Supplementary materials
The mathematical derivation of Equation 8 (denoted as Eq. (a) in the main text)
The metabolizeProb in ISPKS 1.0 was the in silico counterpart to the expectation of the metabolic probability of enzyme (MetE) in the simplified two-site enzyme design. Figure S1 indicates the simplified illustration of a two-site enzyme.
Figure S1. The schematic diagram of the regulation of simplified two-site enzyme design
Referring to assumption 2, the MetE of enzyme with the regulatory site occupied was equal to zero. So, the expectation of MetE was as listed in the Equation 1.
The Punbound within the Equation 1 was the probability that no free solute could bind to the regulatory site of the enzyme within one cycle. In other words, every free solute in cell would have one attempt and all of them failed. The Pone-bind was the probability that one free solute could bind to the regulatory site of enzyme in one attempt. So, the Punbound was derived, as listed in Equation 2.
Then, we substituted the Punbound in Equation 1 with Equation 2 and came up with the Equation 3.
After that, we transformed Equation 3 into natural base, as listed in Equation 4.
As the definition, the intracellular amount of unbound solute (Nin-cell) was the product of the volume of cell (Vcell) and the intracellular concentration of unbound solute (Cin-cell), as listed in Equation 5.
Referring to assumption 1, Cin-cell was proportional to the initial concentration of solute (Cinitial) in the dosing compartment. So, we substituted Cin-cell in Equation 5 with Cinitial, as listed in Equation 6.
Then, we substituted Nin-cell in Equation 4 with Equation 6 and came up with Equation 7.
Finally, we rewrote Equation 7 into a simplified form, as listed in Equation 8.
Implementation of inhibition of enzyme via regulatory site
The enzyme inhibition process is controlled by an enzyme parameter, metabolizeProbFactor, as shown in Equation 9 (denoted as Eq. (b) in the main text).
The metabolizeProbFactor can take values ranging from 0 to positive infinity. If the metabolizeProbFactor is between 0 and 1, the updated metabolizeProb would be smaller than the initial one, mimicking an inhibitory effect. The closer the metabolizeProbFactor is to 0, the more complete inhibition would occur. If metabolizeProbFactor is bigger than 1, the updated metabolizeProb would be bigger in value, mimicking the enhancement of enzyme metabolic function. The bigger the metabolizeProbFactor is, the more substantial the enhancement that would occur.
The derivations for cross-model validation
1) The link through substrate binding
The biological meaning of the Michaelis constant Km was the substrate concentration when V (metabolic rate) was equal to half of Vmax (maximum of metabolic rate). Km could map to the concentration of substrate in the ISPKS 2.0 when half of enzymes were bounded. The concentration was defined as the quotient of Nm divided by Vcell. Nm here represented the intracellular amount of free substrate when half of enzymes were bounded in the ISPKS 2.0. Vcell meant the volume of intracellular space. Therefore, the ratio of Nm to Vcell was proportional to the Michaelis constant Km (as listed in Equation 10).
The process of substrate binding to enzyme’s active site could be divided into two steps, as illustrated in Figure S2.
Figure S2. A schematic diagram of the process of substrate binding
Firstly, the free substrate should get into the neighbourhoods of enzyme’s active site via random walk. This probability (Ain-site) was quantified as the ratio of the siteNactivesite (the number of grids of the neighborhoods of enzyme’s active site) to Vcell (the number of grids of intracellular space), as listed in Equation 11.
Secondly, the free substrate within the neighborhoods of enzyme’s active site attempted to bind to enzyme’s active site. This step was controlled by solute parameter activeSiteAffinity. Combining these two steps, we derived Aone-bind (listed in the Equation 12), which was the probability that a free substrate could bind to enzyme’s active site within one cycle.
Then, we substituted the Ain-site in Equation 12 with Equation 11 and came up with Equation 13.
The Aunbound was the probability that no free substrate could bind to the active site of enzyme within one cycle. So, the Aunbound with the intracellular amount of substrate equal to Nm was derived, as listed in Equation 14.
The opposite of the event that enzyme’s active site was unbound (Aunbound) was the event that enzyme’s active site was bounded (Abound). Therefore, the sum of Aunbound and Abound was equal to 1 (listed in Equation 15). On the basis of Equations 14 and 15, we derived Abound (listed in Equation 16).
Then, we performed first order Taylor expansion on Equation 16 and derived a linear approximation of it (listed in Equation17).
Next, we substituted the Aone-bind in Equation 17 with Equation 13 and simplified the expression (listed in Equation 18).
Finally, since siteNactivesite was a constant for a specific model and the ratio of Nm to Vcell was proportional to Km (as listed in Equation 10), we derived that the Michaelis constant Km was proportional to the inverse of in-silico parameter activeSiteAffinity (listed in Equation 19).
Since UGT and SULT shared the same intracellular environment, the proportionality constant between Km and 1/activeSiteAffinity should be the same for two kinds of enzymes. Therefore, we map the relationship between the substrate binding-related parameters of UGT and SULT (as listed in Equation 20).
2) The link through metabolic capacity
The process of formation of metabolite in the in silico system is presented in Figure S3.
Figure S3. A schematic diagram of the formation of metabolites
We linked Vmax to metabolizeProb and numEnzyme by considering the metabolite formation rate when all enzymes were bounded. In the wet-lab system, Vmax was achieved when every enzyme was made full use of. The in-silico counterpart of this idea was that every enzyme would be bounded through all the cycles. Under this condition, the process of substrate binding would no longer limit the biotransformation of enzyme. Therefore, the formation rate of metabolites for a single enzyme was governed only by enzyme parameter metabolizeProb.
On the other hand, the system parameter numEnzyme (the number of enzymes) would also affect the metabolic capacity of enzymes in terms of the system performance. Taking all these into consideration, we derived that Vmax was proportional to the product of numEnzyme and metabolizeProb (listed in Equation 21).
Since UGT and SULT shared the same intracellular environment, the proportionality constant between Vmax and the product of numEnzyme and metabolizeProb should be the same for the two kinds of enzymes. Therefore, we derived the relationship between metabolic capacity-associated parameters of UGT and SULT (as listed in Equation 22).
3) The link through inhibitory extent
To link Ki to ISPKS 2.0 parameters, we considered the inhibitory extent – the ratio of inhibited metabolite formation rate to uninhibited rate – at the concentration of Km.
In the perspective of the wet-lab system, inhibitory extent (I) was directly defined as the ratio of Vi (inhibited metabolic rate) to V (uninhibited metabolic rate) under the concentration of Km, as listed in Equation 23. Substituting the Vi and V in Equation 23 with Equations 24 and 25, we derived the method 1 to calculate I (listed in Equation 26).
For the in-silico system, inhibitory extent (I) was similarly defined as the ratio between expected metabolizeProb values with and without inhibition when half of enzymes’ active sites were bounded (listed in Equation 27).
The expression of expected metabolizeProb with inhibition was listed in Equation 28.
The Rbound was the probability that the regulatory site of enzyme was bounded after one cycle. The process of regulatory site binding consisted of three steps, as illustrated in Figure S4.
Figure S4. A schematic diagram of the regulation of enzyme function
The algorithm of the first two steps was similar to active site binding. In the first step, the free substrate needed get into the neighborhoods of enzyme’s regulatory site via random walk. This probability (Rin-site) was derived as the ratio of the siteNregsite (the number of grids of the neighborhoods of enzyme’s regulatory site) to Vcell, as listed in Equation 29.
In the second step, the free substrate within the neighborhoods of enzyme’s regulatory site attempted to bind to enzyme’s regulatory site. As illustrated in Figure S4, this step was governed by solute parameter regSiteAffinity. Combining these two steps, we derived Rone-bind (listed in Equation 30), which was the probability that a free substrate could bind to enzyme’s regulatory site within one cycle.
Then, we substituted the Rin-site in Equation 30 with Equation 29 and came up with the Equation 31.
In the third step, the bind-to-regulatory site substrate was trying to get released. This step was controlled by solute parameter regSiteReleaseProb, as illustrated in Figure S4. Taking all these three steps into consideration, we derived the expression of Rbound (listed in Equation 32).
Nm was calculated on the basis of Equations 13 and 16, as listed in Equation 33.
On the other hand, according to the two-site enzyme design (listed in Equation 9), the inhibited metabolizeProb (metabolizeProbi) with enzyme’s regulatory site occupied was derived, as listed in Equation 34.
Finally, combining Equations 27, 28, 31, 32, 33, and 34, we derived method 2 to calculate inhibitory extent I (as listed in Equation 35).
Comparing the values of inhibitory extent from method 1 and method 2, we could map inhibitory constant Ki to in-silico parameters.
3