Where Xm is the population size at stationary phase. From integrating equation (8), with x = xo at t = 0 as initial conditions: (9) Equation (9), known as the logistic equation, gives a sigmoidal variation of biomass which empirically represents two phases of cell growth, the exponential and stationary phases, during which most PHB is produced. The biomass concentration depends on the initial and maximum attainable cell concentrations with respect to time, which varies with fermentation conditions and the microorganism used. The basic kinetic model for growth and non- growth associated PHB production under nitrogen limitation was proposed by Leudeking-Piret (Pazouki et al., 2008), where the product formation rate depends upon both the growth rate and biomass concentration in a linear fashion, equation 10: (10) where α and β are the associated and non-associated growth constants respectively. In order to evaluate the constants (α and β) we divide equation 9 by x and assume k =, yielding: When the experimental cell growth data are plotted as against k the result is a straight line where α is equal to the slope while the β is equal to the intercept (data not shown). P and x represent the PHB polymer and dry cell weight concentration. The biomass concentration with respect to time depends upon the initial and final biomass concentrations. Theoretically, this model is based on three assumptions: (a) is constant (b) death-rate of the cell is negligible, and (c) also, the lag phase period is negligible. In a batch culture, the combined of Monod and Luedeking–Piret model can be used to describe quite simply, with three set of mass balance equations, the cell mass (equation 7), product concentration (equation 10) and the limiting substrate (equation 12) with respect to time (t): (12) , the rate specific substrate consumption, is related toµ according to: (14) where known as substrate yield coefficient which is defined as:
(15) Also, can be utilised to develop a relation between biomass and substrate concentrations, by assuming it has a constant value: (16) where is the cell biomass concentration in the stationary phase, by substituting equation (16) in equation (7), the cell balance equation becomes: (17) By re-arranging the equations, the cell balance equation becomes: (18) By integrating equation 18 X = X0 at t = 0 as the initial condition, the resulting solution is; (19) Equation (19) is an analytical relation, but non-explicit for X, which given all growth related parameters (, and , are known. X0 and S0 are given, and Xm is calculated from equation (16) with the conditions X = X0 and S = S0. Equation 19 can be used to evaluate the growth parameters by linear regression on batch culture data. Partially integrating to solve for product formation, from equation 12; P−P0= α. (X-X0) +β. (20) The difficulty in the integration of equation 20 is the last term since the solution for X is not explicit., therefore, by integrating by parts: (21) In order to obtain an analytical solution to equation 21, it was temporarily assumed that t is the dependent variable. Integrating with respect to X between X and X0, the following solution was obtained; (22) Then, by substituting equation (22) in (20), the result is; (23) (24)
Steinwandter, 2014 Chicory root hydrolysed media 3.8 1.7 43 Batch fermentation Baei et al., 2011 Sugarcane molasses 8.9 1.3 0.17 14.6 Shake flasks This work A 6.1 4.58 0.12 75 Shake flasks B 6.3 4.6 0.11 73 C 7.2 3.9 54 D 6.7 2.9 0.04 43.3 1. Overview of PHB synthesis by C.necator with different systems and carbon sources.