To give you a more intuitive understanding of the phase separation process and the effect of phase separation on the rate of enzymatic reactions, we have made an interactive interface. You can choose the different strengths of interactions between separation elements and different enzyme concentrations to see how the phase separation proceeds and how the rate of the enzymatic reaction changes.

The computation of enzymatic reaction rate changes in the user-interface is based on formula (2.17), which we made some modifications to. First, we assume that every protein cluster forms a real phase, which means 𝜒_𝑖=1 for all 𝑖’s. Then, we get

If we denote by 𝑆₁ the specific surface area of protein clusters (the ratio of surface area to volume) after phase separation and 𝑆₀ the specific surface area of protein all protein particles before phase separation, one will find the term 𝑞^1/3₀∑^𝑛_𝑖=1𝑞^2/3_𝑖 is exactly equal to 𝑆₁/𝑆₀. We can use this relationship to extend the application of equation (3.1) to the case where the shape of the protein phase is arbitrary (i.e., not necessarily a sphere), just by replacing the term 𝑞^1/3₀∑^𝑛_𝑖=1𝑞^2/3_𝑖 with 𝑆₁/𝑆₀. The phase-separated images are two-dimensional, where we can only obtain the ratio of the perimeter to the area of the protein phase (denoted by 𝐶₁ and 𝐶₀ after and before the phase separation respectively) rather than the surface area to volume ratio. We can roughly estimate the relationship between the two ratios using the relationship 3𝐶₁/𝐶₀=2𝑆₁/𝑆₀. Therefore, we can rewrite (3.1) as

The value of 𝐶₁ can be calculated via image processing programs, while the value of 𝐶₀ can be inferred from the size of protein particles after scaling to the same scale as the image where we calculate 𝐶₁.

Since there is not much prior knowledge of the solubility difference of the substrate between in the cytoplasm and in the phase, we conservatively assume that 𝑘^∘_𝑓=1. As discussed previously, the term 1−𝑒^{−𝜆𝑉₀1/3} is the orientation factor. The orientation factors of different chemical reactions vary greatly, and the orientation factor data of enzymatic reactions are lacking. But as a proof of concept, we don’t need to be that precise and hence we can roughly estimate the orientation factor to be 1×10^-3.