## The Phase Diagram

An important tool for analyzing microbial growth is obtained by plotting the growth data on a phase diagram. The latter provides information of substantial biological significance, which is impossible or very difficult to extract directly from the growth curve. A qualitative description of such a phase diagram is presented in Figure 4 (where the x-axis representing the cell concentration is presented on a logarithmic scale). While Figure 4a presents qualitatively a phase diagram obtained

Figure 4 (a) Qualitative description of a phase diagram for typical microorganisms growth based on theoretical models such as Baranyi and Roberts (1994) or Vadasz and Vadasz (2005): (b) Qualitative description of a phase diagram for typical microorganisms growth based on experimental data of O'Donovan and Brooker (2001). 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.

from theoretical models, Figure 4b presents a corresponding phase diagram obtained from the experimental data of O'Donovan and Brooker. Let us compare the information that can be extracted from the growth curve presented in Figure 2 with the information revealed in the phase diagrams presented in Figure 4. The cell concentration at any given time is obviously the most important information that can be easily extracted from Figure 2. However, the information about the specific growth rate is represented by the slope of the curve and is difficult to extract from Figure 2. In addition, the source of the lag is concealed in Figure 2. On the other hand, the phase diagram in Figure 4 presents the specific growth rate versus the cell concentration, a relationship that can be usually (in the case of autonomous models) plotted directly from the equation without the need to solve it. The latter occurs because a typical autonomous equation governing microbial growth has the form x/x=fx), where the left-hand side represents the specific growth rate and the right-hand side is a known function of the cell concentration. Then, as can be easily observed from Figure 4, the specific growth rate can be extracted directly from this diagram even prior to solving the equation. In most cases, if the experimental data are available at a sufficiently high sampling frequency the experimental phase diagram provides useful information about the specific growth rate. Since above the x-axis the specific growth rate is positive, x/x > 0, it implies that along a curve above the x-axis on the phase diagram the values of x increase in the positive time direction. Therefore, the positive time direction is in the direction of increasing values of x as described by the arrows in Figure 4a. In addition, the maximum of this curve represents an LIP that its value can be easily extracted from the phase diagram. Lastly, additional information that is biologically extremely important and can be extracted from the phase diagram is related to the lag. Obviously, the lag is linked to very small values of specific growth rate.

Therefore, since the small values of x/x are located at the lower part of the diagram near the x-axis it implies that a lag will exist if the initial data points are located close to the x-axis. The importance of using phase diagrams can be summarized in the following: (1) the specific growth rate is directly revealed, (2) the arrows' direction represents the time direction, (3) regions close to the x-axis have small specific growth rates identifying the neighborhood of the lag or stationary phases, and (4) maxima on the curves on a phase diagram represent LIP. This tool of presenting results on phase diagrams will be used in this article to extract the significant biological information and interpret it.

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