A Generalized Kinetic Model for Compartmentalization of Organometallic Catalysis

Compartmentalization is an attractive approach to enhance catalytic activity by retaining reactive intermediates and mitigating deactivating pathways. Such a concept has been well explored in biochemical and more recently, organometallic catalysis to ensure high reaction turnovers with minimal side reactions. However, a scarcity of theoretical framework towards confined organometallic chemistry impedes a broader utility for the implementation of compartmentalization. Herein, we report a general kinetic model and offer design guidance for a compartmentalized organometallic catalytic cycle. In comparison to a non-compartmentalized catalysis, compartmentalization is quantitatively shown to prevent the unwanted intermediate deactivation, boost the corresponding reaction efficiency (𝛾), and subsequently increase catalytic turnover frequency (𝑇𝑂𝐹). The key parameter in the model is the volumetric diffusive conductance (𝐹 ) that describes catalysts’ diffusion propensity across a compartment’s boundary. Optimal values of 𝐹 for a specific organometallic chemistry are needed to achieve maximal values of 𝛾 and 𝑇𝑂𝐹. Our model suggests a tailored compartment design, including the use of nanomaterials, is needed to suit a specific organometallic catalysis. This work provides justification and design principles for further exploration into compartmentalizing organometallics to enhance catalytic performance.


INTRODUCTION
Compartmentalization has been well documented in biochemical literature as one method for achieving efficient in vivo tandem catalysis by encapsulating enzymes in well-defined microand nano-structures. [1][2][3][4][5][6][7] By controlling the diffusion of species in and out of compartment boundaries, nature is able to retain reactive or toxic intermediates, increase local substrate concentration, and mitigate deactivating or competing pathways. [1][2][3][4][5][6][7] For example, carboxysome microcompartments enhance the rate of CO2 fixation by encapsulating the cascade of carbonic anhydrase and ribose 1,5-bisphosphate carboxylase/oxygenase to generate high local concentration of CO2 and exclude deactivating O2 within their polyhedral structures. 8,9 Also, the last two steps of tryptophan biosynthesis -the conversion of indole-3-glycerol-phosphate to indole and then to tryptophan -takes advantages of the substrate-channeling effect bestowed by compartmentalized subunits of tryptophan synthase. 10,11 Here, a hydrophobic tunnel between the two subunits retains the indole intermediate, which prevents its free diffusion and participation in deactivating side reactions. 10 With billions of years of evolution, compartmentalization appears the mainstay of biology to manage the complex network of biochemical reactions that are frequently competing and incompatible with each other in a homogenous solution.
The success of natural compartmentalized enzyme cascades inspires the development of bio-mimetic synthetic catalysis with organometallic chemistry being the latest frontier. Multiple groups have employed well-defined spatial organization at the nano-and microscopic levels to construct in vitro biocatalytic and organometallic cascades with enhanced catalytic performance. 2,3,[12][13][14][15][16] Encapsulating NiFe hydrogenase in virus capsids improves its proteolytic and thermal stability as well as enhances the rate of H2 production. 12 Confining a biochemical cascade of βgalactose, glucose oxidase, and horse radish peroxidase in metal-organic frameworks led to an enhancement of reaction yield in comparison to a freely diffusing analogue. 13,14 The extent to which reaction yields are enhanced in confined enzyme cascades is reported to correlate with the distance among active sites, suggesting that spatial organization or localization of catalysts is beneficial in tandem or cascade reactions. 15 In addition to biocatalysis, recently compartmentalization of organometallic catalysts has been experimentally demonstrated. [17][18][19][20][21][22][23] For example, our group employed a nanowire-array electrode to pair seemingly incompatible CH4 activation based on O2-sensitive rhodium (II) metalloradical (Rh(II)) with O2-based oxidation for CH3OH formation. 17,24 The application of a reducing potential to the nanowire array electrode created a steep O2 gradient within the wire array electrode, such that an anoxic region was established at the bottom of the wires. As a result, a catalytic cycle was formed in which the airsensitive Rh(II) activated CH4 in the O2-free region of the wire array electrode, while CH3OH synthesis proceeded in the aerobic domain. Moreover, by substituting a planar electrode (no anoxic region) for the nanowire array, CH activation and CH3OH generation are negligible. 17 The retainment of the ephemeral Rh(II) intermediate by the nanowire electrode for catalytic CH4-to-CH3OH conversion 17, 24 encourages us to further explore the design principles of compartmentalizing organometallic cascades for higher turnovers with mitigated deactivation pathways.
We envision that a theoretical framework for organometallic catalysis will expand the use of compartmentalization for organometallic chemistry. In biochemistry, mathematical modeling of confined enzyme cascades has been well developed and offers the design principles in natural systems 11,25 and for engineered bio-compartments. 11,16,25,26 The models pinpoint a key parameter, volumetric diffusive conductance ( ! ), which describes the diffusion propensity across a compartment's boundary. ! is determined by a compartment's surface-to-volume ratio and its boundary's permeability. 26,27 An optimal value of ! tailored to the specific biochemical reactions are needed in order to achieve better reactivity in comparison to the non-compartmentalized alternative. Similarly, we note that further development of compartmentalized organometallic chemistry demands a quantitative design principle applicable towards a model catalytic cycle that includes oxidative addition (OA), isomerization/migratory insertion (Iso/MI), and reductive elimination (RE) along with undesirable deactivation pathways ( Figure 1A). Yet there has been a paucity of theoretical treatment despite progress in experimental demonstration. 17-23 22 Such a lack of theoretical treatment motivates us to establish a general kinetic model and quantitatively investigate how compartmentalization will affect the competing reaction pathways and the corresponding turnover of the desired organometallic catalysis.
Here we report a general kinetic model and offer design guidance for a compartmentalized organometallic catalytic cycle. We took advantage of the established theoretical frameworks in biochemistry 16,25,26 and applied such kinetic frameworks to a model compartmentalized organometallics with competing deactivation pathways ( Figure 1A), 28 and an analogous noncompartmentalized cycle ( Figure 1B). We examined three metrics in the catalytic cycle in Figure  1C: 1) reaction efficiency ( ) that gauges the percentage of intermediates funneled towards desirable catalytic turnover over deactivation pathways, 2) the flux of catalytic intermediates out of the compartment to be deactivated ( " ), and 3) turnover frequency ( ) that measures the steady-state catalytic rate despite intermediate deactivation.
A compartmentalized system can significantly outperform a homogeneous counterpart with respect to and with a lower value of " , at ! values smaller than the intrinsic kinetics of the organometallic cycle in question. We showcased how the developed model can serves as a guiding design principle for specific organometallic catalysis for maximal and . The established kinetic model can be adapted to suit a plethora of catalytic cycles or materials-based compartments, offering a framework to be expanded on for advanced compartmentalization of chemical catalysis.

Establishing a general kinetic framework of compartmentalization for an organometallic catalytic cycle
We modelled a three-step catalytic cycle consisting of oxidative addition (OA), isomerization/migratory insertion (Iso/MI), and reductive elimination (RE) steps in the context of a compartmentalized system with multiple deactivation pathways in the solution ( Figure 1A) We assign volumetric diffusive conductance ( ! ) to quantitatively describe the extent of mass transport, predominantly diffusion-based, between the compartment and the surrounding bulk solution. As a measure of diffusion across the compartment's boundary, ! equals the product of compartment boundary's permeability ( ) and its total surface area ( ), while normalized by the volume ( ) of the corresponding compartment and Avogadro's number ( ' ) ( Figure 1C). 26 ! describes a molecule's tendency to contribute a diffusion flux across the compartment under a given concentration difference across the compartment's boundary. As is linearly proportional to the species' diffusion coefficient ( ) and inversely proportional to the distance of diffusion path across the boundary, 35 , , and depend on not only the compartment's geometric dimensions (for , , and ) but also the materials' property of the compartment (for ). Since ! governs ( ) ), and the deactivation rate of intermediates − ( " ). Moreover, in both compartmentalized and non-compartmentalized scenarios, we aim to analyze the rate of reaction (in the form of ) and the efficacy of transforming the substrate into targeted product (in the form of ) that is defined as the percentage of intermediates funneled towards desirable catalytic turnover. 16,26 In both cases, is calculated as the ratio between the formation rate of product and the consumption rate of substrate . In the case of pseudo-first-order kinetics towards in oxidative addition (m = 1), , ",23# , and 23# in a compartmentalized system can be expressed as, in which, While the , ",23# , and in a non-compartmentalized scenario, denoted as 4 , ",23#

4
, and 23# 4 are expressed as, The  Figure 2B) and lower value ( Figure   2C); alternatively when $ is much larger than &$ and the deactivation step is less relevant, plateaus towards unity with concomitant increase in . Despite the dominant role of $ , whether or not the system is compartmentalized strongly affects the values of , " , and ( Figure 2D-F). While the trend is generally applicable for all values of ! , a specific case when ! = 320 s −1 , corresponding to the nanowire array electrode for CH4-to-CH3OH conversion in our previous work, 17 illustrates under which situation the advantages of compartmentalization will be observed. As the value of $ increases, reaction efficiency (red trace in Figure 2D) increases in a sigmoidal fashion when $ approaches the value of ! with compartmentalization, while in a non-compartmentalized case (black trace in Figure 2D The above noted effects can be mathematically justified based on our derived equations. When the value of ! is similar to or even larger than &$ or &% ( ! ≳ &$ or &% ), This will lead to ≈ 4 , i.e. the reaction efficiency is not significantly altered with compartmentalization in comparison to the non-compartmentalized case.
Alternatively, when ! ≪ &$ or &% , we have This leads to The equations noted above suggest that optimal, near-unity reaction efficiency , high , and low " values would be obtained when ! ≪ $ and % , which is consistent with our observations in Figure 2.
Lastly, we explored " and as functions of ! and $ when accounting for .,-,-0-,1 , as a function of ! and $ , and ! alone in this alternate scenario. In comparison to plots in Figure   2C and 3C, Figures S5B and 6B also predict to decrease exponentially with ! , and increase with $ , with a difference in actual value of when accounting for .,-,-0-,1 . Figure S7 displays ",23# and as a function of $ alone, compared to the corresponding non-compartmentalized metrics for model .,-,-0-,1 are also plotted. Similar to Figure 3B, compartmentalized " ( Figure   S7A) is predicted to be much smaller than non-compartmentalized " at high $ . However, at low $ , it is predicted that a compartmentalized system will have greater " , when accounting for .,-,-0-,1 , suggesting that compartmentalization may be marginally should be considered when $ < &$ and % < &% , when the intrinsic catalytic reactivity cannot outcompete the deactivation pathway. The efficacy of compartmentalization will be observable, as long as the compartment's volumetric diffusive conductance ! is much smaller than &$ or &% ( ! ≪ &$ or &% ). Nonetheless one interesting conclusion from our analysis is that maximal efficacy of compartmentalization (reaction efficiency → 1) demands ! to be smaller not only than the rate constants of deactivation steps ( &$ and &% ) but also than the rate constants of steps in the catalytic cycle ( $ and % ). This requirement for maximal stems from the fact that a "leaky" compartment with large ! is not sufficient to conserve the yielded intermediates and is prone to deactivation. Practically, such a requirement is indeed a blessing for organometallic chemistry. As typical organometallic studies do not commonly characterize the deactivating side reactions, there lacks detailed kinetic information for compartment design, as the values of &$ or &% were needed to determine the range of desirable ! values. Though we posit that designing a confined catalytic cycle to have ! < $ and ! < % is sufficient for a compartment to "revive" a proposed, unfunctional catalytic cycle, future design of compartmentalization can be simplified.
The feasibility of obtaining the range of ! from the kinetics of the proposed catalytic cycle offers more guidance for the materials design for the compartment. As ! equals the product of compartment boundary's permeability ( ) and its total surface area ( ), while normalized by the volume ( ) of the corresponding compartment, 26 multiple synthetic handles could be applied to achieve a desirable ! value. A less permeable interface at the boundary of compartment as well as smaller surface-to-volume ratio will help to reduce the mass transport hence the value of ! .
Characterization techniques that help determine encapsulation geometry and assess permeability, such as electron microscopies and chromatographic methods should be welcomed for more detailed mechanistic investigations in experimental demonstration. [46][47][48][49] One interesting result from this argument is that a compartment of extremely small dimension, for example of nanoscopic scale, may not be necessarily beneficial, since nanoscopic dimensions can create an equivalently "leaky" compartment when normalized to the compartment volume. Careful design is recommended before experimental implementation.
Last, we cautioned that our established model only considers the mass transport of catalysts and assumes an unconditionally fast supply of substrate and quick removal of product . While such assumptions have their real-life correspondence under certain circumstances (vide supra), the established model is incapable of accounting for the possible mass-transport limitation from substrate and products, which could be induced by a small ! value recommended by the model.
Given that, we cautioned that a lower bound of ! exists for optimal performance in practical applications, and an unnecessarily small value of ! could be detrimental to the compartment design.

CONCLUSION
Here we have developed a kinetic framework for compartmentalizing organometallic catalysis, using a classical three step cycle consisting of OA, Iso/MI, and RE in that order. Under the same kinetic and diffusive parameters, the kinetic model predicts that key reaction metrics, derived from solving steady state equations of catalytic species, are significantly enhanced versus a homogenous counterpart. Furthermore, we demonstrated that careful design of a structured material to produce ideal volumetric diffusive conductance ( ! ) values in relation to kinetic parameters is a viable approach to optimization by plotting key reaction metrics as functions of ! and rate constants # and $ . From this, we conclude that confinement essentially induces a reaction to compete with influx and outflux instead of deactivation, provided deactivating media are adequately barred from the compartment. As diffusion into and out of a compartment can be tuned by confinement geometry, this offers a clear handle for optimization that freely diffusing systems do not possess. We also derived an additional kinetic framework accounting for both total catalyst concentration and catalyst in the bulk (model .,-,-0-,1 ), which ultimately yielded the same conclusions as when not accounting for total catalyst concentration. Lastly, we offered insight into accounting for various approaches to compartmentalization, where rigorous definition of confinement will be instrumental. The results from this study will assist in the a priori design of compartmentalized organometallics for enhanced catalytic performance.