Universal Design Principle to Enhance Enzymatic Activity using the Substrate Affinity

ABSTRACT

Design principles to improve enzymatic activity are essential to promote energy-material conversion using biological systems. For more than a century, the Michaelis-Menten equation has provided a fundamental framework of enzymatic activity. However, there is still no concrete guideline on how the parameters should be optimized to enhance enzymatic activity. Here, we demonstrate that tuning the Michaelis-Menten constant (K_m) to the substrate concentration (S) maximizes enzymatic activity. This guideline (K_m = S) was obtained by applying the Brønsted (Bell)-Evans-Polanyi (BEP) principle of heterogeneous catalysis to the Michaelis-Menten equation, and is robust even with mechanistic deviations such as reverse reactions and inhibition. Furthermore, K_m and S are consistent to within an order of magnitude over an experimental dataset of approximately 1000 wild-type enzymes, suggesting that even natural selection follows this principle. The concept of an optimum K_m offers the first quantitative guideline towards improving enzymatic activity which can be used for highthroughput enzyme screening.

MAIN TEXT

Introduction

Enzymes are responsible for catalysis in virtually all biological systems,^[1,2] and a rational framework to improve their activity is critical to promote biotechnological applications. Since the early 20^th century, a reaction mechanism where the enzyme first binds to the substrate (E+S → ES) before releasing the product (ES → P) has been used as the conceptual basis to understand enzyme catalysis (Scheme 1).^[3-6] The reaction rate of this mechanism is given by the Michaelis-Menten equation: Here, the reaction rate (v) is expressed as a function of a rate constant (k₂), the Michaelis-Menten constant (K_m), and the substrate (S) and enzyme (E_T) concentrations. K_m can be interpreted as a quasi-equilibrium constant for the formation of the enzyme-substrate complex, defined as: with rate constants defined based on the mechanism shown in Scheme 1. k₂ is the rate constant for releasing the product from the enzyme-substrate complex (ES → P), routinely expressed as k_cat in the enzymology literature. These parameters are experimentally accessible by fitting the theoretical rate law (Eq. (1)) with experimental data^[7-10] and are subsequently registered in databases such as BRENDA^[11] and Sabio-RK.^[12] In principle, the accumulated data may help rationalize and improve the activity of existing enzymes.

Scheme 1.

The standard reaction mechanism of enzyme catalysis.

However, there is no concrete understanding on how these parameters influence enzymatic activity. For example, increasing k₂ may enhance activity according to Eq. (1), or diminish it due to a larger K_m (Eq. (2)).^[13] Thus, the mutual dependence between k2 and K_m complicates their influence on the enzymatic activity ( v ), hindering the rational design of enzymes towards biotechnological applications such as the synthesis of commodity chemicals,^[14] antibiotics,^[15] or pharmaceuticals,^[16] increasing the nutritional content of crops,^[17] and restoring the environment.^[18]

In this study, we analyzed the Michaelis-Menten equation to clarify the relationship between the enzyme-substrate affinity (K_m) and the activity (v). The key ingredient of our mathematical analysis is the Brønsted (Bell)-Evans-Polanyi (BEP) relationship,^[19-23] which models the activation barrier as a function of the driving force. This is a well-known concept in heterogeneous catalysis, and in conjunction with the Arrhenius equation,^[24] can be used to evaluate the mutual dependence between k₂ and K_m to quantitatively. This allowed us to calculate the optimum value of K_m required to maximize enzymatic activity (v), a finding which is supported by our bioinformatic analysis of approximately 1000 wild-type enzymes.

Results

Construction of the Thermodynamic Model

In principle, an ideal enzyme with low K_m and large k2 can be realized if both k1 and k2 are increased simultaneously. However, this is physically unrealistic, because the driving force which can be allocated to k₁ and k₂ is limited by the free energy change of the entire reaction. Within this thermodynamic context, maximum activity is realized by optimizing the distribution of the total driving force between the first (E +S → ES) and second (ES → P) steps shown in Scheme 1. To quantitatively evaluate the relationship between the driving force and the activity, we have used the BEP relationship^[19-23] to convert driving forces ( ΔG ) into activation barriers ( Ea ), and the Arrhenius^[24] equation to convert activation barriers to rate constants.

The thermodynamic model which served as the basis of our calculations is shown in Fig. 1. In a classical Michaelis-Menten reaction, the enzyme and substrate first form an enzyme-substrate complex (E+S → ES) before producing the product in the second step (ES → P). This mechanism is conceptually similar to reactions that occur on a heterogeneous catalyst surface, where the substrate molecule first binds to the catalyst surface before being converted into the product.^[19-23] The Gibbs free energies for the formation of the enzyme-substrate complex and the product are denoted as ΔG₁ and ΔG₂, respectively. By definition, their sum must equal the total free energy change of the reaction ΔGT:

Figure 1.

The free energy landscape corresponding to the mechanism shown in Scheme 1. Each reaction in the mechanism is labeled by its corresponding rate constant. The free energy landscape below indicates the free energy changes (ΔG₁, ΔG₂) and activation barriers (E_a1, E_a1r, E_a2).

Reaction Coordinates

From these thermodynamic constraints, we will use the BEP relationship^[19-23] to obtain activation barriers (Ea), and then the Arrhenius^[24] equation to obtain rate constants, which ultimately yields quantitative insight on the relationship between k₁, k₂, and K_m. Based on the BEP relationship, the activation barrier corresponding to k₁ can be written as: where represent the activation barriers when the elementary reaction is in equilibrium (ΔG₁ = 0). They are positive constants which reflect the inherent favorability of this elementary step. α₁ is a positive constant coefficient which indicates the sensitivity of the activation barrier with respect to the driving force. Recently, Kari et al have shown that fungal cellulases indeed satisfy such linear free energy relationships between the activation barrier and the driving force.^[25] Next, activation barriers can be converted to rate constants based on the Arrhenius equation^[24] as follows: Here, A₁ is a pre-exponential factor, and R and T are the gas constant and absolute temperature, respectively. Using Eqs. (4) and (5), k₁ can be expressed as: where and were used to aggregate factors independent and dependent on the driving force, respectively (see Supporting Information, Appendix 1 for details). k_1r and k₂ can also be written similarly as: using notations similar to those defined for k₁ (See Appendices 2 and 3 for details). Substituting these rate constants into Eq. (2) yields the following expression for K_m: where K was defined as Finally, based on Eqs. (8) and (9), the enzymatic activity (v) can be expressed as: To illustrate how Eq. (10) captures the tradeoff relationship between k₂ and K_m, numerical simulations were performed (Fig. 2A). Hereafter, all simulations will be performed at α₁ = α₁r = α₂ = 0.5, which is a common assumption used to make baseline models in heterogeneous catalysis.^[22,26-28] Physically, this means that when the driving force of an elementary reaction is increased by 1 kJ/mol, its activation barrier decreases by 0.5 kJ/mol. In reality, typical experimental values of α range between 0.3 and 0.7 for artificial catalysts,^[29-31] and the experimental value reported for cellulases is 0.74.^[25] Therefore, the influence of α deviating from 0.5 will be discussed in detail in Fig. 5D.

Figure 2.

Thermodynamic landscapes (A) and their corresponding activity shown in the form of Michaelis-Menten plots (B). The K_m values are indicated as vertical dashed lines in (B). Increasing the driving force of the first step increases activity at low substrate concentrations but lowers the activity at high substrate concentrations. Therefore, the thermodynamic landscape of an optimum enzyme depends on the substrate concentration (S). The free energies of the enzyme-substrate complex (ΔG₁) were −25, −20, and − 15 kJ/mol for the black, blue, and red lines, respectively, and that of the total reaction (ΔG_T) was −40 kJ/mol. All numerical simulations in this study were performed at E_T = 1 µM, (1/µM/s and 1/s units, respectively).

Fig. 2A shows three possible thermodynamic landscapes for a reaction with a total driving force of ΔGT = −40 kJ/mol. This parameter was chosen as a representative value based on the fact that the ΔGT of typical biochemical reactions is between −80 ~ + 40 kJ/mol.^[32,33] Similar calculations with different values of ΔGT can be found in Figs. S1-S3. When the first reaction is thermodynamically favorable compared to the second (ΔG₁ < ΔG₂; Fig. 2A, black lines), the activity increases rapidly from low substrate concentrations (Fig. 2B, solid black line), consistent with the small K_m value. However, an enzyme with a small K_m suffers from a small k₂ value, which is evident from the saturating behavior at S > 1 µM. Increasing the driving force of the second step (blue and red lines) leads to a larger k₂ and thus higher activity at large S values (S > 1 µM) compared to the enzyme shown in black. At the same time, however, K_m increases, which decreases the enzymatic activity at low S (S < 1 µM). The difference in activity at low and high substrate concentrations occurs because the substrate participates in only the first elementary step. For example, even if k1 < k2 (ΔG1 > ΔG2), the rates of the two forward reactions (k1E ∙ S and k₂(ES)) can be matched if the substrate concentration (S) is sufficiently large. However, at low substrate concentrations, a small k₁ can no longer be compensated, resulting in the first step being rate-limiting. For this reason, a large k₁ is necessary to increase the enzymatic activity at low substrate concentrations, whereas a large k₂ is more desirable when the substrate concentration is sufficient. The balance in tradeoff changes when the rates of the two forward reactions are equal As the optimum values of k₁ and k₂ are dependent on the substrate concentration (S), the K_m value necessary to maximize the activity must also be dependent on (S).

Analysis of the Activity – Driving Force Relationship

To directly illustrate the influence of driving force ( ΔG₁ and ΔGT ) on enzymatic activity, we performed numerical simulations using Eq. (10) at various substrate concentrations (Fig. 3). At a substrate concentration of 0.1 μM (Fig. 3A), the region of highest enzymatic activity (orange) was observed in the bottom left region. It is reasonable for activity to be higher in the lower half of the panel, due to the more negative ΔGT. A negative ΔG₁ is also beneficial for activity at a low substrate concentration (S =0.1 μ), leading to enzymatic activity being higher in the left half of the panel. At higher substrate concentrations, the overall color within each panel changed from blue to red, because a higher substrate concentration always increases activity (Figs. 3B-3D). At the same time, the ΔG₁ corresponding to maximum activity gradually shifted positively (black dashed lines). This finding is consistent with Fig. 2 which shows that a more positive ΔG₁ is desirable when the substrate concentration is increased. In each panel, the location with the highest activity at a given ΔGT value is shown as a dashed black line. Notably, when the K_m value was calculated at the (ΔG₁, ΔGT) values under the dashed line using Eq. (9), the obtained value was always equal to the substrate concentration S in each panel. In other words, the dashed line is not only the ridge of the volcano plot, but also the contour line showing K_m = S . This suggests that the condition for maximizing enzymatic activity can be represented by K_m = S.

Figure 3.

Enzymatic activity (v) plotted against ΔG₁ and ΔG_T based on Eq. (10). The substrate concentration (S) in each panel was (A) 10⁻¹, (B) 1, (C) 10, and (D) 10² μM, as indicated in the bottom right of each panel. In all panels, the black dashed line corresponding to K_m = S overlaps with the region with the highest enzyme activity.

To examine why K_m = S leads to maximum activity, Eq. (10) was rearranged to give the following expression for the activity (v): in which g₁ is only in the denominator. The derivative of the denominator, denoted as f is: To maximize the activity (v), f must be minimized which is realized at: Considering that K_m is defined as K_m ≡ g₁(1 + K) (Eq. (10), Eq. (13) yields a surprisingly simple formula for the condition of maximum activity when α1 = α1r = α2 = 0.5: Eq. (14) provides the theoretical basis for why maximum activity was consistently observed along the contour line K_m = S in Fig. 3: The combination of (ΔG₁, ΔGT) necessary to maximize activity guarantees K_m = S. This finding is further illustrated in Fig. 4, where the activity (v) is plotted as a function of K_m at different substrate concentrations. In all cases, maximum activity (v) is observed when the binding affinity (K_m) is equal to the substrate concentration (S). Thus, the derivations and simulations so far provide mathematical evidence that having a K_m value equal to the substrate concentration S guarantees maximal enzymatic activity as long as the enzyme follows the Michaelis-Menten mechanism (Scheme 1), and the rate constants follow the BEP relationship with α₁ = α_1r = α₂ = 0.5.

Figure 4.

Volcano plots showing how the activity is expected to change with respect to the Michaelis-Menten constant ( K_m ). As the substrate concentration was increased from 10⁻¹ μM (black) to 10² μM (red), the volcano plot shifted to the upper right. The apex is located at K_m = S, as indicated by the vertical dashed lines of the corresponding color. ΔG_T = −40 kJ/mol and [1/μM/s and 1/s units, respectively] were used for the numerical simulations. Changingthese values did not influence the conclusion that the activity is maximized when K_m = S, as shown in Fig. S4.

Robustness of the Theoretical Model

To confirm the robustness of our finding, we have performed numerical simulations by loosening each of the theoretical requirements. Deviation from the Michaelis-Menten mechanism (Scheme 1) are shown in Fig. 5A-C, and deviation of α values from 0.5 are shown in Fig. 5D. The possibility of reverse reactions (P→S) or inhibition (E + I → EI or ES + I → ESI) are common deviations from Michaelis-Menten kinetics.^[34] The net rate in the presence of a reverse reaction when the substrate and product are in equal concentrations (S = P = 10 µM) is shown in Fig. 5A. In terms of maximizing the activity in the forward direction (S → P), the physically meaningful region is (ΔGT < 0), where the net reaction proceeds in the forward direction. Under this condition, the dashed line corresponding to K_m = S and the solid line corresponding to the true maximum activity (forward minus reverse reaction rates) overlap almost completely, indicating that K_m = S is a good guideline to enhance activity even in the presence of reverse reactions (P → S).

Figure 5.

Influence of (A) Reverse reaction, (B) Competitive inhibition, (C) Uncompetitive inhibition, and (D) BEP coefficient, α on the optimal K_m. The dashed line corresponds to K_m = S, with S = 10 µM. The true optimum K_m for each mechanism is shown as a solid line along with its analytical equation (refer to the SI for the derivations). In panel A, the product concentration (P) was set to 10 µM. The top half of (A) was colored at an arbitrarily low activity because the reverse reaction is more favorable in this region. The large discrepancy between the dashed and solid lines at ΔG_T > 0 is physically irrelevant, because the activity of the forward reaction cannot be discussed when the net reaction proceeds in the reverse direction. In panels B and C, the degree of inhibition (γ ≡ I/K_i) was set to 10. In panel D, the BEP coefficients were set to α₁ = α₂ = 0.2. No analytical solution was obtained for D.

Similar calculations for competitive and uncompetitive inhibition, where the inhibitor binds to either the free enzyme or the enzyme-substrate complex, are shown in Fig. 5B,C. The degree of inhibition , is determined by the inhibitor concentration (I) and the equilibrium constant of inhibition (Ki).^[34] Based on the experimental data of Park et al.,^[35] γ can range from 10⁻⁴ to 10⁴. As γ was less than 10 in approximately 80% of their data, γ = 10 was used here for the numerical simulations. Again, the optimal K_m (solid line) deviates only slightly from the dashed line (K_m = S), and both lines pass through the region of high activity (orange). The K_m values are approximately 1 order of magnitude apart between dashed and solid lines, yet there is only a 57 % difference in activity at a specific ΔGT. This is much smaller than the scale of the entire diagram (10 orders of magnitude), suggesting that adjusting K_m to the substrate concentration S is a robust strategy to enhance the activity, even in the presence of inhibition. A detailed discussion on the parameter dependence (γ, S), as well as for other mechanisms such as substrate inhibition or allostericity can be found in Section 3 of the supporting information. The derivations for the equations of the true optimal K^m can also be found in the same section.

The influence of the second assumption (α1 = α1r = α2 = 0.5) is shown in Fig. 5D. As physical constraints require α1r = 1 − α1 (Appendix 2), only α1 and α2 are independent. In an extreme case where α₁ = α₂ = 0.2, the activity is markedly diminished because rate constants hardly change even if their driving force is increased. However, the dashed line still passes through the region of high activity, and the activity is still less than an order of magnitude away from the true optimum (solid lines). Taken together, these simulations confirm that K_m = S is a robust theoretical guideline to enhance enzymatic activity.

Validation based on Experimental Data

Finally, to evaluate whether K_m = S can rationalize enzymatic properties in nature, we have analyzed their relationship based on the experimental data from Park et al.^[35] The original data consisted of K_m values of wild-type enzymes obtained from BRENDA, and intracellular S values obtained from Escherichia coli, Mus musculus, and Saccharomyces cerevisiae cells, yielding a total of 1703 K_m–S combinations. This dataset was then classified based on the number of entries for each substrate, based on the expectation that a substrate which participates in many reactions is more likely to deviate from Michaelis-Menten kinetics. ATP is the most frequent substrate with 313 entries and is shown in black. Both the raw K_m and S values (Fig. 6A) and their relative distribution (Fig. 6B) shows that S > K_m for ATP. The deviation from K_m = S may be because the Michaelis-Menten mechanism, which is the basis of our mathematical analysis, does not consider scenarios where multiple reactions compete for the same substrate. The next subset shown in blue covers 410 entries and consists of 5 substrates which each appear more than 50 times: NAD⁺, NADH, NADP⁺, NADPH, and acetyl-CoA. These cofactors are less universal than ATP, and S is only slightly larger than K_m. The remaining 980 entries are shown in red. This subset contains 115 substrates such as carbon metabolites and amino acids and appear within the dataset 8 times on average. As the substrate becomes less universal, their K_m and S values become roughly consistent. In particular, the Gaussian distribution fitted to the red histogram (Fig. 6B) has a center at log₁₀ S/K_m = 0.18 and a standard deviation of 1.3, which is reasonable considering that influences from inhibitors or the BEP coefficient can change the optimum K_m by roughly an order of magnitude (Fig. 5). Thus, the dataset from wild-type enzymes supports the theoretical prediction that a Michaelis-Menten constant equivalent to the substrate concentration is favorable for the activity, especially when the substrate participates in fewer reactions and Michaelis-Menten kinetics becomes more accurate.

Figure 6.

Relationship between K_m and S from the dataset reported by Park et al.^[35] The raw values of K_m and S are shown in (A), and their relative values are plotted in (B). Each entry of K_m and S was categorized based on the number of times the substrate appeared in the entire dataset. Black: > 300 (ATP), blue: > 50 (NAD⁺, NADH, NADP⁺, NADPH, and acetyl-CoA), red: < 50 (others). The number of entries was used as a proxy for the validity of the Michaelis-Menten mechanism of the specific substrate. The dashed line in (A) corresponds to K_m = S, and the shaded area shows a deviation of 1 order of magnitude.

Discussion

So far, various criteria^[13,34,36] such as large k2 ( kcat ), small K_m, or large k2/K_m have been proposed to characterize enzymes with high activity, making it difficult to rationally evaluate or engineer the activity of an enzyme. The lack of a universal consensus is largely due to the mutual dependence between k₂ and K_m. As our theoretical model addresses this challenge directly and maximizes the activity within the thermodynamic constraints imposed by k₂ and K_m, we believe that K_m = S is a criterion for high activity which is viable in a wider range of scenarios.

The idea that the Michaelis-Menten constant should be increased at higher substrate concentrations to maximize activity is consistent with the experimental work by Kari et al,^[37] who measured the activity of cellulases with different K_m . When the substrate concentration was increased 6 times, the K_m value of the most active enzyme increased approximately 2.4 times. Considering that K_m can change by roughly 6 orders of magnitude, the experimental trend supports our hypothesis K_m = S, especially when their experimental BEP coefficient of 0.74 is considered. The idea of the optimum binding affinity being dependent on the reaction condition and driving force is also consistent with recent theoretical models of heterogeneous catalysis.^[22,38-40]

As a corollary, our model which quantifies the relationship between K_m and k₂ immediately rationalizes the recently reported free energy relationship between them in cellulases.^[25] Namely, the relationship between K_m and k₂ can be written as: This equation shows that log k₂ and log K_m are linearly correlated by a factor of α₂, and provides a physical basis to the high linearity (R₂ = 0.95) observed for cellulases.^[25] The consistency between our theoretical model and previously accumulated experimental insight suggests that it may be possible to quantitatively rationalize enzymatic properties based on fundamental principles of physical chemistry.

Online Methods

The mathematical formulas were derived by hand, and the step-by-step derivations for the standard Michaelis-Menten mechanism are explained in the main text. The derivations in the presence of inhibition and allostericity are provided in the supporting information. Numerical simulations and bioinformatic analysis were performed using Python 3.9.12. The code used for the analysis can be found in the extended data or accessed directly at github: https://github.com/HideshiOoka/SI_for_Publications.

Acknowledgments

H.O. gratefully acknowledges the support from the JST FOREST program (Grant Number JPMJFR213E, Japan). Y. C. is grateful for the support from the JST ACT-X program (Grant Number JPMJAX20BB, Japan).

Footnotes

• Competing Interest Statement: The authors declare no competing interests.