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Is there a known relation between the mass of a plant to the size of its roots?
For example, can I take the root size to be proportional to the biomass of the plant like in this relation: $$R = R_0 (1+EB) $$, where $R_0$ is the minimal size of the root (i.e. the diameter of the seedling), $B$ is the biomass of the plant, $E$ is some parameter, and $R$ is the length of a typical root?
Your question specifically asks for an equation which gives root radius and your sample equation is for typical root length, neither of which I'm aware of. However, there is a known relation between the above-ground biomass and the below-ground biomass of a plant, which varies with species and environmental factors. This is the root:shoot ratio.
This is a linear model where the constants represent differences in starting conditions (the intercept) and biomass partitioning (slope).
Roots and shoots are functionally interdependent and these two systems maintain a dynamic balance in biomass which reflects relative abundance of above-ground resources (light and CO2) compared with root-zone resources (water and nutrients). Whole-plant growth rate and root : shoot ratio are thus an outcome of genotype × environment interaction, but source of control is ambiguous.
The actual size of the roots - as in their extent - will depend on the soil characteristics, water table, the plant species, etc.
Source: http://plantsinaction.science.uq.edu.au/edition1/?q=content/6-3-1-biomass-distribution
