# Swarming Behavior Emerging from the Uptake-Kinetics Feedback Control in a Plant-Root-Inspired Robot
## 2 Materials and Methods  
### Clues from Plants  
#### 2.1.1 Uptake-Kinetics  
As plants are sessile organisms, they have to develop a series of strategies for survival. The ability to adapt morphological and physiological properties to environmental stimuli is called plasticity and has enabled plants to explore and exploit the environment [27]. The role of roots is to supply nutrients to the whole plant. In order to maximize the probability of success, they have developed several strategies for nutrient exploitation, e.g., increase in root hairs, growth of lateral roots in patches of soil that is nutrient rich (morphological plasticity) or increasing the nutrient uptake rate (physiological plasticity) in conditions of nutrient deficiency [28].  
In soil, nutrients are not always present and distributed in such a way to satisfy the requirements of plants for optimal growth [29]; indeed, plant growth is limited by their availability, quantity, and ratio [30]. Ion charges in plants also need to maintain a balance for the correct evolution of processes such as protein synthesis or ion transportation through the membranes. Experiments evaluating the interaction among ions show in fact that the uptake of Na\({}^{+}\) and K\({}^{+}\) changes according to the presence of calcium in the medium or processes such as the synthesis of organic acids are altered by excessive cation or anion uptake [31]. These observations suggest that the internal concentrations of nutrients in plants need to maintain an equilibrium to prevent process alterations. Various characteristic indices and requirements among macronutrients have also been established [32].  
There is thus the first fundamental property of plant behavior:  
_Property 1:_ Plant growth is driven more by maintaining a balance of the internal nutrient concentrations rather than by collecting the closest and most available nutrient in the soil.  
The consequence of _Property 1_ is the selectivity of ion uptake. Uptake rates have been analyzed in both lower and higher plants by comparing the accumulation of nutrients in roots with the concentration remaining in the external solution. Results have shown that ratios among these two quantities differ for each observed nutrient [31], and thus confirm the selective characteristic of nutrient absorption by plants.  
Nutrient uptake is known to work as a function of the nutrient concentration in soil (at least up to a limiting threshold) with the saturation kinetics mechanism, which is described by the Michaelis-Menten equation [33], and defines the uptake rate, or absorption velocity, as:  
\[I=\frac{I_{max}\times C}{K_{m}+C}. \tag{1}\]In Equation (1), \(I_{max}\) identifies the capacity factor, maximal rate of absorption, which is approached asymptotically when the ion concentration in the medium increases. \(K_{m}\) represents the concentration with half of the maximal rate of absorption and \(C\) is the concentration perceived. It has been shown that the parameters of the uptake-kinetics (\(I_{max}\) and \(K_{m}\)) are strongly influenced by the internal concentration status of the plant. For instance, in both _Zea mays_ (corn) and soybean, it has been shown that with an increasing concentration of phosphorus in plants, both parameters decrease linearly (\(I_{max}\) more rapidly than \(K_{m}\)) [34]. The same behavior has been observed in barley roots when the internal nitrate availability increases [35]. This adjustment of \(I_{max}\) and \(K_{m}\) suggests a second property:  
_Property 2:_ A feedback control modifies the uptake rate of a nutrient according to its internal state in plants.