# Bioconvection in Microalgae: Review of Mathematical Models
## 2 Models
This relatively simple linear equation makes this model computationally efficient and easy to implement in existing gyrotaxis models; its low computational overhead allows for large-scale simulations, such as photobioreactors for algal cultivation, by predicting cell distribution patterns in vertical water columns and by predicting cell behavior in these cultivation systems where light intensity varies. Though it works best for species where the swimming speed is primarily light-dependent, it may need modifications for cells with more complex photoresponses since it assumes a linear relationship between swimming velocity and light intensity, which may not always hold true, and maintains fixed gyrotactic behavior regardless of the light conditions. The second model is named Light-Dependent Center-of-Mass Offset. It modifies how deterministically cells swim by varying their center-of-mass offset, h (i.e., the distance between the cell's geometric center and the center of mass), with light intensity. In this case, the center-of-mass offset varies linearly with light intensity according to  
\[h(I)=-\xi(I-I_{c}) \tag{6}\]  
where \(\xi=h_{n}/I_{c}\) (\(h_{n}\) is the normal offset in the dark). The maximum offset will be in dark (most deterministic swimming), and the zero offset will be at \(I=I_{c}\) (purely stochastic swimming), while the negative offset will be set at high intensities. This second model is better suited for species with well-documented center-of-mass changes, and it is the most accurate for studying cell orientation control in photobioreactors and for understanding cell clustering behavior in order to optimize light exposure in dense cell cultures. Its main limit is the linear relationship assumption between the center-of-mass offset and light intensity, which may be oversimplified since it does not account for potential physiological adaptations over time and does not accurately represent all species' mechanical responses to light; hence, it may require validation for different cell types. Its computational efficiency is moderately complex due to the need to recalculate orientation parameters, and it is more computationally intensive than the first model, but it is still manageable for medium-scale simulations. The third model is named Reactive Phototactic Torque; it introduces an additional torque (\(L_{p}\)) that responds to both light direction and intensity gradients, and it is induced by an asymmetric flagellar beat pattern. The phototactic torque equation is  
\[L\ _{p}=f(I)C\ _{o}\wedge\ (\beta_{1}\pi+\beta_{2}\
abla I) \tag{7}\]  
where \(L_{p}\) is the total phototactic torque vector, \(f(I)\) is the phototactic strength, \(C_{o}\) is the unit vector in the direction of cell orientation, \(\wedge\) is the vector cross-product operator, \(\pi\) is the unit vector indicating global light direction, \(\
abla\ I\) is the gradient of light intensity, and \(\beta_{1}\) and \(\beta_{2}\) are two constants. The model considers two cases:  
1. \((\beta_{1},\ \beta_{2})=(1,0)\): the response to light direction only;
2. \((\beta_{1},\ \beta_{2})=(0,H/I_{s})\): the response to light gradients only (\(H\) is the fluid layer depth). The torque characteristics are as follows:  
- Zero at \(I=0\) and \(I=I_{c}\);  
- Maximum at \(I=I_{c}/2\). In terms of cell behavior, the intensity responses will be as follows:  
- No torque in darkness (\(I=0\));  
- No torque at critical intensity (\(I=I_{c}\));  
- Maximum effect at \(I=I_{c}/2\);  
- Positive torque between \(I=0\) and \(I=I_{c}\). Unlike the first two models, which modify existing cell properties (swimming speed and center of mass), this model introduces a new physical mechanism that can work alongside existing cell behaviors. Moreover, thanks to its flexibility, by adjusting \(\beta_{1}\) and \(\beta_{2}\), it can be investigated how cells respond to either global light direction or local light gradients, or potentially a combination of both.