## The Neoclassical Model Formulation

The first major distinction between the neoclassical model and other models that are frequently used in microbial ecology or food microbiology is that the latter are followed by secondary modeling in fitting the data. Some secondary models are aimed at fitting a quadratic response surface that is assumed to represent the dependence of the growth parameters on environmental factors. The secondary model is important because it represents the impact of the environmental conditions on microbial growth, actually via the metabolic process that is not directly reflected in existing models. The very complicated metabolic processes and the resulting interaction between the cells and the environment make the quadratic response fitting (or alternative secondary models, e.g., Arrhenius-type, Rotkowsky-type, artificial networks, as reviewed by McKellar and Lu) a plausible and practical alternative to an otherwise unknown process (at least not completely well known). The fact still remains that the environmental conditions consist only of potential resources and their quality that may or may not be consumed entirely by the cells and the rate of consumption and utilization is not known a priori. To avoid the latter difficulty, Vadasz and Vadasz adopted a distinct and different approach. Instead of using the environmental conditions as representing a growth potential that may or may not materialize as nutrient (or other resources) consumption and utilization at an unknown rate, a constitutive relationship between the total viable biomass and the cell concentration, that is, M(x) is being used. This constitutive relationship represents the result of the actual nutrient (or other resources) consumption and utilization rather than the potential nutrient (or other resources) availability and quality, as described via the environmental conditions, such as pH, temperature, etc. To describe the former, it is essential to distinguish between the deterministic description of mechanisms within a single cell and the statistically averaged behavior of a cohort of cells in terms of their interaction with their environment and the corresponding interaction between the cells themselves. Let M represent the total viable cell biomass in the solution or per unit volume of the solution. Then we may define the average viable cell biomass as m = M/x. The expression M = mx is therefore an accurate representation (by definition) of the relationship between the total viable cell biomass M and the average viable cell biomass m. By using only this simple representation relating the total viable cell biomass to a statistically averaged value of a 'typical cell viable biomass' (the latter reflecting a representative average cell) simple but intriguing effects are revealed when following simple derivations. For example, evaluating the rate of total viable biomass production during the cell growth process, that is, dM/dt leads to dM d(mx) dx dm r , mi" + x— [1]

dt dt dt dt which can be presented in the following alternative form explicitly for the rate of average viable biomass production:

It is evident from eqn [2] that the rate of average viable biomass production is equal to the rate of total viable biomass production per unit cell minus an additional quantity reflecting the effect of the cells' specific growth rate (dx/dt)/x on the rate of average viable biomass production. When the growth rate is positive, reflecting a positive growth in the population number, the rate of average viable biomass production is reduced compared to the viable biomass production per cell. The opposite occurs when the cell concentration declines, representing a negative growth. Only in the stationary phase, the two rates of viable biomass production become equal, that is, when x = 0 (stationary phase conditions) m = M/ x (where we introduced the Newtonian time derivative notation x = dx/dt, in = dm/dt, M = dM/dt). There is therefore a fundamental distinction between the rate of average viable biomass production represented by it = dm/dt and the average rate of viable biomass production represented by M/x. They become identical only in the stationary phase. This reduction in the average viable biomass is a result of the division process that increases the number of cells without affecting the total viable biomass. This effect was demonstrated generally and in more detail by Vadasz and Vadasz.

In addition to introducing in the model the latter effect by relating the growth of the cell concentration to the total viable cell biomass, this model includes an additional effect that is missing in all other existing models. Most of the existing population models have the form x = P[1 - g(x)]x

where the specific function g(x) takes different forms for different models, for example, g(x) = x/8 for the LGM, g(x) = ln(x)/ln(6) for the Gompertz model, and g(x) = (x/6)K for the Bertalanffy-Richards family of growth curves. Equation [3] can be presented in the following alternative form obtained by dividing it by x and replacing x/x using the identity x/x = d(lnx)/dt d(lnx) dt

Two mechanisms are inherently present in eqn [4]. The first can be identified when considering small values of x such that g(x) ^ 1. In such a case, eqn [4] reduces to d(lnx) dt

producing inertial growth (Malthusian growth) because it can be represented in the following form suggested by Ginzburg and Akcakaya et al.:

which, following integration, leads to d(lnx)/dt = ft = xo/xo that produces the Malthusian growth exponential solution x = xoeftt. Malthusian growth represented by a balance between the left-hand side of eqn [4] with the first term on its right-hand side, reflects the mechanism of inertia of cell growth. A more detailed derivation demonstrating the inertial growth effect was presented by Vadasz and Vadasz. This inertial effect can reflect ''the physiological state of the cells at inoculation'', the latter being introduced by Baranyi and Roberts. The second term on the right-hand side of eqn [4] represents all mechanisms of resistance to growth that produce the inhibition phase on the growth curve. While it is now clear that two mechanisms representing the inertia and resistance to cell growth are already included in existing models, Vadasz and Vadasz propose to include a third mechanism that is not included in any existing model. This mechanism reflects storage effects on microbial growth. Cells have the ability to store nutrient as well as 'potential life' as they grow in size toward division. The neoclassical model that was developed by including the latter storage effects revealed that even when the latter are insignificantly small, the general form of the inhibition mechanism is different than the forms considered in other models so far. This difference is essential in the resulting lag effect, as it represents the controlling mechanism over the lag.

The proposed neoclassical model was derived from first biological as well as physical principles by Vadasz and Vadasz and can be presented in the form of an autonomous system of two first-order differential equations,

where x is the cell concentration, x = dx/dt is the growth rate, z is related (but not identical) to the nutrient (or resource) consumption/utilization rate and Z = dz/dt is related (but not identical) to the nutrient consumption acceleration. The denominator in the third term of eqn [7a] is related to the net average biomass, hence the third term is a representation of the excess nutrient consumption/utilization per unit of viable biomass in the population, ^max is the maximum specific growth rate, 6 is the carrying capacity of the environment, and rm = mi/ mo is the ratio between the viable biomass constant coefficients mi and mo corresponding to a linear constitutive relationship between the total viable biomass and the cell concentration in the form M(x) = mo + mix. The accurate form of the function f(x,xx) affects only nonmonotonic growth regimes such as growth followed by decay (decline) as presented by phase 5 (not shown in Figure 2), or oscillatory growth. Vadasz and Vadasz used a linear approximation for this function to show very good agreement of the model with experimental results of nonmonotonic growth.

For the purpose of monotonic growth, it is sufficient to mention that when the value of the coefficient ko is identically zero, eqn [7b] produces the solution z = zo = const., where zo represents the initial condition of z at t = 0 defined by using eqn [7a] at t = 0, in the form zo Mmax ^ ( 1 ^ rmxo fxo x + y -1

The latter produces monotonic growth, leading to a growth curve that does not exhibit a maximum nor a minimum except at the boundaries at t = 0 or tn. Monotonic growth is regularly obtained in laboratory experiments, whether in batch or continuous environments and is also observed in nature. In the event that the value of ko is not identically zero but rather very small, one can anticipate an original monotonic growth curve over the short to medium timescales, followed at much longer (slower) timescales by possibly nonmonotonic behavior such as growth followed by decay or oscillations. Examples of nonmonotonic features obtained by using eqn [7] were presented by Vadasz and Vadasz and showed to compare well with experimental data. However, for monotonic growth the system of eqns [7a] and [7b] degenerates to one first-order differential equation in the form x

S mm

which was obtained from the second-order system [7] as a mathematically degenerated case. The only memory, which this equation has that it originated from a second-order system is the term zo which depends on two initial conditions, that is, on the initial cell concentration xo as well as on the initial growth rate xo. It is easy to observe that when zo = ^max, corresponding to a specific particular set of initial conditions, eqn [9] produces the particular case of Pearl's LGM. The system represented by eqn [9] is an autonomous system.

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