First, a conceptual model is established. To simplify, we capture the basic feature of phase separation as compartments. We assume there are two boxes with equal volume, one with dichotomous separate composition and the other with uniform mixture. They are shown in fig1 to represent the situations with and without phase separation, respectively. In this instant, the total amount of enzyme and substrate are identical in both boxes. The difference is that enzyme and substrate are distributed homogeneously in the uniform mixture, while they are divided into two parts in the separated case. The volume of the two parts are denoted by χV and (1-χ)V, added up to V, which is the total volume for both boxes. Enzyme is condensed in the χV part, we assume its concentration is enhanced by p. The enzyme concentration then writes as p[E]0. To guarantee the total amount of enzyme remains unchanged, the enzyme concentration in the other part is automatically (1-pχ)/(1-χ) [E]0. Substrate is treated in the same way: q[S] in the χV part and (1-qχ)/(1-χ) [S] in the (1-χ)V part. For convenience, we stress here that p is assumed to a number larger than one in all cases followed, but there is no limit on q.
Fig1: Illustration of a dichotomous separate box and a uniform box.
Next, we couple them with a typical enzyme kinetic reaction model shown in fig2. Considering quasi-steady-state and the conservation of enzyme, the instantaneous reaction rate can be expressed as Hill equation:
Fig2: A typical enzyme kinetic reaction.
Thus the instantaneous amount of product in the two cases are:
To see the effect of how separation affects reaction rate, we take the quotient of the two and define it as Q ̃:
Then if Q ̃>1, separation accelerates reaction; if Q ̃<1, separation decelerates reaction.
We notice with ease that since the effect of enzyme concentration is nearly linear. Thus, even when the enzyme condenses, if the substrate still remains uniform, i.e p>1 and q=1, it has no effect on the reaction rate. In other words, Q ̃=1 once q=1. To conclude, if we aim to alter reaction rate through phase separation, a heterogeneous distribution of substrate must be satisfied.
From another perspective, we can regard equation (4) as a relation between dependent variable Q ̃ and independent variable [S]/K_A under parameters p, q and χ. Here are some typical calculation results:
Based on the calculations, when q>1, enzyme and substrate condense in the same part, the reaction is accelerated; when q<1, enzyme and substrate condense in different region, thus inhibiting the reaction. But in both cases, observable changes only take place when [S]/K_A is small. The smaller [S]/K_A is, the more obvious the changes become.
We also vary the Hill number n to see what differences will emerge. As expected, the larger n is, the more sigmoid the kinetic curve evolves, and the more significant the acceleration become becomes.
To sum up, we have arrived at several rough conclusions, for noticeable enhancement to happen, it requires
1. enzyme and substrate condense in the same droplet
2. a relatively large KM
In addition, positive cooperative binding benefits the acceleration process.
But how to condense enzyme and substrate together? Since the diffusion velocity is generally larger than reaction rate, if the substrate diffuses normally down the concentration gradient, normal distribution will almost eliminate any heterogeneous distribution of substrate instantaneously. We simulate this process by coupling reaction-diffusion equations with the previously described formula to model phase separation.
Fortunately, if the substrate has a higher solubility in the dense oil phase, once the enzyme droplets form, substrate can be attracted into the droplet. Hence, enzyme and droplet both condense in the same droplet. It is possible to anticipate a significant acceleration in reaction as predicted by previous conceptual model since p>1 and q>1. In practical computation, the substrate is incorporated in the free energy density expression and diffused by chemical potential as well. The complete equations are specified as followed:
Apparently observed from the simulation results, when substrate has a higher solubility, the reaction rate is indeed increased. Incidentally, positive cooperative binding also demonstrates a positive effect in this simulation.
Generalized from the conceptual model and dynamic simulation, we are finally able to attain the two conditions for accelerating reaction:
1. substrate or intermediate has a higher solubility in droplet phase
2. the reaction has a relatively large KM