Flexible Catalysis: Adaptive Dynamics

• Flexible catalysis is defined as catalytic function arising from dynamic conformational shifts, structural compliance, and mobility that overcome the limitations of static active-site models.
• It encompasses systems from enzymatic and biomimetic catalysts to heterogeneous and electrostatically driven reactors, where adaptability optimizes substrate binding, transition-state stabilization, and product release.
• Computational models and experimental studies reveal that strategic flexibility can optimize rate-limiting steps and balance the trade-offs inherent in classical rigid catalysis.

Flexible catalysis denotes catalytic regimes in which catalytic performance is shaped by conformational change, structural compliance, mobility, spatial heterogeneity, or externally imposed modulation rather than by a single static active-site geometry. In the literature, the concept appears across enzymology, heterogeneous catalysis, single-atom catalysis, electrostatic and oscillation-driven catalysis, and adaptive catalytic materials. A recurrent theme is that classical transition-state stabilization remains necessary, but is often insufficient: catalytic turnover also depends on substrate capture, barrier crossing, product release, catalyst regeneration, and the ability of the catalyst or catalytic environment to reorganize across those steps (Parra et al., 1 May 2025).

1. Conceptual scope and defining features

Classical catalysis is usually framed in terms of lowering an activation barrier without changing the equilibrium free-energy difference between reactants and products. Flexible catalysis extends that picture by treating catalytic function as a property of an evolving system. In proteins, this evolution may be conformational; in nanoparticle systems it may be morphological or positional; in interfacial reactors it may be field-driven; and in dynamic nonequilibrium settings it may be programmed by time-dependent control parameters (Rivoire, 2022).

A central distinction in this literature is between productive flexibility and mere structural softness. Several works argue that flexibility is not generically beneficial. In enzymatic systems, too little frustration makes the catalyst overly rigid and suppresses functionally important motions, whereas too much frustration produces excessive heterogeneity and unproductive motion; catalytic performance is optimized at an intermediate “sweet spot” of local frustration (Parra et al., 1 May 2025). Likewise, a minimal lattice model shows that one obvious form of flexibility—fluctuation away from transition-state-matched geometry—does not solve the basic rigid-catalyst trade-off, whereas a substrate-triggered conformational switch can (Rivoire, 2022). A related misconception is that flexibility implies loss of structural identity. The palladium nanocrystal system discussed below is explicitly “liquid-like” in morphology yet remains crystalline and preserves its initial crystal orientation during motion (Lu et al., 2018).

2. Conformational flexibility in enzymatic and biomimetic catalysis

In biology, flexible catalysis is often formulated in terms of a rugged free-energy landscape with multiple catalytically competent substates. A theory of intracellular enzyme catalysis with time-dependent substrate concentration $[S](t)$ replaces the single Michaelis–Menten cycle by two catalytically active routes: a slower extended cycle and a faster truncated cycle. The switch is controlled by competition between conformational relaxation and substrate capture, yielding a critical substrate concentration

$$[S_c] = \frac{k_{e_1\to e_0}^{0}\, e^{-\beta \Delta G_1}}{4\pi D_S a_S}.$$

Larger $\Delta G_1$ prolongs trapping in the metastable state $e_1$, lowers $[S_c]$, and favors the fast route; smaller $\Delta G_1$ has the opposite effect (Jana et al., 2011).

A broader protein-science view links this behavior to local frustration. In that framework, selected frustrated interactions create “asperities at the bottom of the folding funnel,” populate conformational substates, and channel thermal fluctuations into functionally important motions. The review literature surveyed here states that catalytic sites are enriched in highly frustrated interactions and gives several concrete examples: in Adenylate Kinase, ATP-induced steric and electrostatic frustration lowers the opening barrier from about 20 kcal/mol to 13 kcal/mol and accelerates product release; in nDsbD, oxidation creates local frustration that enables cap-loop opening; in Guanylate Kinase, mutations in distant high-strain regions decrease both flexibility and catalytic activity (Parra et al., 1 May 2025).

A minimal theoretical model of bond cleavage sharpens the mechanistic claim. For a rigid catalyst, the same interactions that stabilize the transition state also hinder product release, leading to a Sabatier-type lower bound on the barrier reduction factor,

$$a \ge \frac{1}{2}+\max\!\left(0,\frac{h_s^+-h_s^-}{2h_s^+}\right).$$

Under the same model, a catalyst with a substrate-triggered discriminative switch can become barrier-free when

$$\epsilon_{cs}=0 \qquad\text{and}\qquad h_s^+ \le \delta\epsilon_{cs}=\epsilon_c \le h_s^-.$$

In that regime the catalyzed process scales with $a=0$, because strong binding is switched on only in productive bound states and switched off for release (Rivoire, 2022). This suggests that the catalytic value of conformational change lies less in generic motion than in state-dependent modulation of binding free energy.

3. Structural compliance, mobility, and adaptive active sites in heterogeneous systems

Flexible catalysis in heterogeneous systems includes catalysts that move, deform, or locally reorganize under reaction conditions. A striking example is provided by approximately 10 nm palladium nanocrystals on amorphous SiN$_x$. In vacuum at pressures around $$[S_c] = \frac{k_{e_1\to e_0}^{0}\, e^{-\beta \Delta G_1}}{4\pi D_S a_S}.$$0 Pa or in N$$[S_c] = \frac{k_{e_1\to e_0}^{0}\, e^{-\beta \Delta G_1}}{4\pi D_S a_S}.$$1 at $$[S_c] = \frac{k_{e_1\to e_0}^{0}\, e^{-\beta \Delta G_1}}{4\pi D_S a_S}.$$2 Pa, the particles remain immobile up to 300 °C. In O$$[S_c] = \frac{k_{e_1\to e_0}^{0}\, e^{-\beta \Delta G_1}}{4\pi D_S a_S}.$$3 at $$[S_c] = \frac{k_{e_1\to e_0}^{0}\, e^{-\beta \Delta G_1}}{4\pi D_S a_S}.$$4 Pa, they move dramatically at 300 °C but not at room temperature, 100 °C, or 200 °C. The motion is accompanied by etched channels in the carbonaceous layer, and EELS line scans show the C K-edge only outside the channel, demonstrating catalytic carbon removal along the trajectory. During migration, the particles undergo elongation–contraction oscillations with eccentricity varying between about 0.65 and 0.93, yet nano-beam diffraction shows that they remain face-centered cubic Pd and preserve their crystal orientation. The same system also exhibits “contagious mobilization,” where a mobile particle decokes a dormant coked one, and fusion followed by fission, which statistically counteracts sintering (Lu et al., 2018).

Single-atom catalysis provides a more localized form of structural flexibility. A comparison of Fe-N$$[S_c] = \frac{k_{e_1\to e_0}^{0}\, e^{-\beta \Delta G_1}}{4\pi D_S a_S}.$$5 and Fe-N$$[S_c] = \frac{k_{e_1\to e_0}^{0}\, e^{-\beta \Delta G_1}}{4\pi D_S a_S}.$$6 model sites on graphene/Ir(111) shows that both sites have essentially the same formal oxidation and spin state—high-spin Fe$$[S_c] = \frac{k_{e_1\to e_0}^{0}\, e^{-\beta \Delta G_1}}{4\pi D_S a_S}.$$7 with $$[S_c] = \frac{k_{e_1\to e_0}^{0}\, e^{-\beta \Delta G_1}}{4\pi D_S a_S}.$$8—and very similar 3d occupancies and d-orbital centers. Nevertheless, the CO adsorption energies differ by more than 0.6 eV: $$[S_c] = \frac{k_{e_1\to e_0}^{0}\, e^{-\beta \Delta G_1}}{4\pi D_S a_S}.$$9 The origin is structural compliance: the Fe-N$\Delta G_1$0 site can lift the Fe atom by about 0.68 Å from the N$\Delta G_1$1 plane. That distortion costs 0.18 eV but strengthens the Fe–CO bond by an additional 0.61 eV, primarily through Fe $\Delta G_1$2–CO $\Delta G_1$3 back-bonding. When Fe-N$\Delta G_1$4 and Fe-N$\Delta G_1$5 are constrained to similar planar geometries, the CO binding energies are essentially identical, $\Delta G_1$6, showing that static electronic descriptors alone are insufficient (Planer et al., 12 Mar 2026).

These cases support a common interpretation: in heterogeneous catalysis, flexibility often enters not by abandoning structural order, but by accessing productive local rearrangements that a rigid site or anchored particle cannot realize.

4. Externally programmable and environment-responsive catalysis

Flexible catalysis also includes systems in which the catalyst is not structurally soft in the usual mechanical sense, but dynamically reconfigured by fields, oscillatory forcing, or reversible environmental transfer. In a microfluidic click-reaction reactor, an oriented electric field localized near two parallel gold electrodes serves as the sole catalyst for a surface-bound azide–alkyne cycloaddition. Under continuous flow of 0.1 M ethynylferrocene in acetonitrile at 50 $\Delta G_1$7L/min for 30 min, net electrostatic catalysis is observed for applied voltages $\Delta G_1$8 V. At 0.75 V, the electrostatic route gives a surface loading similar to the Cu(I)-catalyzed benchmark; at 1.5 V the yield is 21% higher than Cu(I), and at 2.0 V it is 198% higher. The effect is polarity dependent: at $\Delta G_1$9 V the clicked-product signal is much larger than at $e_1$0 V, with the peak area 73% smaller under negative polarity. Continuous flow also matters: at 1.0 V, the yield under continuous flow is 87% larger than under stopped-flow conditions (Sevim et al., 2022).

A more general nonequilibrium theory treats catalysis under rapid periodic modulation of an external control parameter $e_1$1. In the high-frequency regime, the catalyst behaves as if governed by effective rates

$e_1$2

and anomalous performance is controlled by the local geometric criterion

$e_1$3

Within this framework, oscillation-driven catalysis can invert a spontaneous reaction and increase turnover frequency without relying on a single low-barrier static landscape. The paper’s worked examples explicitly show inversion of a reaction with $e_1$4 and enhancement of current from $e_1$5 and $e_1$6 to $e_1$7 under rapid oscillation (Zhang et al., 2023).

A third environment-responsive mode is phase-switchable catalysis. Ultrasmall noble metal clusters confined in ionic organic cages can be transferred reversibly between water and ethyl acetate by anion exchange: LiTFSI converts hydrophilic MC@I-Cage-Cl into hydrophobic MC@I-Cage-TFSI, and KCl reverses the transfer. The cluster size remains below 1 nm during transfer, with Au losses below 5 ppm. For Pt@I-Cage-Cl, ammonia borane hydrolysis in water reaches $e_1$8 within 1.3 min at 300 K, corresponding to TOF = 115 min$e_1$9. Because Pt clusters can be moved into ethyl acetate to interrupt the reaction and returned to water to restart it, the same catalyst acts as a reaction-switchable on/off system (Zhang et al., 2019).

5. Spatial heterogeneity, confinement, and support-controlled flexibility

A general spatiotemporal theory formalizes flexible catalysis in terms of active-site distributions and time-dependent accessibility. In the Active Catalytic Space framework, the instantaneous catalytic output is

$[S_c]$0

and the cumulative turnover number is

$[S_c]$1

Here $[S_c]$2 is the site distribution, $[S_c]$3 is a dynamic modulation factor, and $[S_c]$4 is the local intrinsic rate. The framework is illustrated by a single-atom Pt limit, a Pt surface with $[S_c]$5 and $[S_c]$6 facets, a bilirubin oxidase model with series-coupled T1 and TNC sites, and a Ru homogeneous catalyst with explicit modulation $[S_c]$7 (Crespilho, 2 May 2025). This suggests that flexible catalysis can be analyzed as a space-time overlap problem: performance is maximized when site density, accessibility, and local kinetics are simultaneously favorable.

Confinement-based materials design expresses a related idea at the support level. Mesoporous organosilica produced by direct co-condensation of TEOS with MTES or PTES tunes internal hydrophobicity while preserving ordered mesoporosity. For protein trapping, the phenyl-functionalized material C18-M7-Ph adsorbs $[S_c]$8 lysozyme, compared with $[S_c]$9 for MCM-41 C18 and about $\Delta G_1$0–$\Delta G_1$1 for the methyl-functionalized materials. For supported Cu catalysis of benzyl azide with 4-bromo-1-butyne in water, however, the best performer is the more hydrophobic methyl-functionalized support: Cu/C18-M15-Me gives >99% yield, compared with 87% for Cu/C18-M7-Me, 78% for Cu/C18-M7-Ph, and 11% for Cu/MCM-41 C18 (Osta et al., 2019). The paper attributes this to an “optimum hydrophobic nanoenvironment” and a “tight fit” inside the pores, emphasizing that flexible catalysis may be engineered by tuning confinement and local solvent compatibility rather than altering the active metal alone.

6. Design principles, computation, and unresolved issues

Minimal mechanical models formalize the geometry–flexibility trade-off. In an elastic-network model of catalysis, optimal catalysts are transition-state complementary in geometry but not maximally stiff in all degrees of freedom. For the default parameter set $\Delta G_1$2, $\Delta G_1$3, $\Delta G_1$4, $\Delta G_1$5, and $\Delta G_1$6, the locally optimal catalyst has

$\Delta G_1$7

More generally, the optimal catalyst body is rigid in shape, $\Delta G_1$8, while the optimal interaction stiffness $\Delta G_1$9 is finite and depends on reactant and product concentrations; as $$a \ge \frac{1}{2}+\max\!\left(0,\frac{h_s^+-h_s^-}{2h_s^+}\right).$$0 with $$a \ge \frac{1}{2}+\max\!\left(0,\frac{h_s^+-h_s^-}{2h_s^+}\right).$$1, $$a \ge \frac{1}{2}+\max\!\left(0,\frac{h_s^+-h_s^-}{2h_s^+}\right).$$2, whereas increasing $$a \ge \frac{1}{2}+\max\!\left(0,\frac{h_s^+-h_s^-}{2h_s^+}\right).$$3 or $$a \ge \frac{1}{2}+\max\!\left(0,\frac{h_s^+-h_s^-}{2h_s^+}\right).$$4 favors lower $$a \ge \frac{1}{2}+\max\!\left(0,\frac{h_s^+-h_s^-}{2h_s^+}\right).$$5 (Rivoire, 2019). This again supports the view that productive flexibility is cycle-dependent: capture, transition-state stabilization, and release cannot usually be optimized by a single static interaction strength.

Because flexible catalysis often generates large ensembles of intermediates and pathways, computational frameworks increasingly treat catalysis as an emergent network rather than a single cycle. Autonomous reaction-network exploration argues that catalysis should be identified a posteriori from a network in which a species reappears unchanged and mediates a faster route than the uncatalyzed alternative. The same framework emphasizes that conformer ensembles, side cycles, decomposition pathways, and catalyst restructuring are central rather than peripheral, especially for flexible homogeneous and heterogeneous catalysts (Steiner et al., 2021).

Adaptive computational workflows address a different layer of flexibility: the catalyst-discovery process itself. In methane-to-methanol catalyst screening, a dynamic classifier that monitors geometry optimization on the fly achieves >50% reduction in wasted failed-job time, with <2% false negatives, and transfers from one reactive intermediate to others, including chemically distinct intermediates and an unseen metal center (Duan et al., 2022). This does not alter catalytic chemistry directly, but it changes how flexible catalytic landscapes are explored computationally.

Several controversies remain. The role of conformational change in enzymatic catalysis is described as “highly controversial” in the minimal theory of barrier-free catalysis (Rivoire, 2022). Model systems with environmental TEM, microfluidic OEEF reactors, or idealized lattice and elastic-network catalysts establish mechanism, but they do not by themselves resolve questions of substrate scope, energy efficiency, industrial scalability, or the generality of a given flexible-catalysis mechanism across materials classes (Lu et al., 2018). A cautious synthesis of the literature is therefore that flexibility is neither a universal virtue nor an epiphenomenon. It is a controllable catalytic variable whose benefit depends on whether it selectively improves the slow or inaccessible parts of the catalytic cycle while preserving turnover and structural integrity.