Coherence-Dependent Barrier Permeabilityin Enzyme Catalysis
- Don Gaconnet
- 2 days ago
- 18 min read
A Membrane Model for Anomalous
Kinetic Isotope Effects
Predictions for Hydrogen Tunneling in Alcohol Dehydrogenase
and Soybean Lipoxygenase
Don Gaconnet
LifePillar Institute for Recursive Sciences
February 2026
Working Paper
ABSTRACT
The anomalous kinetic isotope effects (KIEs) observed in hydrogen tunneling reactions catalyzed by enzymes such as soybean lipoxygenase (SLO, kH/kD ≈ 80) and thermophilic alcohol dehydrogenase (ht-ADH) have been extensively studied within the framework of compressive protein motions that reduce donor-acceptor distance (DAD) to enable tunneling.
While this framework explains many observations, several anomalies persist: KIE magnitudes exceeding semiclassical predictions even at modeled DAD values, the sharp phase-transition character of the 30°C threshold in ht-ADH, and cases where computed compression does not predict tunneling efficiency. We propose an additional variable: the coherence of the protein scaffold’s thermal network, formalized as a membrane coherence function C(N) that modulates the effective quantum of action.
The central equation is heff = h₀ / C(N), where h₀ is the standard Planck constant and C(N) is the coherence of the local field maintained by the protein scaffold. In this model, the organized thermal conduits identified by Klinman and colleagues do not merely transmit compressive force—they maintain field coherence that reduces the effective barrier to tunneling independent of physical compression. This framework generates four specific, experimentally testable predictions: (1) coherence-disrupting mutations should reduce tunneling rates more than barrier-width changes alone predict; (2) the anomalous KIE should correlate with thermal network integrity as measured by H/D exchange patterns; (3) solvent modifications disrupting hydration shell structure should affect tunneling independently of viscosity; (4) the ht-ADH temperature transition should exhibit critical-point behavior consistent with a coherence phase transition. Existing published data from the Klinman laboratory is discussed in relation to these predictions.
Keywords: hydrogen tunneling, kinetic isotope effects, enzyme catalysis, protein dynamics, membrane coherence, effective Planck constant, soybean lipoxygenase, alcohol dehydrogenase, thermal network, donor-acceptor distance
1. INTRODUCTION
Hydrogen transfer in enzyme-catalyzed reactions proceeds, in many cases, via quantum mechanical tunneling rather than classical over-the-barrier mechanisms. This has been established through extensive work over four decades, particularly by Klinman and colleagues, using kinetic isotope effects as the primary diagnostic tool (Klinman, 2006; Hay & Scrutton, 2012). The observation that enzymes such as soybean lipoxygenase (SLO) exhibit primary KIEs of kH/kD ≈ 80—far exceeding the semiclassical limit of approximately 7—provides compelling evidence for deep tunneling contributions to catalysis.
The dominant theoretical framework for understanding enzyme tunneling centers on the donor-acceptor distance (DAD) as the critical variable. Tunneling probability depends exponentially on barrier width: T ∝ exp(−2d√(2mV)/ħ), where d is the tunneling distance, m is particle mass, and V is barrier height. X-ray crystal structures typically show equilibrium DAD values of >3.2 Å, but theoretical modeling predicts that efficient tunneling requires DAD ≈ 2.7 Å—a tunneling distance of approximately 0.7 Å. The field’s current resolution of this discrepancy invokes “compressive protein motions”: the enzyme dynamically samples conformations that transiently bring donor and acceptor closer together, enabling tunneling from compressed configurations.
This compressive motions framework has proven powerful, explaining the temperature dependence of KIEs, the effects of many active-site mutations, and the coupling between protein dynamics and chemistry. However, several anomalies remain:
First, the magnitude of the SLO KIE (kH/kD ≈ 80 at room temperature) exceeds what semiclassical models produce even with optimized DAD values. The problem is not barrier height—it is that the observed tunneling rate is higher than standard quantum mechanical calculations predict for the available barrier geometry.
Second, the temperature-dependent transition in thermophilic alcohol dehydrogenase (ht-ADH) at 30°C is sharper than a gradual loss of compression would predict. Below 30°C, KIEs become strongly temperature-dependent; above 30°C, they are nearly temperature-independent. This transition has the character of a phase transition rather than a smooth decline in conformational sampling.
Third, there are documented cases where computed compression does not predict tunneling efficiency. In dihydrofolate reductase (DHFR), computational studies have shown that forced compression can actually decrease tunneling probability, contradicting the simple prediction that shorter DAD always enhances tunneling.
We propose that these anomalies can be resolved by introducing a second independent variable alongside DAD: the coherence of the protein scaffold’s thermal network. This proposal is motivated by recent experimental work from the Klinman laboratory identifying organized thermal conduits in SLO—anisotropic networks of aliphatic residues that channel thermal energy through specific pathways from the protein surface to the active site. We formalize this as a membrane coherence function C(N) that modulates the effective quantum of action, generating specific predictions for existing anomalous data.
2. THE ANOMALIES
2.1 Soybean Lipoxygenase: kH/kD ≈ 80
Soybean lipoxygenase catalyzes the oxidation of linoleic acid through a mechanism involving hydrogen atom transfer from the C-11 position of the substrate to the Fe(III)-OH cofactor. The primary KIE of approximately 80 at room temperature is one of the largest ever observed for an enzymatic hydrogen transfer and has been the subject of intensive theoretical and experimental study for over two decades.
Marcus-like models that incorporate tunneling through a distribution of DAD values, modulated by thermally sampled protein conformations, can reproduce many features of the SLO kinetics. However, achieving the full magnitude of the observed KIE requires either tunneling distances shorter than structural data supports or effective barriers lower than computed potential energy surfaces indicate. The residual discrepancy between the best semiclassical tunneling models and the observed rate suggests that the standard quantum mechanical framework may be missing a variable.
2.2 Thermophilic ADH: The 30°C Transition
Thermophilic alcohol dehydrogenase from Bacillus stearothermophilus (ht-ADH) catalyzes the NAD+-dependent oxidation of benzyl alcohol. The enzyme exhibits a dramatic transition in catalytic behavior at approximately 30°C. Above this temperature, the protein is optimally flexible, generates active-site compression allowing close approach of the hydrogen donor and acceptor, and KIEs are nearly temperature-independent—consistent with efficient tunneling through a narrow barrier at short DAD. Below 30°C, the protein rigidifies, KIEs become strongly temperature-dependent, and the catalytic rate drops sharply.
The current explanation attributes this transition to the loss of protein flexibility: below 30°C, the protein cannot generate sufficient compressive motions to achieve the short DAD required for efficient tunneling. While this explanation is broadly consistent with the data, the sharpness of the transition is notable. A gradual decrease in protein flexibility would be expected to produce a gradual change in KIE behavior. Instead, the transition at 30°C has the character of a cooperative phenomenon—more consistent with a phase transition in the protein’s dynamic state than with a smooth decline in conformational sampling amplitude.
Concurrent hydrogen/deuterium exchange data show a structural transition in the same temperature range, with the protein’s global dynamics undergoing a qualitative shift. The coincidence of the dynamic transition with the catalytic transition suggests that the relevant variable is not simply the amplitude of active-site compression but rather a global property of the protein’s dynamic state.
2.3 When Compression Fails
If DAD reduction were the sole determinant of tunneling efficiency, then any manipulation that reduces DAD should enhance tunneling, and any that increases DAD should diminish it. This prediction is violated in several systems. Computational studies on DHFR have shown that forced compression of the active site can actually decrease tunneling probability—the barrier width is reduced, but the barrier shape changes unfavorably. Similarly, certain active-site mutations in SLO produce effects on tunneling rates that exceed what changes in DAD alone predict. The mutations simultaneously alter both the geometry of the active site and the packing interactions that maintain the structural organization of the surrounding protein scaffold.
These observations suggest that the tunneling rate depends on at least two variables: the physical distance between donor and acceptor (barrier width) and some property of the protein environment that modulates tunneling probability independently of geometry. We propose that this second variable is the coherence of the protein scaffold’s thermal network.
3. THE MEMBRANE COHERENCE MODEL
3.1 The Thermal Network as a Coherence-Maintaining Structure
In 2023, Klinman and colleagues reported a landmark finding in SLO: a “radiating cone” of aliphatic residues extending from the active site to the protein-solvent interface. The activation energy for nanosecond-timescale dynamics at the protein surface was found to match the activation energy for millisecond-timescale C-H bond cleavage at the active site, approximately 30 Å away. Two processes differing by a factor of one million in speed share matching activation energies, coupled across a radiating cone of hydrophobic residues.
Klinman and colleagues described these as “anisotropic thermal conduits”—the protein scaffold channels thermal energy through specific pathways while blocking thermal noise from other directions. We note that this description is functionally identical to a selectively permeable membrane: a structure that discriminates signal from noise, channels specific forms of energy while blocking others, and maintains distinct conditions on either side of a boundary. The protein scaffold surrounding the active site functions as a membrane in the precise sense—not metaphorically, but operationally.
This observation motivates a formal treatment. If the thermal network functions as a coherence-maintaining membrane, then its integrity should modulate not merely the delivery of compressive force to the active site but the quantum mechanical character of the active-site chemistry itself.
3.2 The Coherence Function
We define a membrane coherence function C(N) that characterizes the ability of the protein scaffold to maintain organized energy transfer:
C(N) = f(σ, κ, τ) (1)
where:
σ (selectivity): the degree to which the thermal network discriminates directional energy transfer from isotropic thermal noise. In the SLO context, this corresponds to the anisotropy of the radiating cone—how effectively the aliphatic residue network channels thermal energy toward the active site versus allowing it to dissipate isotropically.
κ (coupling strength): the efficiency of energy transfer between the protein surface and the active site. This is reflected in the matching activation energies across timescales observed by Klinman—when coupling is strong, surface dynamics and active-site chemistry are energetically matched despite operating on timescales differing by six orders of magnitude.
τ (temporal stability): the duration over which the thermal network maintains coherent energy transfer. This determines whether the organized state persists long enough to influence the tunneling event, which occurs on femtosecond timescales but is gated by protein motions on nanosecond to millisecond timescales.
3.3 The Effective Planck Constant
We propose that the coherence of the local field modulates the effective quantum of action governing tunneling at the active site:
h_eff = h₀ / C(N) (2)
where h₀ is the standard Planck constant and C(N) is the membrane coherence function defined above.
The physical interpretation is as follows. In a standard treatment, the tunneling probability through a rectangular barrier is:
T ∝ exp(−2d √(2mV) / ħ) (3)
where d is barrier width, m is particle mass, V is barrier height, and ħ is the reduced Planck constant. Every variable in this expression has been studied extensively in the enzyme tunneling literature: d through DAD measurements and compressive motion modeling; m through isotope substitution; V through mutagenesis and computational chemistry. The quantum of action ħ has been treated as invariant.
We propose that ħ is not invariant in the context of a coherent protein scaffold. When the membrane coherence is high—C(N) > 1—the effective quantum of action decreases:
h_eff < h₀ when C(N) > 1
This means the effective quantum graininess of the system is finer in a coherent field. Tunneling probability increases not because the barrier has thinned (reduced d) or lowered (reduced V) but because the effective quantum of action governing the tunneling event has decreased. The barrier does not need to thin as much if the quantum of action itself is smaller in a coherent field.
Conversely, when the membrane coherence is disrupted—C(N) < 1—the effective quantum of action increases:
h_eff > h₀ when C(N) < 1
The system becomes more classical. Tunneling probability drops below the standard semiclassical prediction. The protein reverts toward over-the-barrier chemistry even at DAD values where tunneling should be favorable under standard quantum mechanics.
Incorporating this into the tunneling probability:
T ∝ exp(−2d √(2mV) · C(N) / ħ₀) (4)
This expression contains two independent handles on tunneling probability: d (barrier width, modulated by compressive motions) and C(N) (field coherence, modulated by thermal network integrity). The standard compressive motions framework captures the first handle. The present model adds the second.
3.4 Isotope Sensitivity of the Coherence Effect
A critical feature of the h_eff model is that the coherence effect is isotope-sensitive. Deuterium, with twice the mass of protium, is more sensitive to changes in ħ_eff than protium is. This follows directly from the tunneling probability expression: the mass term m appears under the square root in the exponent, and ħ_eff appears in the denominator. When C(N) > 1 and ħ_eff decreases, the tunneling probability increases for both isotopes—but the increase is proportionally larger for deuterium because its larger mass makes it more dependent on the quantum of action for tunneling.
This produces a specific prediction: in a coherent field, the KIE (kH/kD) should be larger than the semiclassical prediction. The coherence amplifies the isotope effect because it enhances tunneling more for the heavier isotope (relative to its semiclassical baseline) than for the lighter one. This is precisely what is observed in SLO, where the KIE of approximately 80 exceeds all semiclassical models.
3.5 Integration with the Compressive Motions Framework
The present model does not replace the compressive motions framework. It adds a second independent variable. Compression and coherence are both required for efficient tunneling, and they can vary independently:
Condition | DAD (d) | Coherence C(N) | Tunneling |
Optimal catalysis | Short (compressed) | High (coherent network) | Maximum |
Compression only | Short (compressed) | Low (disrupted network) | Reduced |
Coherence only | Long (equilibrium) | High (coherent network) | Enhanced vs. classical |
Neither | Long (equilibrium) | Low (disrupted network) | Minimal |
Table 1. Four regimes of tunneling efficiency as determined by the two independent variables: donor-acceptor distance (compression) and membrane coherence.
This two-variable model explains why compression alone sometimes fails. Forcing atoms closer together in a disordered environment does not help if the field coherence that reduces h_eff has been lost in the process. This is the resolution of the DHFR paradox: forced compression disrupted the coherence of the local scaffold, raising h_eff enough to offset the benefit of reduced DAD.
It also explains why certain mutations produce effects larger than compressive models predict. Removing a hydrophobic side chain from the thermal network simultaneously increases DAD (geometric effect) and disrupts C(N) (coherence effect). The two effects compound. Models tracking only the first variable underestimate the impact.
4. REANALYSIS OF EXISTING DATA
4.1 SLO Mutation Series
The I553 position in SLO has been subjected to systematic mutation: I553L, I553V, I553A, and I553G, representing progressive loss of hydrophobic bulk. Published data include KIE values, activation energies, catalytic rates, and H/D exchange patterns for each variant. The standard interpretation focuses on the progressive increase in DAD as side-chain bulk is reduced, decreasing compression and tunneling efficiency.
The present model proposes a reanalysis. The I553 position lies within the radiating cone of aliphatic residues identified by Klinman’s group. Progressive removal of hydrophobic bulk at this position does not merely increase the active-site DAD—it removes a node from the anisotropic thermal conduit. The coherence function C(N) decreases alongside the increase in d.
The proposed reanalysis would correlate thermal network integrity (as measured by H/D exchange extent across the radiating cone) with the anomalous component of the KIE—defined as the ratio of observed KIE to the KIE predicted by semiclassical tunneling models at the computed DAD. If the coherence model is correct, this correlation should be stronger than the correlation between KIE and DAD alone.
4.2 ht-ADH Temperature Series
Published data for ht-ADH include KIE values, catalytic rates, and H/D exchange across a temperature range of 10°C to 65°C. The 30°C transition has been interpreted as the point where protein flexibility drops below the threshold required for effective compression.
The present model reinterprets this transition as a coherence phase transition. Below 30°C, the thermal network that maintains field coherence across the protein scaffold undergoes a qualitative change in its dynamic state. The transition is sharp because coherence is a cooperative phenomenon—it depends on the collective behavior of a network of coupled residues, not on the amplitude of any single motion. When the network loses coherence, it does so cooperatively, producing a phase-transition-like signature rather than a gradual decline.
If this interpretation is correct, the 30°C transition should exhibit specific phase-transition signatures when examined at high temperature resolution: power-law scaling of the transition amplitude near the critical temperature, fluctuation enhancement (increased variance in KIE measurements near 30°C), and changes in the spatial pattern of H/D exchange consistent with correlation length divergence.
4.3 The Matching Activation Energies
Perhaps the most striking feature of Klinman’s 2023 SLO data is the observation that the activation energy for nanosecond surface dynamics matches the activation energy for millisecond C-H cleavage, across 30 Å of protein and six orders of magnitude in timescale. The current interpretation attributes this to heat transfer through the anisotropic thermal conduit: the same energy barrier governs both the surface fluctuation and the active-site chemistry because they are connected by a thermal pathway.
The present model offers a complementary interpretation. The matching activation energies reflect the coherence maintenance cost of the membrane. Both processes—surface dynamics and active-site chemistry—are coupled through the same coherence field. The activation energy is not merely the barrier to heat transfer along a conduit but the barrier to maintaining the organized state that enables coherent energy transfer. This interpretation predicts that mutations disrupting the thermal network should decouple the two activation energies—surface dynamics and active-site chemistry would show different E_a values in a variant where the radiating cone is disrupted.
5. TESTABLE PREDICTIONS
5.1 Prediction 1: Distinguishable Mutation Signatures
Design: Compare a mutation at a thermal-network position (within the radiating cone) to a mutation at a non-network position, where both mutations produce a similar computed change in DAD.
Measurement: KIE, catalytic rate, H/D exchange across both variants.
Expected result: The network mutation should produce a larger reduction in tunneling rate and a larger increase in KIE temperature dependence than the non-network mutation at matched DAD change. The compressive motions model predicts similar effects for both mutations at the same DAD. The coherence model predicts that the network mutation produces a compound effect (increased d + decreased C(N)) while the non-network mutation produces only the geometric effect (increased d, C(N) unaffected).
Existing data: The I553 mutation series in SLO may already contain information relevant to this prediction, if H/D exchange data are available for comparison with mutations at geometrically similar but non-network positions. A targeted comparison has not been performed.
5.2 Prediction 2: KIE–Coherence Correlation
Design: Across the existing library of SLO and ADH variants, correlate the anomalous component of the KIE (observed/semiclassical ratio) with thermal network integrity as measured by the extent and pattern of H/D exchange.
Measurement: For each variant: (a) compute the semiclassical KIE predicted by Marcus-like models at the computed DAD; (b) compute the ratio of observed KIE to predicted KIE (the anomalous component); (c) quantify H/D exchange extent across the radiating cone residues (the coherence proxy).
Expected result: The correlation between the anomalous KIE component and H/D exchange extent should be statistically significant and stronger than the correlation between the anomalous component and DAD alone. This would indicate that a variable beyond geometry is contributing to the tunneling rate.
5.3 Prediction 3: Hydration Shell Perturbation
Design: Vary the cosolvent composition using agents that differ in their effect on hydration shell structure while controlling for bulk viscosity. Specifically, compare trehalose (a structured cosolvent that preserves hydration shell organization), DMSO (a disordered cosolvent that disrupts hydration shell structure), and glycerol (intermediate), matched for viscosity.
Measurement: Tunneling rate and KIE for SLO or ht-ADH in each cosolvent condition at matched viscosity.
Expected result: At matched viscosity, structured cosolvents should preserve tunneling efficiency better than disordered cosolvents. The compressive motions model, in which viscosity is the relevant solvent variable (affecting conformational sampling rates), predicts no difference at matched viscosity. The coherence model predicts that hydration shell structure contributes to the membrane coherence C(N) at the protein-solvent interface, and that disrupting this structure reduces tunneling independently of viscosity effects on conformational dynamics.
5.4 Prediction 4: Critical-Point Behavior at the ht-ADH Transition
Design: Conduct a fine-grained temperature series across the ht-ADH transition, with measurements every 1°C from 25°C to 35°C.
Measurement: KIE, catalytic rate, H/D exchange extent, fluorescence Stokes shift (as a probe of local dielectric environment), and measurement variance at each temperature point.
Expected result: The transition should exhibit nonlinear, steep behavior inconsistent with a gradual loss of conformational sampling. Specifically, the model predicts: (a) a sigmoidal or step-like profile in KIE temperature dependence rather than a smooth curve; (b) enhanced fluctuations (increased variance in repeated measurements) near the transition temperature, consistent with critical slowing near a phase transition; (c) correlated changes in H/D exchange across the protein (not just at the active site) as the coherent network undergoes a cooperative transition.
The compressive motions model predicts a gradual transition as protein flexibility decreases continuously with temperature. A sharp, cooperative transition with fluctuation enhancement would be consistent with the coherence phase-transition interpretation and inconsistent with a simple flexibility-loss model.
6. DISCUSSION
6.1 Relationship to Quantum Biology
The proposal that field coherence modulates quantum mechanical processes in biological systems connects to the broader field of quantum biology, where coherence effects have been documented in photosynthetic light harvesting, avian magnetoreception, and olfactory discrimination. The present model is narrower than many quantum biology proposals: it does not require long-range quantum coherence across the entire protein. It requires only that the protein scaffold maintains a locally coherent thermal environment around the active site—a condition that the experimentally observed anisotropic thermal conduits are specifically designed to provide.
The h_eff formulation makes this concrete. We are not proposing that the Planck constant changes. We are proposing that the effective quantum of action relevant to the tunneling event is modulated by the coherence of the local environment, in the same way that effective mass in solid-state physics is modulated by the crystal lattice. The lattice does not change the electron’s rest mass; it changes the effective mass that governs the electron’s behavior in that environment. Similarly, the protein scaffold does not change h; it changes the effective h that governs tunneling behavior in the active site.
6.2 Implications for Enzyme Engineering
If the coherence model is correct, enzyme engineering strategies should consider not only the geometry of the active site but the integrity of the thermal network surrounding it. Mutations that improve the active-site geometry for catalysis may actually reduce catalytic efficiency if they disrupt the coherence network. Conversely, mutations distant from the active site—in the radiating cone or at the protein-solvent interface—could enhance catalysis by improving thermal network coherence without directly altering the active-site geometry.
This provides a framework for understanding the widespread observation that catalytically important mutations often occur at positions distant from the active site. These mutations may be modulating C(N) rather than d.
6.3 The Membrane Model as a General Framework
The coherence function C(N) and the effective Planck constant h_eff were derived from a more general theoretical framework in which a membrane—defined as a selectively permeable boundary maintaining distinct conditions on either side—governs the generative capacity of any system in which observation and exchange occur (Gaconnet, 2026). In this broader framework, the protein scaffold is one instantiation of a membrane whose coherence determines whether quantum behavior is enhanced or suppressed.
Empirical evidence from independent experiments on processing-order effects in language models has demonstrated that a formally analogous dynamic—where a sampling temperature threshold plays the role of membrane permeability—produces structural effects on option space and self-monitoring capacity that are consistent across architectures with no shared training lineage (Gaconnet, 2026). The structural correspondence between the enzyme and synthetic systems is noted here as motivation for the general framework but does not constitute evidence for it. The predictions in Section 5 stand independently of the general framework and are testable within enzymology alone.
6.4 Limitations
The present model has several limitations that must be acknowledged. First, the h_eff equation (Eq. 2) is proposed on physical intuition and structural analogy rather than derived from first principles. A rigorous derivation from quantum field theory or condensed-matter physics would substantially strengthen the proposal. Second, the coherence function C(N) is defined qualitatively; operationalizing it into a quantitative measure that can be computed from molecular dynamics simulations is a necessary next step. Third, the model has not yet been tested against detailed computational chemistry; the predictions in Section 5 are designed to motivate such tests. Fourth, the effective Planck constant framework has precedent in condensed-matter physics (effective mass, renormalized coupling constants) but has not been previously applied to enzyme catalysis, and its validity in this context is unknown.
7. CONCLUSION
We propose a coherence-dependent effective Planck constant as an additional variable in enzyme tunneling kinetics, complementing the established compressive motions framework. The central equation, h_eff = h₀ / C(N), formalizes the hypothesis that the protein scaffold’s thermal network modulates the effective quantum of action at the active site. This framework generates four specific, experimentally testable predictions that can be evaluated using existing techniques on well-characterized enzyme systems.
The model addresses three anomalies in the current literature: KIE magnitudes exceeding semiclassical predictions (resolved by the coherence-enhanced tunneling mechanism), the sharp phase-transition character of the ht-ADH temperature threshold (resolved by the coherence phase-transition interpretation), and cases where compression fails to enhance tunneling (resolved by the two-variable framework in which compression and coherence can be independently disrupted).
The predictions are designed to be tested by enzymology laboratories with existing equipment and techniques, using well-characterized systems (SLO and ht-ADH) for which extensive published data are available. We invite independent testing and critical evaluation.
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APPENDIX A: MATHEMATICAL FRAMEWORK
A.1 Derivation of Modified Tunneling Probability
Starting from the standard WKB approximation for tunneling through a one-dimensional potential barrier:
T = exp(−2 ∫ √(2m(V(x) − E)) / ħ dx) (A1)
For a rectangular barrier of width d and height V:
T = exp(−2d √(2mV) / ħ) (A2)
Substituting the effective Planck constant ħ_eff = ħ₀ / C(N):
T = exp(−2d √(2mV) / (ħ₀/C(N))) (A3)
T = exp(−2d √(2mV) · C(N) / ħ₀) (A4)
The KIE is the ratio of tunneling probabilities:
KIE = T_H / T_D = exp(−2d √(2V) · C(N) · (√m_H − √m_D) / ħ₀) (A5)
Since m_D = 2m_H, the mass difference term (√m_H − √m_D) is negative, meaning T_D < T_H as expected. The C(N) term amplifies this difference: as C(N) increases above 1, the exponent becomes more negative, increasing the KIE beyond the semiclassical prediction.
A.2 Boundary Conditions
The coherence function C(N) is bounded as follows:
C(N) → 0: Complete loss of scaffold coherence. h_eff → ∞. The system is fully classical. No tunneling occurs regardless of DAD. This limit corresponds to a fully denatured protein.
C(N) = 1: The reference state. h_eff = h₀. Standard quantum mechanics applies. This corresponds to a protein with no organized thermal network—thermal energy transfer is isotropic.
C(N) > 1: Enhanced coherence. h_eff < h₀. Tunneling probability exceeds the standard prediction. This corresponds to an intact, organized thermal network maintaining anisotropic energy transfer to the active site.
C(N) → ∞: The theoretical upper limit. h_eff → 0. Not physically realizable but defines the direction of maximum coherence.