# Sustainable Microalgae and Cyanobacteria Biotechnology
## 2 Applications of Biotechnology on Microalgae and Cyanobacteria Utilization
### Biofuels from Microalgae
It was assumed that autotrophic and heterotrophic growth in _Euglena gracilis_[91] and _Spirulina platensis_[92] occur simultaneously and independently in cells growing in mixotrophic condition. In general, by mathematical terms, the mixotrophic growth is the sum of the growths in autotrophic and heterotrophic conditions, and can be expressed as:  
\[\frac{dX_{M}}{dt}=\frac{dX_{A}}{dt}+\frac{dX_{H}}{dt} \tag{1}\]  
where \(X_{M}\) are cells growing in mixotrophic conditions, \(X_{A}\) are cells growing in autotrophic conditions, and \(X_{H}\) are cells growing in heterotrophic conditions. However, it is difficult to know how many cells are growing in autotrophy or heterotrophy at any time during the cell growth. To simplify the conceit, cells growing in autotrophic mode can be defined as an \(\alpha\) fraction of the total cell biomass and a \(\beta\) fraction of cells growing in heterotrophic metabolic conditions. In other words, Equation (1) may describe as. \[\frac{dX_{M}}{dt}=\alpha\frac{dX_{M}}{dt}+\beta\frac{dX_{M}}{dt} \tag{2}\]  
The values of \(\alpha\) and \(\beta\) can be calculated on the basis of a ratio of the autotrophic growth rate \(dX_{A}/dt\) and heterotrophic growth rate \(dX_{H}/dt\), both of which proceed, or are suggested to proceed, during mixotrophic cultivation \(dX_{M}/dt\). \[\alpha=\frac{dX_{A}/dt}{dX_{M}/dt} \tag{3}\]  
\[\beta=\frac{dX_{H}/dt}{dX_{M}/dt} \tag{4}\]  
The values of \(\alpha\) or \(\beta\) are 0.0 when growth occurs in the absence of organic carbon and light energy, whereas, the summation of \(\alpha\) and \(\beta\) are 1.0, signifying growth in mixotrophic conditions. In other words, the fractions \(\alpha\) and \(\beta\) during mixotrophic growth vary as follows [93]:  
(i) For the autotrophic fraction  
The condition \(\alpha\leq 1\) at initial time (\(t_{o}\)) signifying that light is unlimited and the organic carbon source consumption is negligible. The condition \(0<\alpha<1\) at any time (\(t_{i}\)). The condition \(\alpha\approx 0\) at final time (\(t_{f}\)) light irradiance is limiting and growth is almost entirely dependent on organic carbon source consumption. In other words:\[Y_{X/S}=\frac{X_{i}-X_{0}}{S_{0}-S_{i}}\] \[\int_{X_{Mi}}^{X_{Mi+1}}\Delta X_{M}=\frac{Y_{X/S}(S_{0}-S_{i})}{1-\alpha_{i}} \int_{t_{i}}^{t_{i+1}}\Delta t_{i} \tag{5}\]  
where \(Y_{X/S}\) is the yield of biomass (\(X\)) produced by substrate (\(S\)) consumed. (ii) For the heterotrophic fraction  
The condition \(\beta=0\) at time = \(t_{o}\) that light is unlimited and organic carbon source consumption is negligible  
The condition \(0<\beta<1\) at any time = \(t_{i}\)  
and with the condition \(\beta\approx 1\) at final time = \(t_{f}\) signifying that light irradiance is limiting and growth depends on organic carbon source consumption. In other words:  
the light saturation constant, \(K=\frac{Y_{Qi}A}{Y}\), biomass produced per energy unit and illuminated area, this constant can be used as photobioreactor design criteria, because it involves the light energy absorbed by a specific microalgal specie and the amount of light irradiated in a specific photobioreactor geometry. \[\int_{X_{Mi}}^{X_{Mi+1}}\Delta X_{M=}\frac{Kl_{o}}{1-\beta_{i}}\int_{t_{i}}^{ t_{i+1}}\Delta t_{i} \tag{6}\]  
And then, \(\beta_{i}\) at any time can be calculated  
\[\beta_{i}=1-\frac{Kl_{o}}{X_{M_{i+1}}-X_{M_{i}}}(t_{i+1}-t_{i}) \tag{7}\]  
_Kinetic of Biofuels Production by Microalgae_  
Microalgae can be grown in three ways: photoautotrophically, heterotrophically, and mixotrophically [94; 95], as shown in Table 5. Due to the lower production costs, photoautotrophic cultivation using sunlight as an energy source is preferred over the other two types of cultivation [94]. In the course of mixotrophic cultivation, microalgal cells absorb both CO\({}_{2}\) and organic carbon [94; 96]. Several reports on the cultivation of microalgal species in mixotrophic conditions with various organic carbon sources have been published. In general, the amount of biomass increases significantly under mixotrophy.