranyi & Roberts growth model. International Journal of Food Microbiology 114: 357-365, with permission from Elsevier.

concentration x reaches a value which is, by a certain percentage, above the unstable stationary point xs, that is, xA — Pxs, P > 1. An alternative definition in terms of y is: yA — 7ys, 7 >1. Substituting for xs from eqn [45] leads to xa = ßxj 1 -

However, the solution for any x, and in particular for xA, is given in the short term by eqn [42], that is,

^max

Mmax Arc

By equating [50] to [51] produces the following result for the lag duration Ard, which is consistent with Baranyi and Roberts growth model:

^max

The new definition overlaps with the classical one if we equate the right-hand sides of eqns [52] and [48] leading to a value of P for which both definitions produce an identical result in the form

As an example, we shall use now the experimental data from O'Donovan and Brooker that we presented in Figure 4b, which correspond to the parameters ^max — 0.7 X 10_3 s_1, xo — 1.1458 X 105cfuml_1 and Xo — 1.4 X 10"2cfu(mls)_1 consistent with Baranyi and Roberts model. The latter produces a value of zo — 1.2219 X 10_7s_1. Substituting these data into eqn [48] yields the value for the classical lag Aci — 3.434 h. The value of P evaluated by substituting the same parameter data into eqn [53] is P — 2.0002. With this value of P the lag based on the new definition [52] is identical to the classical definition, that is, Ard — 3.434 h. Alternatively, the same data produce a value of 7 — 1.0595. The meaning of the latter is that the value of the cell concentration at the lag point is by about 100% above the unstable stationary value of xs or by about 5.95% above the unstable stationary value of ys. The equivalence between the classical and the new definitions ofthe lag is not necessarily required. The new definition of the lag is linked to the fact that the solution moves away from an unstable stationary point asymptotically in a similar fashion as it moves toward a stable stationary point. It is impossible to allocate a set value to the time needed for the solution to reach the stationary phase. All one can do is to set a time value for the solution to get close up to a certain percentage (say 99%) of its stationary phase (steady state) value. Similarly, it is impossible to allocate a set time value for the solution to depart from an unstable stationary point. All that can be done is to set a time value for the solution to get by a certain percentage away from its corresponding unstable stationary point. The latter may be done in terms max max of the cell concentration x, hence the corresponding definition of fl, or in terms of y = ln(x), hence the alternative definition of 7. Practically, for any set of experimental data of growth taken as cell count versus time one may evaluate the lag duration by using eqn [52] for a set value of fl. If one insists that the lag duration be identical to the classical definition then the choice of fl has to be evaluated by using eqn [53]. If the experimental data are taken at a sufficiently high sampling frequency, one may evaluate also the growth rate X followed by plotting the specific growth rate z = X/x versus the cell count x to produce a phase diagram that reveals information which is otherwise concealed.

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