Optimal macromolecular density in the cell

Dill et al. (1) have reported very interesting estimates of the physical limits of different cell processes, including protein folding and metabolism. They hypothesized that cells have an optimal protein density that maximizes the speed of biochemical reactions. Indeed, reaction rates increase with increasing the concentration of catalytic units, such as enzymes and ribosomes, which are mainly composed of protein. However, high protein densities can slow diffusion, reducing reaction rates. Assuming that reactions are diffusion limited; that the total enzyme concentration is proportional to the volume fraction occupancy c ∼ φ, a hard-sphere approximation for the diffusion constant D of approximately (1 − φ/ φ_c)² with φ_c equal to 0.58; and that reaction rates are proportional to the concentration of reactants and the diffusion constant r ∼ cD; Dill et al. (1) obtained an optimal volume fraction occupancy φ* of 0.19, which is close to the protein density observed in cells. However, this calculation is too simplistic and contains some incorrect assumptions. First, in a crowding media, the effective concentration of reactants increases faster than φ, because the volume available to any particle of finite size is smaller than the total volume. In a first approximation, c ∼ φ/ (1 − φ), and further corrections may be needed when φ gets close to 1. Second, the intracellular media does not behave like hard spheres—more like colloidal glass formers—and the diffusion constant is better described by the exponential low D ∼ exp(−αφ) (2). Assuming again that reaction rates are proportional to the concentration of reactants and the diffusion constant r ∼ cD, and using an α of 5.8 as measured for fibroblast cells (3), we obtain:

which has a maximum at the optimal volume fraction occupancy φ* of 0.22. Further corrections are required to take into account that not all reactions are diffusion limited (4), obtaining optimal volume fraction occupancy values in the range of φ* values from 0.22 to 0.8, with the lower and upper bounds corresponding to all reactions being diffusion-limited or not diffusion-limited, respectively. In conclusion, the intuitive ideas of Dill et al. (1) are correct, but the corrections reported here are required to obtain a more precise accounting of the laws governing the intracellular media.