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Ⅱ. Modeling for Enzymatic Reaction in Phase Separation System

Enzymatic Reaction Kinetics

Outline

After phase separation is formed, we can exploit the difference in solubility of material inside and outside the liquid droplet phase to create a difference in the rate of enzymatic reactions inside and outside the phase. For example, luciferase is a class of enzymes that oxidize the substrate and produce light. Firefly luciferase converts luciferin in the presence of ATP-Mg²⁺ and O₂ to generate oxyluciferin and light, while renilla luciferase uses coelenterazine as its substrate without requiring ATP. Both of the two enzymes and their substrates have higher solubility in the liquid droplet phase than in the aqueous cytoplasm, which leads to different reaction rates inside and outside the phase. Luciferase can be applied to monitor phase separation by expressing them in cells with phase separation systems and detecting light intensity in different regions of the cell. This also opens up the possibility of using a light-controlled phase separation system to regulate the rate of general enzymatic reactions.

Modeling Outline

In our model, we mainly focus on the most common single-step reactions which can be expressed by the following chemical reaction equation, where “S” stands for the substrate, “P” stands for the product and “E” stands for the enzyme:

We denote by 𝑣 the rate of this enzyme. Based on simple collision theory, we know that the reaction rate is determined exclusively by the frequency of effective collisions between molecules. In our simplified model, that is the frequency of effective collisions between the substrate “S” and the enzyme “E”. An effective collision requires three independent events to occur simultaneously: two molecules collide, the energy of the collision is high enough, and the collision’s orientation is favorable. The probabilities that the three events occur are denoted as the frequency factor 𝑍, the energy factor 𝑓 and the orientation factor 𝑃. Then we can write:

Therefore, the following discussion mainly focus on how phase separation affects 𝑍, 𝑓 and 𝑃. We denote the values of 𝑍, 𝑓, 𝑃 and 𝑣 after phase separation by 𝑘_𝑍⋅𝑍, 𝑘_𝑓⋅𝑓, 𝑘_𝑃⋅𝑃 and 𝑘_𝑣⋅𝑣 respectively. We have:

Quantitative Description of Phase Separation State

Generally, two methods are often used to regulate the entry of the enzyme to the phase. In one method, the phase is formed independent of light induction. As shown in Figure 1, light-controlled phase separation element is linked to the phase separation element in the phase, which recruits the other phase separation elements linked to the enzyme out of the phase in the presence of light.

Figure 1. The phase is formed independent of light induction. Enzyme enters the phase in the presence of light.

In the other method, the enzyme is also linked to one of the two phase separation elements, but only when the light is present will it gather with another component and form the phase, as shown in Figure 2. Since we usually only consider the reaction process when the substrate is added after phase formation, we do not care about the process of enzyme concentration change in this case, and it can be considered that the distribution of the enzyme is constant after phase formation.

Figure 2. The phase is formed only in the presence of light. The process by which the enzyme enters the phase and the process of phase formation are synchronized.

No matter which method is used, enzymes are freely distributed in the cytosol before light is induced. In reality, the protein in which the enzyme is located usually folds into a nearly spherical shape in the cell, although in the above figures we draw it like a cockroach. As shown in Figure 3, we denote by 𝑟₀ the radius of a single enzyme particle and 𝑉₀ its volume. After light induction, suppose n separate liquid droplets containing enzymes are formed (in Figure 3, 𝑛=3). The radius and the volume of the 𝑖-th liquid droplet is denoted by 𝑟_𝑖 and 𝑉_𝑖, respectively.

Figure 3. Spherical demonstration of single enzyme particles and enzymes in phases.

We denote by 𝑉_𝑡:=∑^𝑛_𝑖=1𝑉_𝑖 the total volume of the enzyme particles and 𝑞_𝑖≔𝑉_𝑖/𝑉_𝑡 the fraction of the volume of the 𝑖-th droplet in the total volume of all droplets. For the sake of convenience, we also write 𝑞₀:=𝑉₀/𝑉_𝑡 , which is exactly the reciprocal of the number of enzyme particles. Obviously, we have the restriction that ∑^𝑛_𝑖=1𝑞_𝑖=1.

It cannot form a real “phase” with only a small number of enzyme particles; if the amount of enzyme is small, just a cluster of particles will be formed. In order that the 𝑖-th liquid droplet represented in Figure 3 forms a real liquid droplet phase, it should reach a smallest volume that is required for a phase to form. We denote that critical volume by 𝑉^∘. We use an indicator function 𝜒_𝑖 for each enzyme particle cluster to indicate whether it forms a real phase, which is defined as below:

According to the the relation that q₀=V₀/V_𝑡, it is not difficult to check that the definition of 𝜒_𝑖 given in (2.4) is equivalent to the following definition:

On the Change of the Frequency Factor 𝑍

In this section, we will discuss how phase separation affects the frequency factor. More concretely, we need to derive how the value of 𝑘_𝑍 in Equation (2.3) is determined.

The substrate particles move randomly in the cytosol, waiting for a chance to collide with the phase formed by the enzyme. Generally, the distribution of substrate particles and phases in the cytosol is uniform in probability. Therefore, the probability of substrate particles colliding with a protein phase should be proportional to the surface area of the protein phases. Before phase separation, the total surface area is the sum of the surface areas of all enzyme particles, which is

After phase separation, the total surface area is the sum of the surface areas of all phases, which is

The multiple of the increase of the frequency factor, that is, the ratio of 𝑆₂ to 𝑆₁, is equal to

In summary, we can conclude that

On the Change of the Energy Factor 𝑓

The energy factor 𝑓 describes the probability that the energy of the substrate particle exceeds the lowest energy required for a chemical reaction to occur when colliding into the enzyme. When phase separation is not formed, 𝑓 actually does not change because that probability described above is irrelevant to whether several enzyme particles are gathered together or not. However, when phase separation is formed, things would be different. For example, if the substrate is hydrophilic while the protein phase formed carries hydrophobic properties, then when a substrate particle hits the protein phase, it will be subjected to a resisting force, thus the value of 𝑓 is lowered, which means that 𝑘_𝑓 &lt 1. Conversely, if the substrate is also hydrophobic as the protein phase is, then the value of 𝑓 will increase, resulting in a 𝑘_𝑓 that is greater than 1. Since the volume of the phase is generally much larger than the substrate particles and the nature of the phase does not change much with volume, we may regard 𝑘_𝑓 as a constant for fixed type of phase separation protein and substrate. We should emphasize that the value of 𝑓 changes only when phase separation is formed. Therefore, from the point of view of average, the value of 𝑘_𝑓 should be expressed as follows:

In (2.10), 𝑘^∘_𝑓 can be viewed the coefficient of energy factor changing for a single cluster, and 𝑘_𝑓 is the overall energy factor changing coefficient, which represents the multiple of 𝑘_𝑓 increase after the phase separation system is adapted.

On the Change of the Orientation Factor 𝑃

In this section, we will discuss how phase separation affects the orientation factor. More concretely, we need to derive how the value of 𝑘_𝑃 in Equation (2.3) is determined.

Suppose several enzyme particles (or may be just one particle) form a cluster (not necessarily a phase) with radius r. Once a substrate particle enters the cluster, it will stay in that cluster for a while. Naturally, the expected time the substrate stay in the cluster is proportional to the radius of that cluster, denoted by 𝜆𝑟_𝑖, where 𝜆 is a parameter that can be determined by the actual value of the orientation factor within a fixed time range Δ𝑡. In the time range of Δ𝑡, the substrate has chance to have an effective collision with any enzyme particle in that cluster. The chance that the substrate has an effective collision with an enzyme particle in the next period of time of a certain length does not depend on how long it has stayed in the cluster. Therefore, it can be inferred from the theory of stochastic processes that the number of times the substrate particles have effective collision with the enzyme each time it enters the cluster, denoted by 𝑁_𝑖, follows the Poisson distribution with parameter 𝜆𝑟_𝑖, i.e., for any non-negative integer 𝑘, 𝑃(𝑁_𝑖=𝑘)=𝑒^{−𝜆𝑟_𝑖}(𝜆𝑟_𝑖)^𝑘/𝑘!. Then, the probability that an substrate particle collides into the 𝑖-th cluster in the proper direction given the condition that it collides to the 𝑖-th cluster at least once, denoted as 𝑝_𝑖,np is

However, if the enzyme particles form a real phase in the cluster where they gather, then any substrate particle that enters the phase will be trapped in it. In this case, we can regard the probability of the substrate particles have have at least one collision with the enzyme in the proper direction in the 𝑖-th cluster given the condition that it collides into the 𝑖-th cluster, denoted by 𝑝_𝑖,p as 100%, that is,

Combining (2.11) and (2.12) together, we get

The contidional probability given by (2.13) is under the condition that the substrate collides into the 𝑖-th cluster instead of the other clusters. If the condition reduces to that the substrate collides into some cluster, not knowing which cluster in specific, then the conditional probability, written as 𝑝_𝑖′, should be

Sum over 𝑖, we get the probability that an substrate particle collides into any one enzyme particle in the proper direction of the 𝑛 clusters at least once given the condition that the substrate collides into some cluster is

Without phase separation, the probability that a substrate particle collides into any enzyme particle in the proper direction given the condition that it collides into some enzyme particle should be 1−𝑒^{−𝜆𝑉₀^1/3}. Thus, we have

Insights from the Mathematical Results

Combining (2.3), (2.9), (2.10) and (2.16), we can write 𝑘_𝑣 as

Equation (2.17) is a product of three terms. The first term depends on the amount of enzyme which is quantified by 𝑞₀. To simplify our mathematical derivation, we may first assume that the volumes of all liquid droplets exceed the critical volume 𝑉^∘ and form phases. In this case, the third term in (2.17) reduces to ∑^𝑛_𝑖=1𝑞^2/3_𝑖, thus we have the observation that the first term 𝑞^1/3₀/(1−𝑒^{−𝜆𝑉₀^1/3}) is determined exclusively by the amount of enzyme, while the other two terms only depends on the pattern of phase separation, quantified by 𝑛 and 𝑞_𝑖, with no relation to the amount of enzyme. This provides convenience and basis for us to discuss them separately.

First, we look at the term 𝑞^1/3₀/(1−𝑒^{−𝜆𝑉₀^1/3}). By taking derivatives, one can easily check that this term increases with 𝑞₀. Since 𝑞₀ is the reciprocal of the number of enzyme particles, this observation suggests that the greater the amount of enzyme expressed, the smaller the effect of phase separation on the enzymatic reaction. If the amount of enzyme is so big that 𝑞₀ is less than the value which makes 𝑞^1/3₀/(1−𝑒^{−𝜆𝑉₀^1/3})=1, then phase separation not only does not promote the enzymatic reaction, but inhibits it. This can be explained by the understanding that when the concentration of the enzyme is too large, phase separation causes the enzyme to be encapsulated inside the phase and cannot be contacted with the substrate, and that this effect overlies the increase in the orientation factor by phase separation which will be discussed later. However, the enzyme concentration cannot be too low, otherwise the phenomenon of enzymatic reaction will not be observed. The proper range of enzyme concentration that can be chosen is shown in Figure 4. One should be aware that the horizontal axis of Figure 4 indicates the concentration of the enzyme, which is in an inverse proportion to 𝑞₀.

Figure 4. Choose proper enzyme concentration.

The second and the third terms actually depend on the allocation of the total enzyme volume to each droplet, and the difference of each 𝑞_𝑖 would bring a lot of trouble to theoretical studies. To avoid such trouble, we may assume that all 𝑞_𝑖’s are the same, i.e., 𝑞_𝑖=𝑞₁ for 𝑖=1,⋯,𝑛, and 𝑛=1/𝑞₁. The product of the second and the third terms is 𝑘^∘_𝑓/𝑞^1/3₁, which reaches its maximum when 𝑞₁ takes its minimum value. Since we have assumed that every droplet exceeds the critical volume, the minumum of 𝑞1 should be 𝑞₀𝑉^∘/𝑉₀. This gives the maximum value of 𝑘_𝑣 in this case, denoted by 𝑘⁽¹⁾_𝑣:

The contidional probability given by (2.13) is under the condition that the substrate collides into the 𝑖-th cluster instead of the other clusters. If the condition reduces to that the substrate collides into some cluster, not knowing which cluster in specific, then the conditional probability, written as 𝑝_𝑖′, should be

Now we consider the case that no droplet forms a phase, and the same as the former case, we only consider the condition that all droplets are of the same size. Then we have the value of 𝑘_𝑣 in this case to be:

The latter term (1−𝑒^{−𝜆𝑉_t^1/3𝑞^1/3₁})/𝑞^1/3₁ is a decreasing function of 𝑞₁. Therefore, in this case 𝑘𝑣 takes it maximal value when 𝑞₁=𝑞₀. Denoted this maximal value by 𝑘⁽²⁾_𝑣, then we have

Now we can come to consider when the phase separation system is really useful for enhancing the enzymatic reaction. That is, we need to find out in what case will 𝑘⁽¹⁾_𝑣>𝑘⁽²⁾_𝑣 hold. From the mathematical expression, we can easily obtain that 𝑘⁽¹⁾_𝑣>𝑘⁽²⁾_𝑣 holds if and only if

Then, let us discuss about the insights for we can get from (2.21). If the substrate is fixed, then whether phase separation enhances the reaction exclusively depends on the type of phase separation protein. This is because that 𝑉^∘ which denotes the smallest volume to form a phase is an intrinsic property of the phase separation protein. However, in most cases, the choice of phase separation protein is limited, and it is easier for one to change the type of substrate. Generally, a substrate with greater difference in solubility between in the protein phase and in the cytosol has a greater value of 𝑘^∘_𝑓, making it likelier for (2.21) to hold. If it fails to choose the proper type of phase separation protein and substrate to achieve the requirement of Inequality (2.21), then it is impossible for the phase separation system to accelerate the overall rate of the enzymatic reaction.

In conclusion, we get the following insights for wet experiment from modeling:

1. In the case where the enzymatic reaction phenomenon is sufficiently observed, the lower the concentration of the enzyme, the better the effect of phase separation on the promotion of the enzymatic reaction.

2. A suitable phase separation element and a substrate having a sufficiently large difference in solubility between in and out of the phase should be selected to satisfy Inequality (2.21) by making the value of 𝑘^∘_𝑓 big enough.

3. Try to make the phases formed more and smaller, but not smaller than the critical volume 𝑉^∘.