Introduction

The wide range of applications of microorganism growth in ecological modeling such as soil microbiology or wastewater treatment are just examples motivating the practical importance of the topic, adding to the already essential theoretical motivation that spans over a large number of disciplines. The diversity of models being used for predictive growth in microbiology and microbial ecology are being reviewed. Historically, the first attempt to propose a predictive model for population dynamics was due to Malthus (1798), followed by Verhulst (1838), who introduced the 'logistic growth model' (LGM) and Pearl (1927), who showed that the LGM can reproduce accurately experimental growth data in some circumstances. The general form of the LGM is x = ^max(1 - x/6)x, where x represents the cell count (or cell concentration), Mmax is the maximum specific growth rate, 6 is the carrying capacity of the environment, and x is the growth rate. The Newtonian notation (:) = d(-)/dt was introduced here for simplicity, where t represents the time. 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.

Nevertheless, it is widely agreed that the lag phase, a typical phase in microorganisms growth, cannot be estimated accurately by most existing models. Additionally, most of the existing models cannot capture typical phases or growth regimes such as growth followed by decay (decline/death phase) or oscillations. The existence of a decay phase where the cell count declines is a typical phase in batch growth, which has been well documented experimentally and is included in most textbooks as a classical growth phase. Reports on oscillations in microbial growth are less common but Vadasz et al. presented both experimental evidence as well as theoretical arguments demonstrating the existence of damped oscillatory growth even for a single species, that is, a particular strain of yeast. Why are all these details important and relevant to microbial ecology and why is modeling needed for investigating this field? While the original population dynamics models of Malthus (1798) and later Pearl (1927) seem to be influenced by the parallel discoveries in physics by Newton's in Principia (1687), the acceptance of modeling as a necessary component of research in the field of ecology was less enthusiastic. Pearl's (1927) LGM faced substantial criticism and was not widely accepted, mainly because it represented a very special case of microbial growth. The circumstances and arguments laid down at Pearl's door were almost identical to those raised against Galileo Galilei (1632) by the supporters of the Aristotelian viewpoint. We will attempt to introduce the reasons for the controversies in physics and their eventual resolution while focusing on the impact these similarities have on the field of microbial ecology. As an example let us consider the logistic (LGM) solution for the growth of yeast cells based on Pearl (1927) compared with experimental data reported by Carlson (1913). We have redrawn Pearl's tabulated data in Figure 1a, clearly confirming his claim of an excellent match. In addition, plotting the natural logarithm of the cell concentration versus time yields the graphical result presented in Figure 1b. From this figure it is clearly observed that the regular inflection point (RIP) that can be identified in Figure 1a at x = 0.56 disappears when presenting the result in terms of ln(x) versus time. The LGM cannot capture a logarithmic inflection point (LIP). The significance of the existence of an LIP is essential in linking the latter to the appearance of a lag in the growth process of some populations as presented for a typical microorganism growth curve in Figure 2. In the limit of short times and if xo/6 ^ 1 the LGM equation above reduces to Malthus's equation X = fix with ^max = ft, which produces the solution x = xoeMmait or in terms of the natural logarithm ln(x/6) = ln(xo/6) + ^maxt. The

Qualitative growth curve

Regular inflection point at: (xls) = 0.5

Experimental data, Carlson (1913) — Analytical (LGM), Pearl (1927) --- Analytical, Malthus (1798)

Regular inflection point at: (xls) = 0.5

Experimental data, Carlson (1913) — Analytical (LGM), Pearl (1927) --- Analytical, Malthus (1798)

Stationary limit

5 10

Figure 1 Comparison of the LGM solution with experimental results for yeast cells based on Pearl after data from Carlson (1913) (here redrawn using the tabulated data from Pearl, 1927). (a) Cell concentration (normalized and dimensionless) versus time; (b) the logarithmic curve of the cell concentration versus time. Reproduced from Vadasz P and Vadasz AS (2002) The neoclassical theory of population dynamics in spatially homogeneous environments - Part I: Derivation of universal laws and monotonic growth. PhysicaA 309(3-4): 329-359. Vadasz P and Vadasz AS (2002) The neoclassical theory of population dynamics in spatially homogeneous environments - Part II: Nonmonotonic dynamics, overshooting and oscillations. PhysicaA 309(3-4): 360-380, with permission from Elsevier.

Stationary limit

LGM, Pearl (1927) No LIP

No logarithmic inflection point ! |

Experimental data, Carlson (1913) — Analytical (LGM), Pearl (1927) Analytical, Malthus (1798)

5 10

Figure 1 Comparison of the LGM solution with experimental results for yeast cells based on Pearl after data from Carlson (1913) (here redrawn using the tabulated data from Pearl, 1927). (a) Cell concentration (normalized and dimensionless) versus time; (b) the logarithmic curve of the cell concentration versus time. Reproduced from Vadasz P and Vadasz AS (2002) The neoclassical theory of population dynamics in spatially homogeneous environments - Part I: Derivation of universal laws and monotonic growth. PhysicaA 309(3-4): 329-359. Vadasz P and Vadasz AS (2002) The neoclassical theory of population dynamics in spatially homogeneous environments - Part II: Nonmonotonic dynamics, overshooting and oscillations. PhysicaA 309(3-4): 360-380, with permission from Elsevier.

latter solution representing Malthus limit is presented in Figure 1. Similarly, the stationary limit obtained for long times, when t n, is also presented in Figure 1. Malthus's and the stationary solutions represent the two limits of the LGM. It is interesting to compare these results with the results obtained from classical physics for the Newtonian solution of a falling mass subject to gravity and friction drag forces as presented in Figure 3. The Galilean limit is obtained for short time solutions,

Stationary phase Inhibition phase

LIP

^hmseattmi

"l

Exponential

i < i

logarithmic

I ' I

phase

Lag phase i V7 i

Ard cl t

Time, t

Figure 2 Qualitative description of a typical growth curve for microorganisms. 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.

Newtonian, Galilean, and Aristotelian solutions

- Newtonian

----Aristotelian

Galilean

Newtonian, Galilean, and Aristotelian solutions v = vT (Aristotelian limit) - - - - 'f - - -

- Newtonian

----Aristotelian

Galilean

Figure 3 Newtonian solution for a falling mass subject to gravity and friction drag, compared with the Galilean and Aristotelian limits for a drag coefficient to mass ratio (c/m) = 1s"1 and g = 9.81 m s"2.

Figure 3 Newtonian solution for a falling mass subject to gravity and friction drag, compared with the Galilean and Aristotelian limits for a drag coefficient to mass ratio (c/m) = 1s"1 and g = 9.81 m s"2.

conditions when t ^ m/c, where m/c is the mass-to-damping-coefficient ratio. The Aristotelian limit is obtained for long times, when tn. One of the controversies between the Aristotelian and Galilean viewpoints was related in principle to the arguments supporting the Aristotelian or the Galilean limits, respectively. With hindsight it is obvious that they were both correct but they differ only on the validity range of their solutions. To be fair to the Galilean viewpoint, the latter also supported Galileo's pioneering use of quantitative experiments with results analyzed mathematically, an approach b

that was not acceptable to the Aristotelian viewpoint. While the comparison between Figures 1b and 3 shows stunning similarities and the controversies that arose in connection with the LGM seem quite similar to the arguments raised between the Aristotelian and Galilean viewpoints, this article shows that the situation in microbial ecology is a bit more complicated.

Excellent reviews of the methodology developed in the field of predictive microbiology are presented by McMeekin etal., and McMeekin and Ross. They mention, among others, the importance of predictive modeling by using mechanistic models because mechanistic models are derived from theoretical principles, provide interpretation of the underlying mechanisms and can be refined as knowledge of the system increases. We prefer to refer to such models that are linked to some underlying biological mechanisms as 'generic', rather than 'mechanistic'. In their review McMeekin et al. also indicate that the lag time duration seems erratic; lag times are less reliably predictive by existing models. This has usually been attributed to the effect of the prior history of cells on the duration of the lag time. Swinnen et al., and Baty and Delignette-Muller presented additional outstanding reviews of predictive modeling of the microbial lag phase. We counted at least 20 different models that are being used in predictive microbiology or in the field of ecology to describe population growth (14 of them being consistently used in the last decade). The 20 models we refer to are: Malthus, LGM, Gompertz, Allee, Richards, Baranyi, Smith, model with delay, structured models, age or otherwise, bi-logis-tic, generalized logistic, generalized Gompertz, modified logistic, varying carrying capacity logistic, modified Gompertz, three-phase linear, Monod model, Ginzburg model, von Bertalanffy, and unified generic growth model. A large number of the models listed above are descriptive rather than predictive.

The major problem in developing and using a model in a predictive manner is to demonstrate that the model can capture at least qualitatively all the features that are being revealed in experimental data. When focusing on microbial growth, a typical growth curve in terms of the logarithm of the cell concentration versus time consists of the following phases as described in Figure 2: phase 1 - lag phase (which may or may not appear), phase 2 - logarithmic exponential growth phase (which may or may not appear, e.g., experiments that follow logistic growth will not exhibit a logarithmic exponential growth), phase 3 -inhibition phase, phase 4 - stationary phase, phase 5 -decay (or decline/death) phase (not shown in Figure 2). The curve presented in Figure 2 represents monotonic growth. A monotonic growth curve is defined as any growth curve that does not reach a maximum, nor a minimum, except at t = 0 or asymptotically as tn. Examples of nonmonotonic growth are followed by decay and oscillations. All the qualitative features that are listed above

(and described in Figure 2 except the decay phase) can in principle be captured by one first-order autonomous differential equation, except for the lag phase (phase 1) or the decay phase (phase 5). On the other hand, all the models listed above, except for the Baranyi and Roberts model, are first-order autonomous systems. As a result, they have no ability to capture a lag, nor the decay phase. To address the latter problem, these models introduce artificially a lag in the form of a lag-exponential model as reviewed by Baty and Deignette-Muller and as presented by Augustin, Rosso, and Carlier. The latter authors artificially introduce the solution y(t) = ln[x(t)] = yo for t < A, y(t) = ln[x(t)] = fx) for t> A where x(t) is the cell concentration, yo = lnxo = (lnx)= is the logarithm of the initial cell concentration and A is the lag duration, while fx) is a particular growth solution that stabilizes at the carrying capacity level. The problem with this artificially introduced lag is that it transforms the lag phase, which is an inherent growth result (effect), into a cause. The lag becomes a parameter in the model that needs to be established upfront, instead of being a result (effect) that is obtained because of other specific biological causes. Therefore, the model becomes descriptive rather than predictive.

Baranyi and Roberts, who developed one of the most popular models in predictive microbiology, were the first to realize that it is impossible to describe microbial growth by using a first-order autonomous system. They also realized that the history of cell growth should be represented by one single parameter revealing ''the physiological state of the cells at inoculation,'' although the definition of the latter in terms of its link to biologically meaningful parameters is vague. Baranyi and Roberts introduced a second-order system and obtained a nonau-tonomous first-order system via an explicitly time dependent adjustment function q(t)/[1 + q(t)] that can be evaluated by following the Michaelis-Menten kinetics. One refers to a model as autonomous if the differential equation describing it does not include coefficients that are explicit functions of time, and the model is nonauto-nomous if these coefficients are explicit functions of time. While usually any nonautonomous system can be transformed into an equivalent higher-order autonomous system by introducing a new variable, the latter transformation typically removes the stationary points from the system, a result that makes little biological sense. In the case of Baranyi and Roberts's model the stationary point is preserved only because their adjustment function saturates to a constant value at a time prior to the growth curve reaching the stationary phase. Except for this adjustment function, Baranyi and Roberts's model is indeed predictive. However, the adjustment function introduces a descriptive component into an otherwise predictive model. Vadasz and Vadasz showed how the Baranyi and Roberts's model can be transformed into an equivalent autonomous system and how its properties can be uncovered by this transformation.

A few important definitions and distinctions are relevant to the derivations that follow in this article. First, the classical definition of the lag duration is presented in Figure 2 by Acl, as the time value at the intersection of the tangent line (line 'b') to the growth curve at the point where the specific growth rate is maximum, that is, at yLIP = ln(xLIP), with the line (line 'a') representing the initial cell concentration, that is, y=yo = ln(xo), where xo is the initial cell concentration. The specific growth rate X/x reaches its maximum when the slope of the growth curve in terms of the logarithm of the cell concentration expressed by d(lnx)/dt = X/x is maximum, that is, when d2(lnx)/dt2 = d(x_/x)/dt = 0 representing the location of the logarithmic inflection point (referred here as LIP). It is typical in food microbiology when the growth curve includes a lag to refer to this maximum slope of the logarithmic growth curve as the maximum specific growth rate ^max. The latter is, however, an approximation and generally there is a distinction between ^LIP = (x/x)LIP defined as [d(lnx)/dt]max where the logarithmic inflection point occurs, and the maximum specific growth rate ^max, which can represent a slope that may not even be present on a particular growth curve. The simplest example is the LGM, where the maximum slope of the logarithmic growth curve always occurs at t= 0 and the latter slope approximates well the value of Mmax only for small values of xo such that xo ^ 6. The reason for this result is based on the definition of the maximum specific growth rate, which for the logistic (LGM) can be obtained from the LGM's governing equation x/x = d(lnx)/dt = ^max(1 — x/6). Obviously, x/xx d(lnx)/dt = ^max only if x/6 ^ 1 for some values of x > xo. Therefore, for the LGM (x/x)t = 0 = [d(lnx)/dt]t = 0 = Mmax only if xo/6 ^ 1. In the case of microbial growth curves that include a lag phase, the initial cell concentration xo is typically much smaller that the carrying capacity 6, and therefore the approximation yULIP = (x/x)LIP ~ ^max is appropriate. Nevertheless, it is essential to distinguish between the two as they represent conceptually distinct factors.

One additional important biological feature is related to experimental evidence relevant under reinoculation conditions. Maier introduces this point by stating the conditions that are necessary to encounter a lag phase in microbial growth as follows: (1) dependence on the type of medium as well as on the initial inoculum size; (2) dependence on the initial growth rate. For example, if an inoculum is taken from an exponential phase culture (high initial growth rate) in trypticase soy broth (TSB) and is placed into fresh TSB medium at a concentration of 106 cells ml—1 (large initial inoculum size) under the same growth conditions (temperature, shaking speed) (similar type of medium), there will be no noticeable lag phase. However, if the inoculum is taken from a stationary phase culture (very small initial growth rate), there will be a lag phase. Similarly, if the inoculum is placed into a medium other than TSB, for example, a mineral-salts medium with glucose as the sole carbon source (a more stressed type of medium), a lag phase will be observed. Finally, if the inoculum size is small, for example 10 cells ml—1, a lag phase will be observed until the population reaches approximately 106 cells ml—1. A similar description was presented by Baranyi and Roberts. Because of the lack of clarity regarding the accurate definition of the ''physiological state of the cells at inoculation'' Baranyi and Roberts linked the latter to ''the previous history of the cells,'' a true statement that might have misleading connotations to delay-type of models, which do not capture correctly the lag phase. Vadasz and Vadasz showed the accurate definition of Baranyi and Roberts's parameter representing the ''physiological state of the cells at inoculation'' by deriving the latter link to the initial growth rate at inoculation.

An autonomous neoclassical model proposed by Vadasz and Vadasz based on their earlier studies was shown to capture all the qualitative features that appear in experiments for monotonic growth of microorganisms, such as lag, LIP, convex and concave curves on the phase diagram, as well as the LGM as a special case. These authors showed that their proposed model fits the experimental data well for five distinct sets of data. In addition, the earlier model derived from first principles was shown to capture additional qualitative features that appear in nonmonotonic regimes of growth, such as growth followed by decay (decline) as well as oscillations.

To summarize this introduction, one may conclude that there appear to be only two models that are general enough to capture all qualitative features of monotonic growth of microorganisms and they are the Baranyi and Roberts's model and the neoclassical model proposed by Vadasz and Vadasz. All other models cannot capture naturally (rather than artificially) the lag phase. Therefore, the main thrust of this review will focus on these two models with particular emphasis to the source of the lag phase. However, prior to doing so it is imperative that we introduce one additional tool for analysis in the form of the phase diagram.

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