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63aedbf204902ad473148fa4	27	Comparing the spectra T 1 (ν) over the entire frequency range for both bulk solutions and slit pores, and for 2 different salt concentrations (c s = 0 and 3 mol/kg), two clear effects of the salt can be noted. First, at high frequencies, where molecular motions dominate, an increase of c s leads to systematically lower values of T 1 as expected for the increased bulk viscosity (area II in Fig. ). This effect is observed both for slit nanopores and bulk systems. Second, at lower frequencies, T 1 (ν) shows clear variations with ν only in the case of a slit pore with c s > 0, and is flat otherwise (area I in Fig. ). This low frequency regime resembles the observation previously made also for Na 2 SO 4 slit pores with c s = 0 (Fig. ), as well as results obtained for Na 2 SO 4 slit pores with c s > 0 (Fig. ) and will discussed in more depth in the discussion section.
63aedbf204902ad473148fa4	28	In presence of a solution with finite salt concentration, ions adsorb at the solid wall. We define the average number of ions adsorbed at the solid walls, N Ads , based on a distance criterion: When the distance between an ion and the solid becomes similar to the lattice spacing we consider it adsorbed. We checked that different definitions of the adsorbed amount do not alter our conclusions. For both NaCl and Na 2 SO 4 surfaces, there are more Na + ions adsorbing than Cl -and SO 2- 4 ions, respectively (Fig. ). As a consequence, both solid surfaces acquire a net positive surface charge, a phenomenon that has been previously reported both experimentally and numerically for NaCl surfaces. In addition, the atomic roughness of both surfaces is modified due to the presence of adsorbed ions. Ions were observed to adsorb at the solid wall for a typical duration similar or larger than the total simulation time of 100 ns, i.e., for much longer time than the typical adsorption duration of water molecules (τ A ≈ 40 -100 ps (Fig. )). The presence of ions in the solution, and their adsorption at the solid surface, induce several changes in the structure and dynamics of water near the interface. For instance, density and orientation profiles of the water molecules at the surfaces are modified (Fig. bc), the typical rotation time τ R of water next to the interface increases (Fig. ), and the typical adsorption time of water at the solid interface decreases.
63aedbf204902ad473148fa4	29	Several properties of the solid wall are affected by the adsorption of the ions, including the surface atomic roughness and the surface net charge. In order to isolate the effects of the surface roughness on T 1 , simulations were performed in a simplified configuration: the pores were made of NaCl solid walls, and a fraction 1-r of the atoms of the last layer were randomly deleted, thus creating a rough surfaces. The total charge of both surfaces was ensured to be equal to zero after atoms were deleted. The pores were then filled with pure water, and the desorption of the surface ions was prevented by means of harmonic potentials maintaining the wall atoms near their original positions. Our results show that varying the roughness r of the pore surface has a strong impact on T 1 , with T 1 being as low as 0.5 s for r = 1 %, as compared to the value of 3 s for reference surface NaCl(100) (Fig. ). The lowest value of T 1 is obtained for r = 1 %, which corresponds to the limit of very few additional NaCl atoms at the solid surface.
63aedbf204902ad473148fa4	30	Our numerical investigation was motivated by FFC results showing lower values of the NMR relaxation time T 1 in the case of Na 2 SO 4 as compared to NaCl salt crusts (Fig. ). However, the value of T 1 obtained using MD simulations for salt concentrations close to the experimental values (c s = 4 kg/mol for NaCl, c s = 1 kg/mol for Na 2 SO 4 ) show a slightly lower value for T 1 in the case of NaCl as compared to Na 2 SO 4 , namely T 1 ≈ 0.6 s vs. T 1 ≈ 0.9 s (Fig. ). This apparent contradiction between experiments and simulations could indicate that the differences observed experimentally for T 1 in Fig. are not primarily induced by differences in surface properties between the two salt crusts, but rather by differences in their respective pore size distributions. This would suggest, that there is a larger proportion of small pores in the Na 2 SO 4 crust as compared to NaCl, which is consistent with the pore size distributions extracted from SEM images (Fig. ) and the pore size dependence of the relaxation rate shown in Fig. . However, in our SEM analysis, we lack data for pore sizes smaller than 1 micrometer and thus no definite conclusion can be made from this study, leaving room for future pore size characterization covering the full pore size range.
63aedbf204902ad473148fa4	31	Our MD simulations revealed the complex interplay between the adsorption of ions at the solid salt walls and the NMR relaxation time T 1 of water. By adsorbing at the solid walls, ions induce a net surface charge. The viscosity of water close to a charged surface is known to differ from its bulk value due, at least in part, to a redistribution of hydrogen bonding induced by the electric field . In addition to the build-up of a net surface charge (Fig. ), the adsorption of ions creates a rough landscape with inhomogeneous local charge distributions (Fig. ). In turn, these inhomogeneities induce a strong 2D water structuring in the x-y plane parallel to the solid surface (Fig. ), in addition to the structuring occurring in along the z direction that occurs even in absence of ion adsorption (Fig. ), which is likely to further impact the dynamics properties of the water molecules. The intricacy between salt concentration, surface atomic roughness, and surface charge inhomogeneity makes it difficult to isolate the relative importance of each phenomenon on T 1 , and simulations with comparatively simpler systems should be performed in the future.
63aedbf204902ad473148fa4	32	Our numerical results obtained for NaCl slit pores show a strong dispersion of T 1 for frequency ν ∈ [3 -100] MHz only when c s > 0 (Fig. , Fig. ). For Na 2 SO 4 , such T 1 dispersion is reported for all the values of c s considered, including c s = 0, with a degree of dispersion that increases with c s . The frequency range ν ∈ [3 -100] MHz corresponds to events of duration larger t > 10 ns, i.e., much larger than the characteristic molecular times. For instance the typical rotation time τ R of water molecules at the solid walls never exceeds 20 ps (Fig. ), and the typical adsorption time τ A never exceeds 100 ps (Fig. ). On the other hand, the maximum characteristic diffusion time of a water molecule in between two adsorption/desorption events is t D ≈ h 2 /D ≈ 30 ns where D = 2 • 10 9 m 2 /s is the diffusion coefficient of water, and h ≈ 7 nm the pore size. Therefore those adsorption-desorption events are likely to be at the origin of the dispersion of T 1 for ν ∈ [3 -100] MHz. This assumption is consistent with the Levitz model, for which the surface relaxation time in Eq. ( ) is associated to a 'slow' contribution induced by successive adsorption and relocation steps of the fluid molecules inside the pore . Strikingly, for NaCl slit pores with h ≈ 7 nm and c s = 0, no such low frequency regime is observed (T 1 is flat for ν < 100 MHz, Fig. ) while alternative adsorption-diffusions steps are present, with a typical adsorption time τ A ≈ 55 ps. Here, we suggest that the absence of low frequency regime for NaCl pores with c s = 0 could be due to the relatively weak interaction between crystal NaCl(100) surface and water, as seen by the absence of strong polarization of the water molecules at the solid surface 5).
63aedbf204902ad473148fa4	33	In this article we combine NMRD experiments with MD simulations to study NMR relaxation time of NaCl and Na 2 SO 4 salt solutions confined within NaCl and Na 2 SO 4 salt crusts, respectively. Experimental measurements show a stronger dispersion of T 1 with the frequency ν for Na 2 SO 4 as compared to NaCl salt crusts. MD simulations of pure water within slit pores reveal some difference between NaCl and Na 2 SO 4 solid surfaces, such as a stronger polarization of water molecules next to the SO 2- 4 of the surface, which in turn lead to lower T 1 values in the case of Na 2 SO 4 pores compared to NaCl. MD simulation of salt solutions confined within slit pores, however, show a complex interplay between ion adsorption at the solid surface and the relaxation time T 1 . Ions, by adsorbing at the interface, induce a net surface charge while also creating a rough atomic landscape, which further reduce the measured values of T 1 . to the solid walls (Eq. 10 from the main text). Results have been obtained for pore size h ≈ 7 nm. Note that for cs > 0, the solid surface is not atomically flat due to adsorption, and therefore the first density layer of water may not be the best definition of the adsorbed layer. Simulations were performed for different values of the Lennard-Jones (LJ) cut-off ranging from r LJ = 0.9 to 1.5 nm using a bulk water system with a number N water molecules and temperature T = 293.15 K. Results show a significant impact of r LJ on the value of T 1 for r LJ < 1.2 nm, where we used N = 2000 (Table ). Simulations with different numbers of water molecules from N = 1000 to N = 3000 show no significant impact of N on T 1 , where we used r LJ = 1.4 nm (Table ). Finally, different values of maximal LINCS iterations N LINCS , the temperature coupling characteristic time t v-rescale , the timestep ts, as well as a different thermodynamic ensemble have been tested, without any significant influence on the value of T 1 (Table ). Each production run was divided into four equal parts of 0.5 ns each. From the four parts, four values of T 1 were extracted, allowing us to evaluate an average value of T 1 as well as a statistical error. For all simulations presented here, the statistical error on T 1 is bellow 1 %.
63aedbf204902ad473148fa4	34	Here we describe the recalibration of a force field for the sulfate (SO -2 4 ) ion in TIP4P/ water. We follow closely the procedure of Mamatkulov et al. , which consists in a double-optimization strategy based on the experimental ion solvation free energy and the experimental activity coefficients. The first step of this strategy consists in optimizing the Lennard-Jones (LJ) parameters of the oxygen atoms of SO -2
63aedbf204902ad473148fa4	35	To calculate the single-ion solvation free energy, a SO -2 4 ion is placed in a cubic box of size L = 2.5 nm containing 512 water molecules. A cut-off distance of r LJ = 1.0 nm is chosen for the Lennard-Jones interactions, and MD simulations are performed with a time-step of 1 fs. The solvation free energy is calculated from the integration of the Hamiltonian of the system H ϕ over a solvation path,
63aedbf204902ad473148fa4	36	where z is the ion valency, e the elementary charge, r = 79 the relative dielectric constant of TIP4P/ water at 300 K , W = 2.837279 the Wigner constant, and L = 2.5 nm is the simulation box size. A second correction related to the compression of the gas reads,
63aedbf204902ad473148fa4	37	where p 1 = 24.6 atm is the pressure of the ideal solution, and p 0 = 1 atm the pressure of the ideal gas phase. We find ∆G fs = 14.9 kJ/mol and ∆G press = 8.0 kJ/mol, and the total single-ion solvation free energy is given by ∆G solv = ∆G sim + ∆G fs + ∆G press .
653ed9e448dad23120a97094	0	Osteoarthritis is the most common joint disease in the United States. It is a degenerative joint disease predominantly in the hands, hips, and knees, leading to a decrease in joint mobility. TRPV1 has 6 transmembrane domains, and a calcium-permeable pore is formed between the 5 th and 6 th domains . TRPV1 agonists bind to 2 residues located between the second loop and the S3 domain (Tyr 511 and Ser 512) and a residue beneath the 5 th domain (Tyr 550), creating the pore and allowing calcium cations to flow in .
653ed9e448dad23120a97094	1	like breakdown products of the cartilage extracellular matrix (ECM) are produced. These DAMPs go through pattern-recognition receptors on synovial macrophages, fibroblasts, or chondrocytes, leading to the production of inflammatory mediators. Inflammation causes angiogenesis, the formation of new blood vessels, resulting in an influx of plasma proteins that can serve as DAMPs, leading to more inflammation. Chronic inflammation directly causes cartilage degradation, but it also results in the induction of proteolytic enzymes, accelerating cartilage degradation in osteoarthritis .
653ed9e448dad23120a97094	2	Additionally, reactive oxygen species (ROS) play a role in inducing inflammatory responses. ROS can activate cell signaling pathways and increase the release of proinflammatory cytokines, enhancing and prolonging the inflammatory responses. . This can create a cycle where inflammation produces more ROS, which sequentially causes even more inflammation . The cycle continues until the cellular defense mechanism against oxidative stress removes the ROS molecules from the cells . Although a greater amount of ROS can help deal with infections effectively, prolonged and excessive inflammatory responses can cause tissue damage, which are major factors in the mechanisms of certain diseases, including osteoarthritis .
653ed9e448dad23120a97094	3	TRPV1 agonists have been shown to reduce inflammation. For example, the injection of capsaicin reduced mortality in rats with induced abdominal sepsis . In addition, the injection of capsaicin in mice with lipopolysaccharide stimulation (LPS)-induced bone inflammation inhibited prostaglandin E2 production by osteoblasts. Prostaglandin E2 induces inflammation and bone resorption, so inhibiting prostaglandin E2 production also inhibits inflammation and bone resorption associated with the inflammation .
653ed9e448dad23120a97094	4	Human umbilical vein endothelial cells cultured in the presence of capsaicin before LPS showed an increase in nitric oxide (NO) production and endothelial nitric oxide synthase (eNOS) phosphorylation and a decrease in LPS-induced cytokine and chemokine production. This led the authors to conclude that TRPV1 reduces inflammation in endothelial cells through the activation of the Ca 2+ /eNOS/NO pathway .
653ed9e448dad23120a97094	5	While capsaicin has been a TRPV1 agonist that has been more thoroughly studied, resiniferatoxin has proven to be a much more potent TRPV1 agonist, being over 1000 times more effective in neuropathic pain treatment . The long-lasting desensitization of TRPV1 with resiniferatoxin allows for a long-lasting analgesic effect, and with studies proving TRPV1's association with inflammation as well, resiniferatoxin might very well be able to reduce inflammation in its activation of TRPV1, as well.
653ed9e448dad23120a97094	6	In a study on the effects of resiniferatoxin on dogs, dogs with osteoarthritis had a persistent lowering in pain and also an improvement in limb use after an intra-articular injection of resiniferatoxin . In dog studies involving the injection of resiniferatoxin to cerebrospinal fluid, measurements proved that resiniferatoxin's half-life in cerebrospinal fluid is between 5 to 15 minutes, which is relatively quick . The scientists who conducted the study also predicted resiniferatoxin would have the same half-life in humans, and this rapid decrease in resiniferatoxin concentration after injection helps limit any possible side effects that could be caused by the resiniferatoxin staying in the human body. The results of these studies set a basis for the clinical trials on humans that came afterward.
653ed9e448dad23120a97094	7	Resiniferatoxin is a capsaicin analog, and both resiniferatoxin and capsaicin bind to TRPV1 and desensitize it through calcium cytotoxicity. Resiniferatoxin is 3 orders of magnitude more potent than capsaicin in pain relief, thermoregulation, and neurogenic inflammation . This is because resiniferatoxin binds to TRPV1 for a prolonged period of time, allowing more calcium cations to flow through than capsaicin does.
653ed9e448dad23120a97094	8	A study was conducted regarding the pain-related genes and non-pain-related genes that capsaicin and resiniferatoxin upregulate and downregulate in rats. The scientists found that resiniferatoxin not only downregulates more target pain genes than capsaicin but also pain genes that weren't targeted, which might be a reason for the overwhelming difference in potency of pain relief between resiniferatoxin and capsaicin . Resiniferatoxin downregulated more than capsaicin off-target pain genes associated with nociception or those related to hypersensitivity. These off-target pain genes Kcnk2, Kcnj5, Gal, Nt5e, Gfra2, Comt, and Ptgdr, which are pain genes associated with nociception, and Kcnk2, Acpp, Nt5e, Kcnt1, which are pain genes associated with hypersensitivity . Resiniferatoxin's ability to downregulate effectively both target pain genes and off-target pain genes reveals a reason behind its potency. However, resiniferatoxin further inhibits desired genes, which alleviates nociception and hypersensitivity. These desirable genes include . Phase 2a Clinical Trial Results for resiniferatoxin in treating knee osteoarthritis pain were recently released by Sorrento Therapeutics in September 2023. In the study, all doses of resiniferatoxin from 7.5 to 20 µg were well tolerated, with few severe adverse events and side effects. The 20 µg dose provided the best results, having the best efficacy and a durability of over 26 weeks after treatment. Future studies like Phase 2 pivotal trials and Phase 3 clinical trials may potentially allow resiniferatoxin to become a key treatment for knee osteoarthritis pain.
6769dc3a81d2151a02b75ef6	0	Applications in clean energy, advanced manufacturing, biomedicine, photonics, and microelectronics increasingly demand new materials with complex structures and heterogeneous composition. In principle, machine learning (ML) offers a strategy to accelerate the discovery of such materials, as it has emerged as a transformational tool for the design of small molecules, bulk inorganic crystals, and even single-component nanomaterials. Deep learning (DL) approaches are particularly well suited to model the behavior of systems with large numbers of parameters, but several obstacles hinder DL from being used to guide the discovery of the complex materials often needed for real-world applications, including nanostructures and composites.
6769dc3a81d2151a02b75ef6	1	First, state-of-the-art approaches for representing materials often fail to capture the structural complexity of nanomaterials, such as multi-shell nanoparticles (Figure ) and nanowire heterojunctions, in which nanostructure controls energy transport (Figure ). Nanomaterials exhibit distinct and often superior properties compared to their bulk counterparts, driven by nanoscale confinement and high surface areas relative to their volume. However, the large number of features required to adequately describe a nanoscale material (e.g., the morphology, dimensions, composition, heterogeneity, doping, defects in each domain; internal interfaces, surface ligands) make training on naive tabular representations computationally inefficient, in part because they neglect physical relationships between features. While bulk crystals can be represented by their unit cell coordinates (e.g., CIF) and small organic molecules by strings (SMILES, SELFIES, ), graphs, or atomic coordinates, such atomistic representations are impractical for complex nanomaterials because their critical features often span length scales of one to >10 6 atoms and cannot necessarily be reduced to periodic subunits. More recent DL approaches encode spatial information as pixels or voxels, but these fixed-resolution representations cannot efficiently capture the structural hierarchy of a wide range of nanomaterials, e.g., those of different sizes.
6769dc3a81d2151a02b75ef6	2	Beyond the challenge of representing nanomaterials, it is also challenging to generate datasets sufficiently large to train DL models that can accurately predict the properties of heterogeneous, multi-component nanostructures. Although highthroughput experimental and computational approaches are growing in their availability and utility, the synthesis and simulation of complex heterostructures is often time-consuming, limiting the scale of available datasets and constraining campaigns to the "small data" regime where DL techniques often struggle. Modern DL models can also have difficulty extrapolating outside of the envelope of their training data, which is necessary for the discovery of novel materials with enhanced properties. Finally, the discovery of fundamentally new materials is complicated by the rough response surfaces of material properties with respect to their composition, necessitating dense and tedious "needle-in-a-haystack" searches across a parameter space. The prediction of materials with targeted properties, or inverse design, would be significantly accelerated by surrogate models that are differentiable so that gradient-based optimization techniques can be used to direct efficient searches. Thus, DL-guided inverse design of complex materials would benefit from the development of large structure-property datasets of complex nanomaterials, new methods to represent them across lengths scales, and differentiable models that are accurate out of their training distribution.
6769dc3a81d2151a02b75ef6	3	In this work, we develop a heterogeneous graph representation for nanomaterials with a variable number of spatial domains, each containing multiple components that can interact within the same domain and across interfaces. We demonstrate that physics-informed graph neural networks (GNNs) built atop such representations can accurately predict properties of nanostructures which are far more complex than any contained in the training dataset. As a model system, we center our investigation on lanthanide-doped upconverting nanoparticle (UCNP) heterostructures (Figure ), whose unique nonlinear optical properties have diverse applications in biological and super-resolution imaging, optogenetics, sensing, photonics, scintillators, secure labeling, 3D printing, and photovoltaics. These applications leverage the ability of UCNPs to absorb multiple near-infrared (NIR) photons and convert them into a single photon of higher frequency, e.g., in the visible and ultraviolet spectrum. Such nonlinear processes are the result of complex networks of energy transfer (ET) interactions between different lanthanide ions (e.g., Yb 3+ , Er 3+ , and Nd 3+ , as in Figure ). To promote advantageous ET interactions and inhibit those that quench emission, nearly all practical implementations of UCNPs use doped heterostructures in which a spherical core is surrounded by one to four concentric shells, with each domain having a distinct combination and composition of lanthanide ions (Figure ). Due to the large numbers of tunable structural and compositional parameters, and the complex network of energy transfer interactions between dopants, optimizing the intensity and wavelength for such complex heterostructures is extremely challenging. Thus, multi-shell UCNP heterostructures present a stringent test for any new DL model and representation.
6769dc3a81d2151a02b75ef6	4	To train these DL models, we generated a dataset of Simulated Upconverting Nanoparticle Spectra for Emission Tuning (SUNSET), consisting of results from ∼6,000 kinetic Monte Carlo (kMC) simulations of nanoparticle photophysics (Figure ). Models trained on SUNSET aim to predict photon emission within a specified wavelength band as a function of UCNP heterostructure. By training on simulations of UCNPs with up to three shells and evaluating on simulations with four shells, we can quantify the capacity for models to extrapolate to larger and more complex heterostructures. We find that our heterogeneous graph representation, informed by UCNP physics and geometry, allows DL models to achieve far higher prediction accuracy than tabular, image, and homogeneous graph representations (Figure ), especially when extrapolating beyond the training data. The differentiability of our heterogeneous GNN also yields gradients of emission intensity with respect to layer thicknesses and dopant concentrations (Figure ), which are not accessible from kMC. Our trained model thus facilitates inverse design of UCNP heterostructure via gradient-based optimization (Figure ), identifying novel superior UCNPs with a range of sizes and up to ten shells. When excited at 980 or 800 nm, these optimized UCNP heterostructures exhibit exceptionally high emission between 300-450 nm, a spectral range useful for inducing photochemistry for optogenetic, catalytic, therapeutic, and 3D printing applications. To validate these predictions, we perform additional months-long kMC simulations which indicate that our model possesses considerable ability to extrapolate far out of distribution and can suggest never-before-seen structures with high accuracy, further revealing novel design principles. These findings demonstrate a path forward for the optimization and discovery of technologically useful UCNPs and offer inspiration for the development of novel DL representations and models which enable inverse design for a broad range of optical nanomaterials.
6769dc3a81d2151a02b75ef6	5	To develop and train DL models that can predict core-shell UCNP photophysics and ultimately enable inverse design of UCNPs with complex heterostructures that exhibit efficient UV and blue emission (300-450 nm), we generated SUNSET: a dataset of over 30,000 multi-shell UCNP spectra calculated with a high-performance kMC simulation package (RNMC) optimized for chemical reaction networks and UCNP photophysics (Figure ). SUNSET consists of four sub-collections (SUNSET- ) that include different dopant ion combinations and surface effects (see Figure ). While each of the sub-collections provides utility for model development and testing, we focus exclusively on SUNSET-1 in our main narrative; discussion of the SUNSET- collections can be found in the Supplementary Information. We focus on SUNSET-1 because this sub-collection includes nanoparticles of variable size and a variable number of shells, which is necessary for training models that have the possibility of extrapolating to more complex heterostructures and thus facilitating impactful inverse design. Further, the nanoparticle structures in SUNSET-1 are substantially larger than in SUNSET- (given that they contain multiple layers, and each layer must be at minimum 1 nm thick in order to be synthesizable), and thus the vast majority of the computational cost of SUNSET went towards the ∼6,000 simulations in SUNSET-1. We note that individual kMC trajectories often took weeks to complete, necessitating the use of high-throughput self-checkpointing workflows. Further details of our workflow infrastructure are given in Section S1.
6769dc3a81d2151a02b75ef6	6	SUNSET-1 utilizes a dopant set of Er 3+ , Nd 3+ , Yb 3+ because this combination of dopants has been used to sensitize upconversion and optogenetic activity with 800 nm excitation, a wavelength that lies in the NIR-I biological imaging window. Segregation of these dopants into different shells of UCNP heterostructures has been shown to dramatically enhance emission. In these systems, Nd is typically included to sensitize the absorption of the 800 nm excitation, Er to upconvert absorbed energy and emit UV/visible light, and Yb to act as a conduit to transfer energy between Nd and Er dopants that would otherwise quench each other via cross-relaxation energy transfer. The nanoparticle heterostructures sampled in SUNSET-1 are variable, with core radii ranging from 1-4 nm and up to 3 shells, each measuring between 1-2.5 nm in thickness, as depicted in Figure . To probe the extrapolatory power of developed models, we simulate explicit 4-shell nanoparticles and hold them out of training data to use as an out-of-distribution (OOD) test set. Thus, SUNSET-1 as a whole has nanoparticle radii spanning from 1-13.6 nm, and the brightest particle has an intensity of ∼ 20,000 cps. I vis-U V most closely follows an exponential distribution, so we use the log(I vis-U V ) as the target label for model training.
6769dc3a81d2151a02b75ef6	7	To train ML models on the SUNSET data, we initially investigated several existing representations for encoding the compositional and dimensional features of each UCNP. As we summarize in Table and discuss in more detail in the "Model Performance" section, we found that standard ML models (e.g., random forest regressors and convolutional neural networks) utilizing tabular and image-based representations exhibited poor ability to extrapolate, with 3-to 10-fold lower accuracy during out-of-distribution testing (OOD) than during in-distribution testing (ID).
6769dc3a81d2151a02b75ef6	8	Seeking representations and models with greater ability to extrapolate to more complex nanostructures, we explored the use of graph-structured representations, which have recently gained prominence due to their ability to effectively capture complex relationships (edges) between entities (nodes). The simplest graph representation of a UCNP is a homogeneous graph in which each node is labeled with the identity and concentrations of a single type of dopant (e.g., Er) in a specific domain of the UCNP (e.g., the core), while edges encode interactions between dopants represented by those nodes, i.e., energy transfer processes. However, we found that GNNs utilizing these homogeneous graph representations exhibited equally poor accuracy for both ID and OOD testing (Table ).
6769dc3a81d2151a02b75ef6	9	Reasoning that the poor performance of homogeneous GNNs was related to inadequate representation of the physical interactions between dopants, we developed a UCNP representation based on a directed heterogeneous graph (Figure ). Unlike the homogeneous graphs, dopant-dopant interactions in our heterogeneous graphs are represented by interaction nodes that connect dopant nodes (via edges), allowing the encoding of additional physical features of the interactions. Two different types of interaction nodes are used, intra-layer and trans-layer, to delineate interactions between dopants within the same geometric region (i.e., core or shell domain) and those in different regions, respectively. It is important to note that dopant nodes are never connected to other dopant nodes, and every interaction node connects exactly two dopant nodes. A self-interaction node, describing interactions between different dopant ions of the same type in the same region, has edges both from and to the same dopant node. The use of a directed graph introduces asymmetry in energy transfer between two dopants (e.g. Yb → Er as compared to Er → Yb). This is important for energy transfer processes that are not reversible, such as non-resonant, phonon-assisted energy transfer that results in irreversible heat dissipation. When establishing features encoded in the nodes, we chose a minimal set of descriptors that are most relevant for UCNPs. Dopant node features include dopant type, dopant concentration (within the respective region), and geometric bounds (inner and outer radii of the core/shell domain they reside in). The interaction nodes contain the interaction type (e.g., Yb-Er, Er-Yb, Er-Er, . . . ) and features derived from the pair of connected dopant nodes -dopant concentrations and geometric bounds. Since we explore only spherical nanoparticles with multiple concentric shells, layer radii fully specify heterostructure geometry.
6769dc3a81d2151a02b75ef6	10	To further account for the effect of distance and region geometry on dopant energy transfers, we introduce a quantity that we call the "integrated interaction." This quantity is derived by integrating a Gaussian function, denoted as N (s; 0, σ), over all pairwise distances, s, between interacting regions, V i and V j , as illustrated in Fig. :
6769dc3a81d2151a02b75ef6	11	) where x i and x j are the doping concentrations in regions i and j, respectively and s is written in spherical coordinates. While the probability of energy transfer between two dopants is actually proportional to distance -6 , we chose to represent this probability as a sum of Gaussian functions because they are continuous at x = 0 and can be integrated multiple times while still capturing the decaying nature of energy transfer with increasing distance between two ions. Adjusting the σ parameters within the Gaussians allows for the modulation of the effective interaction distance of dopants. When used in the ML model, the integrated interaction module is parameterized by n learnable weights (here n = 5) each corresponding to a σ value of one of the Gaussians in the sum.
6769dc3a81d2151a02b75ef6	12	The proposed heterogeneous graph structure lends naturally to the use of GNNs for DL. In our heterogeneous GNN (Figure ), we first embed information from each node into a continuous vector space. We construct the dopant node embeddings by passing the dopant type (Z i ) through an embedding layer, then contextualizing the initial embedding on the dopant concentration (x i ) and the radii (r i0 and r if ) using featurewise linear modulation (FiLM) layers, as shown in Figure . Likewise, to obtain the embedding vector for the interaction nodes (Figure ), we pass the interaction type (τ ij ) through an embedding layer and then condition on the integrated interaction using a FiLM layer. A batch normalization is applied prior to the FiLM layer to shift the distribution of integrated interaction values. Note that "contextualizing" or "conditioning" a vector (i.e., an initial embedding) on another value (e.g. dopant concentration, layer radii, or integrated interaction) with a FiLM layer is a way of combining the information contained in each via an operation that is controlled by many learnable parameters, which often provides better expressivity and performance than a simple concatenation or addition. The resulting dopant and interaction embeddings are then used as the inputs for three iterations of message passing (MP) based on the heterogeneous graph's directional edges, where each MP iteration employs graph attention via the GATv2 operator, after which we use mean aggregation to obtain the global latent representation. Finally, a fully connected neural network (FCNN) is used for label prediction (i.e. predicting the log of the emission intensity over the specified wavelength band) from the global latent representation.
6769dc3a81d2151a02b75ef6	13	To assess the performance of the heterogeneous GNN ("hetero-GNN") described above with respect to other models and representations, we train each on the SUNSET-1 dataset, where our target label is emission intensity from the UV-blue (300-450 nm) wavelength band, and training data include 800-nm excited UCNPs with 0-3 shells. We evaluate the mean squared errors (MSEs) of each model when predicting the intensities of held-out ID samples as well as for OOD nanoparticles with 4 shells (Table ). We compare the hetero-GNN to four well-established supervised learning models: two models (a random forest regressor and a FCNN multi-layer perceptron) using a tabular representation; a CNN using an image representation; and the homogeneous GNN described above. Model hyperparameters are provided in Section S8. Additional model, representation, and feature details are provided in Sections S4, S5, and S6.
6769dc3a81d2151a02b75ef6	14	Of the five models tested, the hetero-GNN exhibited the lowest error for both ID and OOD testing, with MSE values of 13.9 and 22.2 photon counts-per-second (cps), respectively (Table ). The ID loss is 4-fold lower than that of models utilizing tabular representations (RFR and FCNN) and 21.5% lower than the CNN utilizing an image representation. The fact that the image-and heterogenous graph-based models have the best ID accuracies demonstrates how their representations allow them to leverage spatial information to learn relationships between heterostructure and properties and to connect the common behavior of dopant ions of the same type but located in different regions. It is notable, however, that the least accurate model for ID testing was also based on a graph representation. The homogeneous GNN exhibited 6-fold higher MSE than its hetereogeneous analogue, highlighting that the enhanced accuracy of the hetero-GNN is the result of its incorporation of interactions as nodes in the graphs.
6769dc3a81d2151a02b75ef6	15	We believe that the promotion of dopant-dopant interactions to a node-level property specifically improves hetero-GNN performance by elevating the prominence of interaction features during message passing. When included as explicit nodes, the interactions are able to alter the content of the passed messages, allowing for the transmission of richer and more physically relevant information. This parallels the photophysics of UCNPs, in which energy transfer interactions between dopants critically determine the excited state populations of donors, acceptors, and Table An overview of the performance of different models on the SUNSET-1 dataset, assessed using the mean squared errors (MSE, Equation ) of predicted I vis-U V derived from 10-fold cross-validation both for SUNSET-1 in-distribution (ID) test, containing structures with up to three shells, and for SUNSET-1 out-of-distribution (OOD) test, containing structures with four shells. The first value in each cell is the mean squared error (MSE) in photon counts-per-second (cps) multiplied by 10 -3 for easier interpretation. The second value in parentheses is the normalized mean squared error (NMSE) which is the MSE normalized by the sum of squares to yield a relative error (Equation ). The lowest errors for each category, both of which are achieved by the hetero-GNN, are emphasized with bold text. their neighbors, driving nonlinear processes such as upconversion, photon avalanching, and quantum cutting. In contrast, interaction features in homo-GNNs have less influence on predictions because edge properties can only contribute to the attention score, influencing the weighting of the messages being passed between dopant nodes rather than the information contained therein. The heterogeneous graph structure also allows for the embedding of interactions between lanthanide dopants. This embedding introduces valuable inductive bias concerning the distinctness of lanthanide interactions, constraining the model to treat the dopant pairs (e.g., Yb-Er) equally, agnostic to which layers they reside in (e.g. the first or second shell), albeit with varying strengths.
6769dc3a81d2151a02b75ef6	16	The most striking benefit of the hetero-GNN is its ability to extrapolate, in this case to 4-shelled nanostructures not included in its training set. When switching from ID to OOD testing, the MSE for the hetero-GNN model increased by 8.3 cps, or 1.6-fold. This modest increase in loss is in stark contrast to the tabular-representation-based RFR and FCNN models, for which extrapolation resulted in 6-to 10-fold increases in the MSE, respectively. The tabular models lack the geometric and relational information of the graph models and therefore must learn the influence of the dopants in each layer independently. This increases the data demand of these models, making them prone to overfitting and reducing their ability to predict the properties of unseen heterostructures. Even the image-based CNN, which had high ID accuracy, exhibited 2.8-fold greater loss during OOD testing. We ascribe the significantly greater ability of the hetero-GNN to generalize to its graph representation. This conclusion is supported by the fact that the homogeneous GNN also exhibits very little change in loss (+6.5%) when shifting from interpolation to extrapolation. In summary, representing UCNP heterostructures as heterogeneous graphs results in a hetero-GNN that exhibits both high accuracy and extrapolative capacity, in contrast to existing models that are inaccurate (homo-GNN), poor at extrapolation (image CNN), or both (tabular models).
6769dc3a81d2151a02b75ef6	17	While the hetero-GNN exhibits superior performance to the CNN, the image representation has the physically intuitive property that arbitrarily subdividing a given nanoparticle region (e.g. dividing a shell into two smaller shells, where both have the same dopant concentrations as the originally undivided region) has no impact on the model's structural representation or subsequent label prediction. This property, which we call "subdivision invariance", is physically motivated by the fact that region subdivision is arbitrary and leaves the nanoparticle being described completely unchanged. However, neither our heterogeneous graph nor any of the other non-image representations are inherently subdivision invariant. For example, as shown in Figure , subdividing an originally core-only particle into a core and a shell dramatically changes the heterogeneous graph, and thus our hetero-GNN model may predict very different latent representations for physically identical nanoparticles. This is clearly undesirable and may be detrimental to both the learning process and subsequent structural optimization. On the other hand, the voxelization that makes the image representation subdivision invariant simultaneously causes layer dimensions to only be present in the model implicitly, preventing UCNP emission from being differentiated with respect to layer thicknesses and precluding gradient-based optimization of UCNP heterostructure. Thus, any DL model that aims to enable inverse design of nanomaterial heterostructure via gradient-based optimization will need to reckon with the problem of subdivision invariance.
6769dc3a81d2151a02b75ef6	18	Even when an input representation is not inherently subdivision invariant, it is possible to design the DL model built atop the representation to explicitly enforce subdivision invariance such that physically identical structures yield identical latent representations. However, in the context of a graph representation, such explicit enforcement is only possible by avoiding the use of any non-linear operations, which dramatically limits model expressivity and performance.
6769dc3a81d2151a02b75ef6	19	An alternate strategy is to train models to approximate subdivision invariance via data augmentation. Using data augmentation to train approximate invariances in different DL contexts (e.g. image rotation, reflection, etc. in CNNs, molecular rotation and translation in interatomic potentials ) is well established and can enhance model prediction accuracy and robustness. We apply this augmentation strategy to our hetero-GNN model by artificially subdividing the UCNP input with the same labels (emission intensities) but with structural representations modified with random subdivisions. This data augmentation is meant to guide the learned latent representation to exhibit approximate subdivision invariance, which should improve model performance. This strategy is implemented on-the-fly (Fig. ), so that data in each batch are augmented as they appear during training, rather than augmenting the entire dataset before training. Thus, this does not explicitly increase the size of the training dataset, but implicitly multiplies the number of unique heterograph representations of UCNPs seen by the model during training by the number of training epochs. Random subdivisions result in a node with r inner , r outer , being split into two nodes with r inner , r subdivision and r subdivision , r outer , where r inner < r subdivision < r outer . Additional details about data augmentation are provided in Section 5.
6769dc3a81d2151a02b75ef6	20	When the hetero-GNN is trained using on-the-fly-augmentation, its performance improves by 23% on the ID test set (with error falling from 13.8 cps to 10.6 cps) and by 25% on the OOD test set (with error falling from 22.1 cps (2.3%) to 16.5 cps (1.7%)). Additionally, we validated that this augmentation scheme actually trains the model to learn subdivision invariance by evaluating the vector distance between the representation of nanoparticles and their subdivided analogs, observing that the augmented hetero-GNN model more closely represented the subdivided UCNPs in the ID and OOD test sets in embedding space than the nonaugmented model across a range of subdivisions (Figure ). These results underscore the importance of considering subdivision invariance in model training.
6769dc3a81d2151a02b75ef6	21	The hetero-GNN is fully differentiable and takes the features that define the UCNP heterostructure (i.e., layer radii and dopant concentrations) as explicit inputs. For optimization, the hetero-GNN acts as a surrogate model for kMC which is not only orders of magnitude faster, but also provides derivatives of a predicted label with respect to structural features (which are inaccessible with kMC), enabling the use of more powerful gradient-based optimizers to identify UCNP structures that minimize or maximize one or multiple properties.
6769dc3a81d2151a02b75ef6	22	To explore the utility of our differentiable model for inverse design, we use the hetero-GNN trained on SUNSET-1 with augmented data to search for Yb/Er/Ndcodoped UCNPs with the highest UV/blue intensities under 800-nm excitation. To facilitate the discovery of novel heterostructures, we conduct optimizations far beyond the structural distribution spanned by our training data. While the SUNSET-1 training set contains UCNP structures with up to four regions (i.e., a core and three shells) and with a maximum radius of 11.5 nm, our optimizations explore UCNPs with up to ten regions (i.e., a core and nine shells) and a maximum radius of 15 nm. Further, while the core radius is limited to a maximum of 4 nm and shells are limited to a maximum thickness of 2.5 nm in the training data, we remove both of these upper bounds and only limit the overall radius during optimization. However, we do retain a minimum core radius and shell thickness of 1 nm to ensure that the optimized heterostructures are synthesizeable . Given randomly initialized nanoparticle structures, we employ a combination of trust region constrained local optimization and basin hopping global optimization to identify UCNPs with maximized UV/blue emission as a function of maximum allowed nanoparticle radius and the number of distrinct regions in the heterostructure. Section 5 contains additional details of our optimization approach. The maximum intensities identified for UCNPs of different sizes and different numbers of regions (core + shells) are illustrated in Figure . The optimal structure for each distinct radius + region number was then simulated with kMC, for which larger radius particles often required months-long simulations. The colors of the two triangles in each square indicate the kMC-simulated and ML-predicted intensities for the optimized structure. Remarkably, the model demonstrates accurate predictions for the upconversion luminescence of particles with significantly out-of-distribution radii and numbers of regions, including when the UV emissions approach an order of magnitude higher than those in the training set.
6769dc3a81d2151a02b75ef6	23	Optimization results display several trends that are well established in the experimental literature on UCNP heterostructures. Optimized UCNPs generally achieve higher absolute brightness at larger diameters, presumably because they are able to host a greater number of absorbing and emitting dopants. The optimized heterostructures for several representative sizes (Fig. ) show that the domains of these champion UCNPs are in fact heavily doped -or rather, alloyed, with up to 100% lanthanide substitution -to maximize absorption and emission throughput. Rather than spreading dopants homogeneously through UCNPs, the brightest structures partition Er and Nd dopants into separate shells, reflecting the established knowledge that Er and Nd are prone to quench each other via cross-relaxation. . Since the energy absorbed by Nd must be transferred to Er for upconversion, the optimizer produced structures that separate Nd-and Er-rich domains by a thin shell heavily doped with only Yb. . Such layers transmit the energy absorbed by Nd dopants to the upconverting Er dopants via rapid energy migration through the Yb sublattice. To maximize Nd-Yb and Er-Yb energy transfer, the Nd-and Er-containing shells are also heavily doped with Yb. Many of the GNN-optimized structures, particularly ones with fewer layers, are reminiscent of the 3-layered heterostructure refined by Zhong et al. and others. Nd-rich domains are located in outer shells to maximize absorption by a larger number (volume) of the sensitizers. Meanwhile, an Er-rich core is used to promote upconversion by concentrating absorbed energy into a smaller volume and smaller number of Er activators. The fact that gradient-based optimization of the hetero-GNN surrogate model can rapidly learn design rules that have been developed over decades of UCNP research suggests its potential to discover even more complex and functional nanostructures.
6769dc3a81d2151a02b75ef6	24	In addition to validating established domain knowledge, the extrapolated results from hetero-GNN optimization provide the opportunity to understand the behavior of complex UCNPs with a greater number of shells than can be readily synthesized or simulated. For example, a major unanswered question is the optimal number of layers for a UCNP, e.g., are more layers better? The optimization matrix in Figure suggests that for smaller particles (below 10 nm radius), moving beyond 2-3 layers does not substantially improve the brightness, most likely because the shell thicknesses would be thinner than characteristic energy transfer distances. However, larger UCNP heterostructures (r ¿10 nm) do see benefit from complex many-shell architectures, with the brightest 14-and 15-nm UCNPs having 7 and 10-shells, respectively. These manyshelled structures also suggest novel strategies to enhance upconversion efficiency. The most striking characteristic of the optimized 12-, 14-, and 15-nm UCNPs (Fig. .ii-iv) is their interleaving of multiple layers of Nd-and Er-rich shells. Rather than converging on one large layer of Nd sensitizer, the optimized 12-nm UCNP sandwiches a layer of Nd sensitizer between two layers of Er activator (with the appropriate Yb buffer layers, as in Figure .ii), while the brightest 15-nm UCNP exhibits the inverse arrangement (Figure .iv). This sandwich shell arrangement allows energy transfer to occur from two sizes, maximizing the number of donors or acceptors within a given distance while minimizing concentration quenching in those outer shells. Curiously, the 14-and 15nm-radius UCNPs also exhibit motifs in which two Nd-rich shells are separated by an intermediate shell of less concentrated Nd. It is unclear what advantage this motif provides. It is possible that UCNPs may be relatively insensitive to variations near their core (where this motif is observed) since the fraction of dopants is relatively small compared to those in outer shells. This argument may also explain the curious dearth of dopants in the core of the 15-nm UCNP. All of these intriguing structural design motif predictions must be verified and investigated more thoroughly through future experiments (preferably with the aid of precision automated synthesis ) and mechanistic analysis of energy transfer pathways from kMC simulations that the hetero-GNN cannot report on.
6769dc3a81d2151a02b75ef6	25	In a particle utilizing all of these strategies, we find a 6.5x increase in I vis-U V , as compared to the brightest nanoparticle in the training set. Even within the feature distribution, the optimization identifies a particle which utilizes these design rules to achieve a 2x increase in emission intensity as compared to the brightest particle in the training set. These results illustrate that, even within a training distibution but especially far OOD, optimization with a differentiable hetero-GNN can rapidly discover new structures with properties that exceed historical training data and identify novel heterostructure design rules.
6769dc3a81d2151a02b75ef6	26	In order to assess the accuracy of our model prediction during optimization, particularly in the far OOD region, we performed explicit kMC simulations. Particularly for the largest particles, these simulations were extremely expensive, and we terminated many simulations early (after 20%-80% of the requested kMC steps had run) to reduce cost. Overall, the validating kMC simulations took >120,000 CPU-hours on AMD EPYC 7763 and Intel Xeon Gold 6330 CPUs, and individual simulations could take dozens of weeks. All optimizations using our trained hetero-GNN took ≈ 2,000 GPUhours on NVIDIA A100 GPUs. Because optimal particles often emerged early during the optimization process, this GPU-hour figure could probably be reduced by improving our optimization procedure. Coupled with the fact that the kMC simulations could not be directly used for gradient-based optimization, as they are not inherently differentiable, this indicates the massive acceleration in nanomaterial design that can be achieved using DL.
6769dc3a81d2151a02b75ef6	27	While we have here focused on UCNPs, we believe that the heterogeneous graph representation that we have described and implemented could be suitable to predict heterostructure-dependent properties in other multi-layered nanomaterials. Possible applications include engineering the nanophotonic properties of plasmonic and dielectric nanoparticles, the catalytic properties of polyelemental heterostructures, the optoelectronic properties of complex semiconductor nanoparticle heterostructures, the layer-by-layer assembly of nanoparticles for drug delivery, multilayered magnetic nanospheres, and multilayer graphene sheets for diverse energy and mechanical applications. We also note that, given a DL model which can predict multiple properties controlled by heterostructure, our approach could allow for structural optimization while maximizing or minimizing multiple properties simultaneously.
6769dc3a81d2151a02b75ef6	28	Inverse chemical and materials design often requires identifying optimal structures in vast search spaces. DL can dramatically accelerate the optimization and design process, but applications of DL are limited by available data and appropriate representations. In this work, we presented SUNSET, a large dataset of emission spectra for core-shell upconverting nanoparticles simulated using explicit kinetic Monte Carlo simulations. To leverage SUNSET for UCNP design, we developed a new heterogeneous graph representation for nanomaterial heterostructures, which we used to train a heterogeneous graph neural network (GNN). We found that this heterogeneous GNN achieved superior in-distribution (ID) and out-of-distribution (OOD) performance compared to existing representations, including vastly higher accuracy compared with a more traditional GNN using a homogeneous representation. Data augmentation, achieved by artificially partitioning UCNP layers, allowed the heterogeneous GNN to approximately learn the subdivision invariance of UCNP emission, improving both ID and OOD accuracy. Applying gradient-based optimization to the heterogeneous GNN trained with augmented data, we identified new UCNP structures with more than 6.5x higher emission intensity than any UCNP in the training set. Optimized particles further elucidated both previously known and novel heterostructure design rules. Our approach has the potential to considerably improve the rate at which we discover new functional nanomaterials and could provide inspiration for applications of DL to underexplored areas of chemistry and nanoscience.
6769dc3a81d2151a02b75ef6	29	We utilize kinetic Monte Carlo (KMC) to simulate the optical response of lanthanidedoped nanoparticles. We use the high-performance C++ implementation, as implemented in the RNMC software package. Input generation is handled by NanoParticleTools. We utilize Jobflow and FireWorks to build a workflow for high-throughput kMC simulations, enabling the generation of datasets for machine Error (MSE) of the validation set is monitored and the learning rate is reduced on plateau, with a patience of 50 epochs. Early stopping is triggered when the validation MSE has not decreased for 200 epochs. Model performance is reported using the MSE and NMSE, as shown in Equations 3-4 below:
6769dc3a81d2151a02b75ef6	30	where N is the number of UCNPs, Îvis-UV is the predicted UV emission intensity, and I vis-U V is the actual emission intensity. Data Augmentation During training of the hetero-GNN model, we utilize data augmentation as discussed in Section 5. We augment our training data on-the-fly by subdividing the input UCNPs with additional random subdivisions each time an UCNP is seen during training. Subdivided layers retain the dopant composition of their original parent layer. For each UCNP input, we randomly subdivide each parent layer into up to 3 child layers in the augmented UCNP. The subdivision is inserted between 5-90% of the parent layer radii. Optimization For a nanoparticle with N control volumes, we can define the following bounds.
6769dc3a81d2151a02b75ef6	31	0 ≤ x i n ≤ 1, for i ∈ [0 . . l] and n ∈ [0 . . z] (5) 0 < r i ≤ 1, for i ∈ [0 . . l] (6) All dopant concentrations are within the closed interval [0, 1]. The fractional radii is defined on an interval of [0, 1] as a fraction of a the maximum nanoparticle size, r max (which we identify a priori). This is necessary to keep on the same interval as the concentration, since the trust region optimizer defines the same trust region for all independent variables r f raction = r true /r max In addition, we define linear constraints that bound the total concentration within each layer [0, 1] and restrict the thickness of each layer.
6769dc3a81d2151a02b75ef6	32	c min ≤ r 0 ≤ c max (8) t min ≤ r n+1 -r n ≤ t max , for n ∈ [0 . . l -1] (9) We initialize random starting configuration within the distributions outlined for each dataset as a starting point for optimization. We perform local optimization using the trust region constrained optimization as implemented in scipy with an initial trust radius of 1.0. An initial constraint penalty of 1 × 10 3 was applied to strongly penalize constraint violation, ensuring that concentrations stayed within the range of [0, 1.0] and total radius within specification. The criterion used for termination of local optimization is when the trust radius is less than 1 × 10 -8 To search for a globally optimal particle, we repeatedly perturb the local minima and re-optimize the nanoparticle heterostructure. To achieve this, we utilize the basinhopping functionality of scipy with up to 500 steps, a step size of 0.15, and temperature value of 0.25. Following global optimization, validation of the best identified candidate structures are subjected to kMC simulations.
66cb085ca4e53c4876968662	0	Hydrogen bonds (HBs) are among the most important noncovalent interactions in chemistry and biology. For instance, they play a key structural role in the (self-)assembly of supramolecular complexes and the folding of DNA, peptides, and proteins. Furthermore, HBs often act as essential motifs to accelerate reactions in both organocatalytic and enzymatic settings. Despite their importance, relatively few experimental methods exist that can be used to characterize HBs within a quantitative, physical framework. One such method is vibrational Stark effect (VSE) spectroscopy, which enables the measurement of local electric field strengths of specific noncovalent interactions via changes to observables in vibrational spectra. As such, VSE spectroscopy has been used to measure electric fields in solvents, at electrode interfaces, and in membranes and proteins. The VSE describes the influence of an electric field (𝐹 ⃗ ) on a vibrational frequency (𝜈̅ ; in units of cm -1 ) via the dipolar VSE equation
66cb085ca4e53c4876968662	1	because Δ𝛼 is typically experimentally negligible. Several vibrational modes, such as the carbonyl (C=O) stretch, have become very useful VSE sensors because they behave according to eq. 1b. As such, they have enabled the assessment of electric field strengths for HBs and other noncovalent interactions in the condensed phase. The nitrile (C≡N) stretch is the most commonly used vibrational probe, since it appears in an uncluttered region of the infrared (IR) spectrum and because nitriles are easily introduced into biological environments like proteins (via drugs or noncanonical amino acids) or chemical settings like surfaces. Despite its popularity, C≡N frequency tuning can exhibit complicated behavior that does not always follow the VSE (Fig. ). In aprotic environments, the C≡N stretch shows a linear 𝜈̅ /𝐹 ⃗ -behavior as described by eq. 1b. However, in H-bonding environments, anomalous frequency shifts are observed which are inconsistent with eq. 1b. Further, this anomalous behavior cannot be explained by relevant quadratic electric field contributions due to Δ𝛼, that is, eq. 1a also cannot describe the frequency tuning. Instead, a description of nitrile frequencies requires the introduction of an additional variable called the HB blueshift ∆𝜈̅ 𝐻𝐵 , to account for 'C≡N⋅⋅⋅H' interactions: 𝜈̅ (𝐹 ⃗ , ∆𝜈̅ 𝐻𝐵 ) = 𝜈̅ (𝐹 ⃗ ) + ∆𝜈̅ 𝐻𝐵 .
66cb085ca4e53c4876968662	2	correct for this anomaly in the nitrile's 𝜈̅ /𝐹 ⃗ -behavior in Hbonding environments via temperature-dependent experiments, correlations with nuclear magnetic resonance, or molecular dynamics (MD). Recently, we found a new, direct approach to circumvent the issues with nitrile frequencies when we observed that the integrated IR absorption intensity (𝐼 𝐼𝑅 ) of nitriles varies monotonically with the electric field in both aprotic and protic solvents (Fig. ). This additional VSE is explained by the dependence of the transition dipole moment (TDM; 𝑚 ⃗⃗⃗), which governs the IR absorbance, with the electric field according to
66cb085ca4e53c4876968662	3	where 𝑚 ⃗⃗⃗ 0 and 𝐴 are the zero-field transition dipole and the transition dipole polarizability, respectively. Importantly, measuring nitrile TDMs enables quantification of nitrile electric fields in H-bonding environments by using eq. 3a. In addition, jointly interpreting the nitrile's TDM and frequency using eq. 3a and eq. 2, respectively, enables quantification of the anomalous H-bonding blueshift ∆𝜈̅ 𝐻𝐵 . In our recent study, we measured nitrile frequencies and TDMs to directly assess nitrile H-bonding blueshifts for the first time. The new TDM-based method showed that ∆𝜈̅ 𝐻𝐵 can adopt values in a large range from 2 cm -1 to 22 cm -1 in distinct solvent or protein environments. Consequently, we wondered whether the blueshift's magnitude could be a useful metric to describe H-bonding, that is, if ∆𝜈̅ 𝐻𝐵 in eq. 2 could be mathematically modelled. Previous theoretical work explored the complicated vibrational behavior of the C≡N group and suggested that the anomalous 𝜈̅ /𝐹 ⃗ -trend stems from nonnegligible higher order multipole effects or from contributions due to Pauli repulsion. Further, previous work implied that ∆𝜈̅ 𝐻𝐵 may be a HB angle-dependent term (Fig. ), which would be consistent with both proposed physical origins. The lack of intuition for the blueshift's magnitude motivates the need to model ∆𝜈̅ 𝐻𝐵 in a physically interpretable form.
66cb085ca4e53c4876968662	4	As such, we systematically explore HB blueshifts of the C≡N probe herein with the aim to find a simple, analytical expression for this observable. Towards this goal, we combined results from density functional theory (DFT) and the AMOEBA polarizable force field to generate a calibration for the vibrational response of the nitrilecontaining molecule o-tolunitrile (oTN; see Fig. ). In this approach, DFT was used to obtain C≡N vibrational frequencies and TDMs in a large set of purely electrostatic and H-bonding environments (~ 1000 conditions) including point charges (Fig. ) and water and methanol (MeOH) molecules (Fig. ), respectively. Then, the corresponding electric fields exerted on the C≡N were derived from the AMOEBA force field. We attempted to recapitulate the DFTbased frequencies using the VSE (eq. 1a), which was (expectedly) unsuccessful due to the HB blueshift; in contrast, DFT TDMs are well-described by their corresponding VSE equation (eq. 3a), highlighting the different frequency/TDM behaviors that were experimentally observed (Fig. ). We modelled the DFT-derived HB blueshift as a function of HB distance and angle and successfully formulated a quantitative "HB blueshift-vs-HB geometry" relationship. We demonstrate the applicability of Figure . VSE modelling of DFT-based transition dipoles and vibrational frequencies (eq. 3a and eq. 1a, respectively; exact analytical forms are shown in eq. S1 and S2) for oTN's C≡N stretching mode in purely electrostatic environments with point charges and in Hbonding environments with water (A, B, C) and methanol (D, E, F). A, D: Correlation plots between modelled and DFT-based transition dipoles demonstrate eq. 3a accurately describes nitrile environments with purely electrostatic perturbations (red triangles in A and D), water as a HB donor (black circles in A), and methanol as a HB donor (blue squares in D). Fitting parameters for all three environments are reported in Table . B, E: Correlation plots between modelled and DFT-based vibrational frequencies indicate that eq. 1a only applies to nitriles under purely electrostatic perturbations (red triangles in B and E; fitting parameters reported in Table ); when water or methanol are HB donors (black circles in B and blue squares in E, respectively), no correlation (R 2 < 0) is found. The HB blueshift, ∆𝜈̅ 𝐻𝐵 , is determined as the difference between the DFT-predicted frequency and the black line representing ideal correlation, as indicated by the double headed horizontal arrows. C, F: 2D heat plots of ∆𝜈̅ 𝐻𝐵 dependence on heavy atom HB distance and angle [d(C≡N---Owater/MeOH) and θ(C≡N---Owater/MeOH)].
66cb085ca4e53c4876968662	5	In order to find an empirical relation for the HB shift, we chose a DFT-based strategy in which individual positive point charges (125 cases; Fig. ) or individual water or methanol molecules (420 cases each; Fig. ) were placed around oTN's C≡N to model attractive purely electrostatic interactions or H-bonding interactions, respectively. These poses were optimized and normal mode analysis was performed to extract nitrile frequencies and TDMs (b3lyp/6-311++g** level of theory with GD3 dispersion correction). oTN was chosen as our model molecule because it is the sidechain fragment of the nonnatural amino acid o-cyanophenylalanine (oCNF), with which we previously developed and applied the new TDM-based analysis in solvent and protein environments. The charges and molecules were positioned at NC≡N-charge or NC≡N-OHB donor distances (d), respectively, ranging from 5.0 -8.0 Å for point charges and 2.5 -5.0 Å for HB donors, and C≡N-charge and C≡N-OHB donor angles (θ) of 70 -175° were used (Fig. ). The HB distance range was motivated by typical radial distribution functions of HBs, which have a first solvation sphere centered around 2.5 -3.5 Å. The angle range encapsulates HBs which vary from head-on (~ 180°) to sideon (~ 90°). Note that the ideal head-on angle of 180° was not used due to convergence issues in the DFT calculations. The DFT-derived vibrational frequencies (𝜈̅ ) and TDM magnitudes (\|𝑚 ⃗⃗⃗\|) were scaled by 0.9598 56 or 0.4464, respectively, to match the experimental zero-field observables (see Methods Section). Using DFT, we obtained \|𝑚 ⃗⃗⃗\| and 𝜈̅ values for oTN of 0.037 -0.060 D and 2210 -2255 cm -1 , respectively (see xaxes in Fig. ), which are consistent with prior experimental observations for aromatic nitriles (see Fig. ). From the observed ranges it can be seen that purely electrostatic and H-bonding environments give rise to similar values for \|𝑚 ⃗⃗⃗\| (Fig. ), consistent with eq. 3a's indication that \|𝑚 ⃗⃗⃗\| is only a function of 𝐹 ⃗ . For the frequencies (Fig. ), purely electrostatic perturbations produce 𝜈̅ values below the gas phase frequency of 2232 cm -1 , consistent with attractive electrostatic C≡N-charge interactions. In contrast, most H-bonding environments with water give rise to frequencies > 2232 cm -1 , indicative of the H-bonding blueshift.
66cb085ca4e53c4876968662	6	To further underscore the difference in behavior exhibited by the frequencies vs the TDMs, we used the corresponding VSE equations including quadratic electric field contributions (eq. 1a and extension of eq. 3a; see eq. S2 and eq. S1, respectively) to model the DFT-based vibrational observables solely as functions of 𝐹 ⃗ . Towards this goal, we used the polarizable AMOEBA force field to extract the electric field vectors (𝐹 ⃗ ) along the C≡N group for the DFToptimized structures (see Experimental Section for further details). All VSE parameters were allowed to freely vary when fitting the VSE equations against the DFT results (see Tables and). For the TDMs, we found that the VSE modelled the DFT results for purely electrostatic and H-bonding perturbations very well with R 2 > 0.97 (Fig. ). This is consistent with our previous experimental results that TDMs give direct access to the local nitrile electric field in both non-H-bonding and H-bonding environments (Fig. ). Further, this modelling provides a good estimate of the experimental linear field sensitivity of -1.0 𝑚𝐷 𝑀𝑉/𝑐𝑚 (as discussed in SI Section 2). In contrast to the TDMs, the C≡N vibrational frequency shifts are modelled well with eq. 1a for purely electrostatic perturbations but extremely poorly for nitriles with HBs to water molecules (Fig. ). For purely electrostatic perturbations, the correlation between the modelled and DFT frequencies is very good with R 2 of 0.95. This modelling resulted in a Stark tuning rate of \|Δ𝜇 ⃗\| = 0.22 ), which is impressively close to the experimental value of 0.19
66cb085ca4e53c4876968662	7	. However, when Hbonded data points are modelled with eq. 1a using the same parameters, an extremely poor correlation of R 2 = -4.6 is obtained, implying that eq. 1a provides a worse description than just modeling the data with its mean value. The bulk of the deviating data points are located below the line of perfect correlation, i.e. the DFT frequencies are larger than those predicted using eq. 1a. We interpret this deviation (along the x-axis in Fig. ) as the H-bonding blueshift ∆𝜈̅ 𝐻𝐵 (eq. 2; see Fig. ).
66cb085ca4e53c4876968662	8	To verify that this behavior is not specific to water, we used methanol as an alternative HB donor; this is an important test, as methanol is a model for the amino acid sidechains of serine or threonine, and the largest experimentally observed ∆𝜈̅ 𝐻𝐵 occurred for a threoninenitrile interaction. We found that the VSE model's ability to recapitulate the DFT results for TDMs is just as robust as in the case where water is the HB donor (R 2 = 0.96, Fig. ), and highly similar VSE parameters were obtained to those derived for water H-bonding scenarios (Table ). Yet, the correlation of VSE (eq. 1a) and DFT 𝜈̅ values for nitriles with methanol HBs is just as poor as the case for water HBs (R 2 = -4.2, Fig. ).
66cb085ca4e53c4876968662	9	In order to understand the unilateral deviation of the 𝜈̅ values modelled with eq. 1a compared to the DFT frequencies in H-bonding conditions, we hypothesized that ∆𝜈̅ 𝐻𝐵 is a HB-geometry-dependent value, i.e. it depends on the HB-heavy atom distance d(C≡N---Owater/MeOH) and the HBheavy atom angle θ(C≡N---Owater/MeOH) (d and θ in Fig. ). Note that we chose heavy atom-based distances and angles instead of the C≡N---Hwater/MeOH geometry used in other work due to inaccuracies in hydrogen atom positions in MD simulations introduced by frequently used constraint algorithms; furthermore, a calibration with heavy-atoms enables comparisons with protein crystal structures, where protons are very rarely resolved. Extracting the ∆𝜈̅ 𝐻𝐵 values from Fig. and Fig. and the corresponding d(C≡N---Owater/MeOH) and θ(C≡N---Owater/MeOH) from the DFT-optimized geometries, we can visualize the ∆𝜈̅ 𝐻𝐵 geometry dependences for water and methanol HBs as 2D heat plots in Fig. and Fig. , respectively. In both cases, we observe two trends: in going from short (2.5 Å) to long (5.0 Å) distances, ∆𝜈̅ 𝐻𝐵 decreases steadily towards zero, with slightly negative values at intermediate distances (3.5 -4.0 Å) for side-on HBs (see transition from blue to dark blue to blue at angles of 70° -90° and distances of 3.0 -4.5 Å; Fig. ); at the same time, ∆𝜈̅ 𝐻𝐵 decreases while going from head-on (175°) to side-on HBs (70°).
66cb085ca4e53c4876968662	10	Extracting the ∆𝜈̅ 𝐻𝐵 values for head-on or side-on HBs, we can quantify the distance-dependence of ∆𝜈̅ 𝐻𝐵 . We combined the data sets for water and methanol HBs, and we found that head-on HBs (θ = 175°) demonstrate an asymptotic trend (Fig. ) which decays from ~ 50 cm -1 at 2.5 Å to ~ 5 cm -1 at 5.0 Å according to a power law
66cb085ca4e53c4876968662	11	with n1 ≈ -4.0 (R 2 = 0.99). This distance dependence is reminiscent of the energetic contribution from dipolequadrupole interactions, which have a 𝑑 -4 distance dependence. The result is therefore in line with previous interpretations of the ∆𝜈̅ 𝐻𝐵 describing higher (difference) multipole terms not included in the dipolar VSE equations (eqs. 1a/b). When evaluating side-on HBs (θ = 70°), we note a more complicated asymptotic distance dependence with a minimum at roughly 3.5 Å (as noted above), at which point ∆𝜈̅ 𝐻𝐵 is about -5 cm -1 ; this is followed by a gradual increase of ∆𝜈̅ 𝐻𝐵 at larger d, becoming nearly negligible around 5.0 Å. We modelled this distance dependence with a Buckinghamlike function (R 2 = 0.89), ). See Table for the complete list of optimized parameters.
66cb085ca4e53c4876968662	12	We extracted values for the exponential decay constant (b) and the exponent of the power-term (n2) of b ≈ 3.1 Å -1 and n2 ≈ -8.2. The decay constant is in a similar range as values used for intermolecular O---C, C---H, and O---H interactions in force fields (a range of 2.7 -4.6 Å -1 ), suggesting that HB blueshifts in side-on HBs originate from Pauli repulsion; this finding is consistent with previous studies. The power law in the Buckingham potential is typically used with an exponent of -6 to account for attractive dipole -induced dipole interactions. However, the original form of the Buckingham potential also included a d -8 -term accounting for attractive quadrupole -induced quadrupole interactions. When fitting the data in Fig. using two power-terms, both exponents converged to the same value of ~ -8.2, indicating this value is fairly robust; as such, we tentatively assign negative contributions to ∆𝜈̅ 𝐻𝐵 (i.e., redshifts) to induced higher-order multipole interactions.
66cb085ca4e53c4876968662	13	In a similar fashion, we extracted the angular dependence of ∆𝜈̅ 𝐻𝐵 at a HB distance of 3.0 Å, the average HB distance found in solvents and proteins (Fig. ; Table ) and the distance where the side-on HB effect should be close to negligible (see Fig. ). We used the relation
66cb085ca4e53c4876968662	14	to model the data points, and the best fit yielded m = 0.82 (R 2 = 0.89), which accounts for the zero crossing at ~ 70° (Fig. ) by altering the cosine period. This deviation from m = 1 can be understood when taking into account that a side-on HB interacting with the π-orbitals of the C≡N would occur at θ(C≡N---Owater/MeOH) ≈ 70° -80° (Fig. ), and this is the point at which the cosine function should be 0.
66cb085ca4e53c4876968662	15	Here, ∆𝜈̅ 𝐻𝐵,0 is the HB blueshift at a reference distance 𝑑 0 , chosen as the point at which the Buckingham shape crosses zero. Further, 𝑓(𝜃) is the angular term in eq. 6 which modulates the contributions of the head-on and side-on distance dependences of eq. 4 and 5, respectively. We modelled the C≡N frequency for nitriles experiencing purely electrostatic perturbations, HBs with water, and HBs with methanol simultaneously as a function of electric field, HB distance, and HB angle, i.e., using eq. 2 with eq. 7 for the ∆𝜈̅ 𝐻𝐵 term. The resulting "𝜈̅ (𝐹 ⃗ , d, θ) model vs DFT" plot (Fig. ) shows that the VSE (eq. 1a) with the addition of eq. 7 recapitulates the DFT frequencies for purely electrostatic environments just as well as the VSE model alone (Fig. ) but significantly improves the recapitulatability in H-bonding environments. Fitting parameters for the model are reported in Table and Table . B: 2D heat plot of ∆𝜈̅ 𝐻𝐵 (𝑑, 𝜃) with water and methanol as HB donors according to the model in eq. 7. C: 2D heat plot of the residuals between modelled ∆𝜈̅ 𝐻𝐵 (𝑑, 𝜃) (see B) and ∆𝜈̅ 𝐻𝐵 from Fig. C and F (R 2 = 0.96 and RMSD = 2.4 cm -1 ).
66cb085ca4e53c4876968662	16	Specifically, the fitting quality was effectively unaltered for purely electrostatic perturbations (from R 2 = 0.95 to 0.94) but drastically improved in H-bonding environments (from R 2 < 0 to ~0.9). In this fit, the previously optimized VSE and empirical H-bonding parameters remain similar to those obtained in Fig. and Fig. with \|Δ𝜇 ⃗\| = 0.22
66cb085ca4e53c4876968662	17	, a side-on exponential decay constant of b = 2.85 Å -1 , and a cosine period modulation of m = 0.91. When we visualized the dependence of ∆𝜈̅ 𝐻𝐵 (d, θ) in eq. 7 on d(C≡N---Owater/MeOH) and θ(C≡N---Owater/MeOH) as a 2D heat plot (Fig. ), we found a highly analogous profile to those in Fig. and Fig. with a similarly broad range of values adopted (-5 -50 cm -1 ), showing that eq. 7 can recapitulate the DFT HB blueshifts with high accuracy. A 2D heat plot of the residuals between ∆𝜈̅ 𝐻𝐵 (d, θ) (eq. 7) and ∆𝜈̅ 𝐻𝐵 obtained from DFT (Fig. ) has residual values ranging from just -3 to +3 (Fig. ; R 2 = 0.96), further indicating eq. 7 accurately describes the blueshift for many HB distance and angle combinations. Some of largest residuals are found for angles corresponding with side-on HBs, where the Buckingham potential slightly underestimates ∆𝜈̅ 𝐻𝐵 (Fig. ). Even though eq. 7 takes the form of a lengthy expression, only four parameters are needed to sufficiently tune the distance and angle dependence (Table ), and all of them carry physical meaning in terms of describing specific underlying intermolecular interactions.
66cb085ca4e53c4876968662	18	As shown in SI Section 6, we narrowed down Fig. to a relevant regime of commonly adopted HB geometries in solvents for heavy atom distances of < 3.5 -4.0 Å. Based on AMOEBA MD simulations of oTN in water and methanol (see details in SI Section 1), the average HB distance decreases monotonically from 3.35 Å for side-on HBs (70°) towards 2.93 Å when head-on HBs are adopted. Our model (eq. 7) predicts ∆𝜈̅ 𝐻𝐵 ≈ -5 cm -1 for side-on HBs interacting with the C≡N's πorbitals [~ 70° for θ(C≡N---O)]. As the angle and distance concomitantly increase and decrease, respectively, the blueshift increases steadily, plateauing around 26 cm -1 for head-on HBs with θ(C≡N---O) > 170°. Furthermore, we also investigated the HB blueshift in the (rare) case of two simultaneous HBs with a nitrile by comparing DFT blueshifts with values derived using eq. 7 (SI Section 5): we found that summing ∆𝜈̅ 𝐻𝐵 for each HB was an accurate model, implying each H-bonding interaction can be treated independently.
66cb085ca4e53c4876968662	19	Based on the HB geometry-dependent model's ability to recapitulate the nitrile DFT frequencies, we sought to test the model by comparing predicted blueshifts against experimental data for cases with nitriles in H-bonded environments. Towards this goal, we revisited our recent work, in which we introduced the noncanonical amino acid oCNF into photoactive yellow protein (PYP). In this previous work, oCNF was incorporated into PYP in place of endogenous phenylalanines (F), resulting in two PYP variants, F92oCNF and F28oCNF, which were H-bonded and showed distinct ∆𝜈̅ 𝐻𝐵 values with moderate to large magnitudes. In the following, we reanalyze our previously obtained data (namely, IR spectra, crystal structures, and MD simulations) to enable comparisons between experimentally derived HB blueshifts (∆𝜈̅ 𝐻𝐵,𝑜𝑏𝑠 ) and HB blueshifts predicted from MD simulations using eq. 7 (i.e., ∆𝜈̅ 𝐻𝐵, (𝑑, 𝜃)). Starting with F92oCNF (Fig. ), x-ray crystallography showed that the C≡N group is engaged in a head-on HB with the hydroxyl group of threonine 90 (T90), and 100 ns long AMOEBA MD simulations indicated an average C≡N---HO-T90 HB distance and angle of 2.93 Å and 169°, respectively (see Fig. , a representative MD snapshot). Using the HB geometry-dependent model in eq. 7, we derive an average predicted value of 〈∆𝜈̅ 𝐻𝐵, (𝑑, 𝜃)〉 = 27.3 cm -1 , a large value as expected for a head-on HB (Fig. ). To compare this value to experimental results (Fig. ), we also determined the C≡N's peak position due to the VSE alone (eq. 1b); this was done by using the experimentallydetermined zero-field frequency and Stark tuning rate and the average electric field for the H-bonding fraction from MD (-78 MV/cm, ref. ; see SI Section 9 for further details). We obtain a VSE-based vibrational frequency of 2215.5 cm -1 (blue values in Fig. /Fig. and vertical blue line in Fig. ). The experimental IR spectrum of F92oCNF has a peak position of 2241.3 cm -1 , and subtracting the frequency for the VSE alone from the experimental frequency results in a HB blueshift of ∆𝜈̅ 𝐻𝐵,𝑜𝑏𝑠 = 25.8 cm -1 (eq. 2). This experimentally derived blueshift matches very well with the HB geometry-based value of 27.2 cm -1 , as indicated by the similar length of the solid and dashed double headed arrows in Fig. . We note that similar results are obtained when 〈∆𝜈̅ 𝐻𝐵, (𝑑, 𝜃)〉 is calculated from the distribution of ∆𝜈̅ 𝐻𝐵, (𝑑, 𝜃) values obtained by applying eq. 7 to each Hbonding frame of the MD simulation (see distributions for this and the following cases in Fig. ).
66cb085ca4e53c4876968662	20	Further support for our HB geometry-based ∆𝜈̅ 𝐻𝐵, (𝑑, 𝜃) model and the observation of large values for head-on and/or short HBs is provided by a new publication where a nitrile probe was incorporated into different metal organic frameworks (MOFs). The rigid MOF structure enabled the introduction of H-bonding moieties (allylic and aromatic carboxylic acids) near the nitrile. According to DFT, these C≡N---HO interactions occur at average HB and, respectively. The bottom row shows corresponding MD-based simulated IR spectra using eq. 2 (with eqs. 1a and 7) and parameters in Table (and Table ) as a vibrational spectroscopic map (vsm) obtained using the fluctuating frequency approximation (see Methods Section). The solid red vertical lines are predicted peak positions due to the VSE only (using eq. 1b and parameters from ref. ) in Fig. and, and the blueshifts derived from the vsm (∆𝜈̅ 𝐻𝐵,𝑣𝑠𝑚 ) are the difference between the simulated peak position and the frequency of the red line. Data in A is reproduced from ref. . Copyright 2022 American Chemical Society. distances/angles of 2.85 Å/168° (allylic acid; "AA"), 2.80 Å/148° (benzoic acid; "CPh"), and 2.79 Å/150° (isophthalic acid; "DCPh"). As in the case of F92oCNF, ∆𝜈̅ 𝐻𝐵,𝑜𝑏𝑠 and 〈∆𝜈̅ 𝐻𝐵, (𝑑, 𝜃)〉 are in excellent agreement: the experimental/predicted values (in cm -1 ) for AA are 36/31.8, for CPh are 29/28.2, and for DCPh are 33/32.2. It is interesting to note that the nitrile in the MOFs is an aliphatic C≡N, not an oTN derivative, and that the nitrile HB partners are carboxylic acids, not water or alcohols. These differences make the similarity between the experimental and our predicted HB shifts all the more impressive; this comparison suggests that our model can work generally for H-bonded nitriles with different types of HB donors.
66cb085ca4e53c4876968662	21	We next analyzed F28oCNF, where crystallography showed that the C≡N group is solvent exposed and Hbonded to bulk water; MD indicated this interaction has an average HB distance and angle of 3.00 Å and 163° (Fig. ). Using eq. 7, we obtained 〈∆𝜈̅ 𝐻𝐵, (𝑑, 𝜃)〉 = 25.4 cm -1 (Fig. ). However, unlike F92oCNF, we noted a considerable discrepancy between this value and ∆𝜈̅ 𝐻𝐵,𝑜𝑏𝑠 when we analyzed F28oCNF's IR spectra (Fig. ). F28oCNF has an average electric field of -64.9 MV/cm in the MD H-bonding fraction (see ref. ), and the pure VSE effect predicts the C≡N's peak position to be at 2219.1 cm -1 (see values in Fig. and red line in Fig. ). However, in the experimental IR spectrum, we observe a peak position at 2230.9 cm -1 , which leads to ∆𝜈̅ 𝐻𝐵,𝑜𝑏𝑠 = 11.8 cm -1 , only half as large as 〈∆𝜈̅ 𝐻𝐵, (𝑑, 𝜃)〉 (this is visually demonstrated in Fig. by the red line appearing halfway along the dashed double headed arrow). We hypothesized that this discrepancy may be related to F28oCNF's H-bonding with the highly fluctuating solvent environment, in which the C≡N rapidly alternates between H-bonding and non-H-bonding states (SI Section 8).
66cb085ca4e53c4876968662	22	To reconcile the excellent match for F92oCNF and the MOFs but the disparity for F28oCNF and oTN in solvents, we must take into consideration the time scales under which HBs fluctuate for both groups. In F92oCNF (Fig. ), the C≡N is engaged in an intra-protein HB: we detect extended periods of uninterrupted H-bonding and narrow HB distance and angle distributions in AMOEBA MD simulations (see SI Sections 7 and 8), indicating this HB experiences long residence times and minimal geometrical fluctuations. Because of this weakly fluctuating (rigid) C≡N---HO-T90 interaction, the ∆𝜈̅ 𝐻𝐵 distribution (Fig. ) directly reflects on the HB geometry as derived by our model in eq. 7 and Fig. . The same argument holds true for the MOFs, where the HB geometry is locked in place by the framework. These cases are classified as the inhomogeneous limit in IR spectroscopy, i.e. where IR spectra directly reflect the distribution of instantaneous vibrational frequencies. Instead, for F28oCNF and oTN in solvents, the H-bonding with bulk solvent is highly fluctuating, characterized in MD by short H-bonding residence times and broad HB distance/angle distributions (SI Sections 7 and 8). If these fluctuations are faster than the difference in the vibrational frequencies between the fluctuating sub-states (a vibrational frequency difference of ~ 20 cm -1 corresponds to a time scale of ~ 2 ps), the sub-states are not resolved in the IR spectrum but instead motionally narrowed towards one IR band with an averaged peak position (as occurs in coalescence in nuclear magnetic resonance) ; lifetimes for H-bonding and non-H-bonding nitrile states were extracted from MD simulations and qualitatively support this possibility (Fig. and Table ).
66cb085ca4e53c4876968662	23	One way to test the hypothesis of motional narrowing is by applying IR lineshape theory. Accordingly, we used the parameters obtained from DFT to describe the C≡N transition dipole and frequency in terms of electrostatics and HB-geometry (Table and; eqs. 1, 2, 3, and 7) as a model to compute theoretical IR spectra from AMOEBA MD trajectories (referred to as a vibrational spectroscopic map, or "vsm"). Towards this goal, we first calculated the instantaneous C≡N transition dipoles and frequencies from MD simulations (performed with 20 fs time steps over 2 ns in aggregate) for oTN in water and methanol and F28oCNF, and we utilized the well-documented fluctuating frequency approximation (FFA) to calculate MD-based IR spectra. In FFA, a Fourier transformation of the auto-correlation of dipole and frequency fluctuations is used to calculate realistic lineshapes (eq. S3). Comparing the resulting computed IR spectra of oTN in water and methanol (Fig. and, respectively) with those from experiment, we observe a very good recapitulation. In water, the simulated spectra yield one symmetric band for the C≡N stretch, with a peak position (2232 cm -1 ) almost identical to the experimental value; in methanol, the FFA-based spectra show an asymmetric lineshape which occurs due to distinct H-bonded and non-H-bonded fractions absorbing at ~2233 and ~2228 cm -1 , respectively, which are again quite similar for experimental and computed spectra. Importantly, we can take the difference between the vsm frequencies and the previously determined frequencies due to the VSE alone (Fig. G, H; red lines in Fig. ) to determine apparent ∆𝜈̅ 𝐻𝐵,𝑣𝑠𝑚 values of 14.5 and 12.7 cm -1 for water and methanol, respectively, which deviate from the experimentally obtained values by < 0.7 cm -1 , an impressively close match. We used the same approach to calculate the IR spectrum and vsm blueshift for F28oCNF (Fig. ) and again obtain a good match for ∆𝜈̅ 𝐻𝐵 : comparing ∆𝜈̅ 𝐻𝐵,𝑜𝑏𝑠 /∆𝜈̅ 𝐻𝐵,𝑣𝑠𝑚 , we observe values of 11.8/13.6 cm -1 , i.e. a deviation of only 1.8 cm -1 .
66cb085ca4e53c4876968662	24	Overall, the vsm can recapitulate the experimental nitrile spectra with high accuracy. This demonstrates that our HB geometry-dependent model is not only robust in minimally fluctuating settings, but also in (quickly) fluctuating solvent or protein environments when dynamical effects are considered. More specifically, in the cases we tested with fluctuating HBs, the geometry dependent values of 〈∆𝜈̅ 𝐻𝐵, (𝑑, 𝜃)〉 = 23.8 -26.3 cm -1 (Fig. ) are reduced by a factor of roughly 2 to ∆𝜈̅ 𝐻𝐵,𝑜𝑏𝑠 ≈ ∆𝜈̅ 𝐻𝐵,𝑣𝑠𝑚 = 11.8 -15.2 cm -1 (Fig. ). This reduction by a factor of 2 is what is expected for the simplest case when nitrile protic/aprotic sub-populations are interconverting with similarly fast exchange rates such that the geometry dependent value 〈∆𝜈̅ 𝐻𝐵, (𝑑, 𝜃)〉 will be averaged with 0 cm -1 (i.e., the blueshift for the non-H-bonded fraction). This exercise makes clear that knowledge of the dynamics experienced by a nitrile is key to prevent erroneous assessments of the HB geometry based on ∆𝜈̅ 𝐻𝐵,𝑜𝑏𝑠 alone: such dynamics can be evaluated using temperature dependent or two-dimensional IR experiments. To summarize the evaluation of our models for HB blueshifts, we correlated the experimental and predicted values for ∆𝜈̅ 𝐻𝐵 in Fig. . We find that calculating 〈∆𝜈̅ 𝐻𝐵, (𝑑, 𝜃)〉 from our HB geometry-dependent model in eq. 7 works very well for rigid HBs in F92oCNF and MOFs, implying it is possible to extract information on HB geometry directly from HB blueshifts. For fluctuating HBs like F28oCNF and oTN in solvents, HB dynamics have to be considered, as described above: when they are, an excellent agreement between observed and modelled HB blueshifts is obtained (R 2 = 0.95). blueshifts; when the appropriate model is used, the correlation is excellent (R 2 = 0.95). The black diagonal is the perfect correlation with a slope of 1. Black data points for MOFs are taken from ref. .
66cb085ca4e53c4876968662	25	Aiming to provide a simple, empirical description for the anomalous HB blueshift of nitriles, we developed a model that describes HB effects on C≡N frequencies as the sum of the widely used VSE and an additional term, ∆𝜈̅ 𝐻𝐵 . This model describes ∆𝜈̅ 𝐻𝐵 in terms of HB geometry, i.e. HB heavy atom distance d(C≡N---Donor) and angle θ(C≡N---Donor). The physical basis for the distance and angle dependence are a combination of repulsive quadrupolar electrostatic interactions for head-on HBs and an interplay between Pauli repulsion and attractive multipolar interactions for side-on HBs, supporting previous interpretations of the blueshift's origin(s). These findings further expand on theoretical models that have aimed to understand H-bonding in terms of its quantum and/or classical mechanical nature, many of which have pointed towards a dominant (classical) electrostatic character; our study is in line with this latter view. We found an important third contributor to ∆𝜈̅ 𝐻𝐵 , the HB dynamics, also needs to be considered when using the model developed herein. ∆𝜈̅ 𝐻𝐵 values of rigid HBs with long residence times and minimal fluctuations are directly dependent on HB geometry; in contrast, nitrile IR bands for quickly fluctuating HBs experience motional narrowing, altering their lineshapes. Consequently, HB residence/exchange times should be considered when estimating HB geometry via ∆𝜈̅ 𝐻𝐵 . In closing, we emphasize that the nitrile model presented in eq. 7 works well for MOFs which have a different type of nitrile and different HB donors. This suggests that the model developed here is broadly applicable and can be used to characterize HBs for nitriles on diverse substrates, ranging from drugs to amino acids, and in diverse settings, ranging from electrodes to microdroplets to proteins.
624d7ec5855ee52f2ce39798	0	Nucleosides and their analogs are central to the biological and chemical sciences, as they serve a variety of biological functions and represent a growing class of anti-cancer and anti-viral pharmaceuticals. Although the synthesis of nucleosides typically proceeds via N-glycosylation of nucleobases with heavily protected sugar synthons, there is an increasing recognition for the inefficiency of the associated synthetic routes. Consequently, recent years have experienced renewed interest in the biocatalytic synthesis of natural and modified nucleosides. This includes, for instance, the enzymatic preparation of halogenated purine nucleoside synthons, the diversification of alkylated pyrimidine nucleoside analogues, the development of high-yielding flow processes, and the synthesis of the pharmaceuticals islatravir (anti-HIV) and molnupiravir (anti-Covid19) in biocatalytic cascades. All these examples employ nucleoside phosphorylases for key (trans-)glycosylation reactions, which enable the installation of ribosyl-based moieties on pyrimidine and purine nucleobases in one step and without the need for any protecting group chemistry.
624d7ec5855ee52f2ce39798	1	Nucleoside phosphorylases catalyze the reversible phosphorolysis of nucleosides to the corresponding nucleobases and pentose-1-phosphates (Scheme 1). This reactivity can be employed in reverse to transfer the glycosyl moiety from one nucleoside (or a pentose-1-phosphate) to another nucleobase, a reaction typically referred to as a (trans-)glycosylation. While such (trans-)glycosylation processes are well established as synthetic tools, they inherently suffer from thermodynamic limitations, as the final yield of these reactions is dictated solely by the substratedependent thermodynamics of the respective (reverse) phosphorolytic steps as well as the employed reaction con-Scheme 1. Nucleoside phosphorolysis and strategies for apparent equilibrium shifts. NB = nucleobase. ditions. Although some progress has been made to mitigate or exploit the tight thermodynamic control in these systems (e.g., via (by)product precipitation, enzymatic product removal or application of recoverable excess reagent ) and irreversible phosphorolysis of 6-oxopurines is well established, there currently exists no general method for the direct manipulation of glycosylation equilibria.
624d7ec5855ee52f2ce39798	2	During our development of the PUB module for continuous high-throughput phosphate detection in biochemical assays, we serendipitously found that the presence of borate effects a series of inhibitory phenomena during pyrimidine nucleoside phosphorolysis as well as an apparent equilibrium shift caused by biased borate esterification of nucleosides over ribose 1-phosphate (Rib1P). While saccharides are well known to undergo complex equilibrium reactions with borate in aqueous solution, 28,29 comparably little is known about in situ competition of such processes. Similarly, although the literature offers some examples for biocatalytic applications of equilibrium shift phenomena through preferential esterification of one reactant with borate (namely, the enzymatic epimerization of fructose, lactose, galactose, and arabinose ) the exact species involved in these processes remain largely elusive, as do the kinetic implications of this esterification on the enzyme-level. Despite the precedents for borate inhibiting NAD + -dependent enzymes in a noncompetitive fashion, the molecular determinants and mechanisms of these phenomena have remained equally elusive. To shed light on the thermodynamic and kinetic implications of borate ester formation on nucleoside phosphorylase-catalyzed reactions, we herein report a multifaceted analysis of this reaction system with spectroscopic and computational approaches. Furthermore, we examined the synthetic utility of this biased borate esterification, as concentration-dependent apparent equilibrium shifts provide an orthogonal dimension for reaction engineering in biocatalytic glycosylation reactions.
624d7ec5855ee52f2ce39798	3	Our investigation was sparked by the serendipitous observation that phosphorolysis reactions with 5bromouridine (1a) consistently exhibited noticeably lower equilibrium conversions in the presence of moderate concentrations of borate (Figures and). For instance, a reaction containing 400 µM 1a, 4 mM (10 equivalents) phosphate and 40 µg mL -1 (0.45 mol%) of the well characterized pyrimidine nucleoside phosphorylase from Geobacillus thermoglucosidasius (GtPyNP), serving as a model enzyme (see the Supplementary Information for details), reached its equilibrium at 70% conversion after 10 min in glycine-buffered solution (Fig. ). In contrast, the same reaction additionally containing 20 mM borate took almost 20 min to reach an equilibrium at 47% conversion, as monitored by multi-wavelength UV spectroscopy employing principles of spectral unmixing. This effect was not rooted in enzyme inactivation, as GtPyNP showed an identical melting point and fully retained its catalytic activity after prolonged incubation in borate-containing buffers (Figs. and). The observed apparent equilibrium shift persisted in the reverse direction of the reaction (starting from the products Rib1P and 2a, Fig. ) and additional experiments established that the reaction system consistently behaved according to lower apparent equilibrium constants of phosphorolysis (K′), whose magnitude depended on the concentration of borate (Fig. ). Since such drastic equilibrium shifts are unprecedented for phosphorolysis systems, we suspected that the formation of a dominant secondary species would actively remove the nucleoside 1a from this equilibrium in a thermodynamically controlled fashion. Indeed, the conversion data phenotypically following a Boltzmann-type relationship (Fig. ) could be described See the SI for experimental details, raw data and equations. [a] Not confidently accessible by DFT due to their high charge.
624d7ec5855ee52f2ce39798	4	well with a thermodynamic model accounting for the presence of an additional equilibrium system in which 1a is partially transformed to a borate ester (Fig. , see the SI for details and equations). Although borate esters of ribose are known to persist in dilute aqueous solution, these stable esters generally involve the anomeric hydroxyl group, which is absent in 1a. Nevertheless, Kim et al. observed borate esters of the nucleoside-based cofactor NAD + and the natural trinucleotides by mass spectrometry, indicating that analogous species might be formed by recruitment of the 2′and/or 3′-hydroxyl groups. In addition, structurally similar nucleoside boronate esters have been reported by Smietana and others. Based on these precedents, we hypothesized that the cyclic borate ester 1a* would be the predominant species causing the observed equilibrium shift (Fig. ).
624d7ec5855ee52f2ce39798	5	To probe if borate esters like 1a* would be feasible species to effect apparent equilibrium shifts under dilute aqueous reaction conditions, we turned to density functional theory (DFT) calculations and NMR spectroscopy. DFT calculations (using B3LYP/TZP in combination with a COSMO solvent model for water, see the SI for details) suggested that the formation of the 2′-or 3′-borate monoesters of 1a would be highly disfavored processes, while the formation of the proposed five-membered cyclic diester 1a* offers around 40 kJ mol -1 net gain in Gibbs free energy, indicating that this esterification could feasibly proceed against the concentration gradient of water (Fig. ). Indeed, direct evidence for the presence of 1a* could be obtained by NMR spectroscopy. When 1a (10 mM) was incubated with 20 mM borate in glycine buffer at pH 9, a dominant borate-containing species in equilibrium with free borate could be observed by B NMR (Fig. ), while 1 H NMR showed additional signals indicative of a modification of the ribosyl moiety of 1a (Fig. ). Furthermore, slight changes in the coupling constants across the sugar ring implied the introduction of ring torsion, consistent with the configurational changes required in 1a* (Table ). In addition to this ester, we observed two distinct minor components in the mixture, which exhibited identical coupling constants to 1a* and existed in similar concentrations (ca. 1:1.2 ratio). Based on the well-characterized diastereomeric borate ester dimers of methyl apiose described by Ishii and Ono and the report of purine nucleoside solubilization as 2:1 complexes with borate by Tsuji and colleagues, we tentatively ascribe these species as the cis-and trans-isomers 1a** (Figs. and). Overall, the Gibbs free energies obtained experimentally by NMR spectroscopy or equilibrium state calculations based on UV data (Δ ca. -32 kJ mol -1 , see the SI for details) are comparable to those obtained by DFT calculation. Although an analogous six-membered ester (and potentially its dimers) between the 3′-and 5′-hydroxyl groups would introduce similar configurational changes to those ascribed to 1a, DFT calculations revealed that its formation is much less exothermic and consequently disfavored. Similarly, the analogous five-membered borate ester of Rib1P is less favored than 1a (which is requisite for the observed equilibrium shift), as supported by DFT and NMR data (Figs. and). Additionally, we observed no trace of dimers of Rib1P* by NMR, which is likely a result of the highly disfavored formal -5 charge of these species. Consequently, the formation of the borate ester 1a* should be expected to dominate in direct competition during a phosphorolysis reaction, which we could confirm by subjecting phosphorolysis reaction mixtures with increasing borate concentrations to NMR analysis (Fig. ). Further thermodynamic experiments indicated that this biased esterification is primarily driven by entropic effects. Arrhenius plots of the 1a-1a* equilibrium obtained by H NMR revealed a drastic temperaturedependence, favoring the presence of the free nucleoside at higher temperatures (Fig. ). In support of these observations, Arrhenius plots for the borate esterification derived from equilibrium shifts in the phosphorolysis reaction indicated an approach to an energetic balance between 1a* and Rib1P* at higher temperatures (Fig. ). Collectively, these results describe 1a* as the dominant borate ester in this reaction system, responsible for depleting the pool of free nucleoside in dilute aqueous solution at room temperature. Consistent with this conclusion, the 2′-deoxy nucleoside deoxy-1a, incapable of forming 1a* or 1a**, did not show an equilibrium shift during its phosphorolysis in the presence of borate or any reaction with borate discernible by NMR (Fig. ).
624d7ec5855ee52f2ce39798	6	Next, we sought to identify the cause of the apparently reduced reaction rates in the presence of borate by enlisting kinetic studies and molecular dynamics (MD) simulations. This effect was especially prominent with borate concentrations greater than 20 mM and led to decreases in the reaction rate of GtPyNP by more than a factor of four, as illustrated in Figure . Since high borate concentrations primarily yield 1a* in solution (and not the enzymatic substrate 1a), we initially entertained the hypothesis that this phenomenon was a function of the decreased concentration of the free nucleoside substrate and the associated apparent decrease of affinity. However, the Michaelis-Menten kinetics obtained for 1a (with phosphate in excess) proved inconsistent with this hypothesis. While a reduction in available substrate concentration should primarily result in a decrease in the apparent Michaelis-Menten constant KM, we observed no change in KM but instead a sharp decline of the rate constant under saturating substrate concentrations (kcat, Fig. ). With saturating concentrations of both substrates, the observed rate constant kobs exhibited a similar Boltzmann-type decrease as observed for the apparent equilibrium constant, which could be described well by an equilibrium model expressing the observed rate constant as a function of kcat and the esterified fraction of 1a (Fig 2D, see the SI for equations). As this decrease of kobs was further completely absent for deoxy-1a in the presence of borate (Figs. and), we concluded that borate alone does not inhibit GtPyNP, but rather the borate ester 1a. If GtPyNP could bind but not convert 1a, we reasoned that the position of the equilibrium between 1a and 1a* would determine the ratio of potentially active enzyme (1a bound to GtPyNP) versus inactive enzyme (1a* bound to GtPyNP), assuming that catalysis is a rate-limiting step. Given the highly solvent-exposed active site of pyrimidine nucleoside phosphorylases in the open state, we expected that GtPyNP should be able to accommodate the slightly twisted and sterically more demanding borate ester 1a* and allow its equilibration with the free nucleoside substrate 1a while bound to the enzyme. Phenomenologically, such a process would resemble a classical non-competitive inhibition, consistent with our kinetic data. Indeed, MD simulations (using GROMACS with the CHARMM36 force field) based on our recently disclosed crystal structure of GtPyNP in complex with uridine (PDB ID 7m7k, see the SI for details) yielded several insights in support of the proposed model. First, an analysis of the clustered states over 50 ns simulation time indicates that the borate ester 1a* can be bound in analogy to 1a via hydrogen bonds with the amide motif of the nucleobase. Secondly, this analysis also showed that the average state in which the enzyme-1a and the enzyme-1a* complexes resided during the simulation time displayed a quite solvent-exposed active site, feasibly permitting esterification and hydrolysis processes to happen in situ. Thirdly, an examination of the distances between the domains responsible for active site closure indicated markedly reduced molecular motion of the enzyme-1a* complex compared to the enzyme-1a complex (Figs. and). While GtPyNP with bound 1a exhibited oscillatory opening and closing motions on a timescale of around 11 ns (similar dynamics are known for the closely related thymidine phosphorylases), binding of 1a* largely arrested this process. Specifically, binding of 1a* locks the enzyme in an open conformation by displacing a catalytically essential arginine residue, which reversibly adopts an inward position. This appears to halt domain movement and yield an inactive enzyme, while the esterification of the 2ꞌ-and 3ꞌ-OH groups of 1a further obstructs access to the anomeric carbon, collectively preventing productive phosphorolysis. As indicated by the clustered structures, 1a* also slightly obstructs access to the phosphate binding site of GtPyNP. Accordingly, we experimentally observed a decreased affinity for phosphate in the presence of 1a, in addition to the lower kcat values stemming from pseudo-non-competitive inhibition by 1a (Fig. ). Experimentally accessed and computed rate constants for the various processes involved in the catalytic cycle also proved consistent with the hypothesized mechanism. Enzyme opening/closing (ca. 0.1 ns -1 ) occurs on a much shorter time scale than the esterification of 1a (ca. 3 s -1 , Fig. ), catalysis (ca. 6 s -1 ) and substrate binding by GtPyNP (ca. 16 s -1 , see the SI). Thus, the comparably slow substrate release (ca. 0.2 s -1 ) necessitates in situ hydrolysis of 1a* to maintain the observed kinetics. In contrast to the phosphorolysis, the kinetics in the glycosylation direction remained essentially unchanged in the presence of borate (Figs. and), providing further support for the minor role of Rib1P* in this reaction system. Consistent with the entropically driven formation of 1a* from 1a, we observed decreased inhibition of the phosphorolysis reaction at higher temperatures, as evident from Eyring plots obtained with different borate concentrations (Fig. ). Taken together, these results support an inhibitory mechanism phenotypically resembling a non-competitive inhibition, where rapid equilibration of 1a to its borate ester 1a* (both in solution and while bound to the enzyme) reversibly decreases the fraction of catalytically active GtPyNP so that throughput in its catalytic cycle is primarily regulated by the position of the 1a-1a* equilibrium (Scheme 2). Preliminary data for other pyrimidine nucleoside phosphorylases as well as other nucleosides suggests that this inhibitory mechanism is likely not limited to GtPyNP and 1a (Figs. and). With a good understanding of the underlying processes governing the kinetics and thermodynamics of the phosphorolysis of 1a in the presence of borate, we aimed to apply the observed equilibrium shifts to other nucleosides, specifically targeting glycosylation reactions. Assuming that other nucleosides would behave in analogy to our model compound 1a, we expected that conversion shifts in glycosylation reactions would provide a general strategy to improve access to nucleosides from the precursor Rib1P (which could either be supplied as an isolated compound or generated in situ). Indeed, DFT calculations for a Scheme 2. Proposed mechanism for rate decreases in the presence of borate. representative set of nucleosides suggested that a variety of pyrimidine and purine nucleosides should undergo similar esterifications with borate as 1a (Table ), which could be confirmed by 1 H NMR (Δ ca. -32 to -34 kJ mol -1 , Fig. ) and, for two examples, with kinetic experiments (see the SI and Fig. for details). Although we were unable to translate the observed minor differences in Gibbs free energies to conversion shifts in transglycosylations, presumably due to a "kinetic lock" effect (see the SI and Fig. for details), glycosylation reactions with various pyrimidine nucleobases 2a-f nicely reflected the expected behavior and facilitated conversion shifts of 6-17% in favor of the respective nucleoside (Fig. ). A similar effect could be observed for the halogenated purine 2g when subjected to identical conditions with the promiscuous purine nucleoside phosphorylase from G. thermoglucosidasius. Lastly, an illustrative two-factor optimization for the glycosylation of 5-iodouracil (2d) and 5ethynyluracil (2f, both are known for their unfavorable glycosylation thermodynamics) showed how a balance of excess sugar donor and the "pull" effect of the corresponding nucleoside borate ester can be employed to improve the conversions in historically challenging nucleobase glycosylations (Fig. ). For instance, when using 2 eq. of Rib1P, the conversion of 2f to its nucleoside 1f could be improved from 74% to 89% through the application of 50 mM borate, which, conventionally, would have required the application of at least 6 eq. of Rib1P.
624d7ec5855ee52f2ce39798	7	In conclusion, we characterized the equilibrium between ribosyl nucleosides and their corresponding 2′,3′-borate esters in aqueous solution, a phenomenon which facilitates apparent equilibrium shifts during nucleoside phosphorolysis and glycosylation reactions due to a biased esterification of nucleosides over the sugar phosphate Rib1P. This borate esterification also causes decreases of the phosphorolysis rate by pyrimidine nucleoside phosphorylases, most likely via non-productive binding of the nucleoside borate ester to the enzyme and its hydrolytic interconversion to the free substrate. Collectively, the effects described herein shine light on the activity of nucleoside-binding enzymes in the presence of borate and provide an orthogonal dimension for reaction engineering in nucleobase glycosylation reactions.
6753b4db7be152b1d031afe9	0	The polarizable continuum model (PCM) is a popular approach for incorporating solvation into electronic structure calculations that describes electrostatic and polarization interactions between an atomistic solute and its continuum environment. Although accurate modeling of solvation energies requires that this model be augmented by non-electrostatic contributions, what the PCM does correctly is to furnish boundary conditions for an electronic structure calculation that are superior to gas-phase boundary conditions. For typical solutes that are amenable to quantum chemistry calculations, the PCM contributes only modest computational overhead. Linear-scaling PCM algorithms have been developed for much larger solutes, extending the use of PCM boundary conditions to cases where the atomistic region is described using a quantum mechanics/molecular mechanics (QM/MM) formalism. PCM calculations on full proteins have been reported. For large protein models, however, it is not always clear that an isotropic dielectric medium is the appropriate choice of boundary conditions.
6753b4db7be152b1d031afe9	1	where Ω represents a molecule-sized cavity. Whereas solution of Poisson's equation requires discretization of the solute's charge density and electrostatic potential throughout three-dimensional space, the PCM only requires the electrostatic potential to be evaluated on the surface of the cavity that defines Ω, the interface with the continuum. The reformulation of Poisson's equation is exact if the entirety of the solute's charge density is contained within Ω, but is highly accurate even when the tails of the charge distribution penetrate into the dielectric medium, as they do in any QM calculation with a realistic, molecule-sized cavity. Derivation of the PCM starting from Poisson's equation is predicated on the assumption that the dielectric medium is isotropic. To describe anisotropic solvation, such as an air/water interface or a water/biomolecule interface, one can always resort to solution of a general-ized Poisson equation with a permittivity function ε(r) that is defined pointwise throughout three-dimensional space. This approach is more common in plane-wave electronic structure codes, where solution of Poisson's equation is already a part of the standard computational machinery, but in localized Gaussian orbital codes it sacrifices the efficiency advantages of the PCM and the computational overhead to solve Poisson's equation is not negligible.
6753b4db7be152b1d031afe9	2	There have been various efforts to extend PCMs to anisotropic solvation environments without sacrificing simplicity and computational expedience. Some approaches with good formal properties, but which lack the generality that we are seeking, include methods that generalize the dielectric "constant" to a 3 × 3 tensor, as appropriate for liquid crystals, and other methods that modify the Green's function for the Coulomb potential appearing in Poisson's equation, for the case of a two-dimensional interface. However, the most convenient (and potentially general) extension is a "heterogeneous" PCM (HetPCM), in which each atomic sphere that is used to construct the solute cavity is assigned its own dielectric constant. In that way, one might hope to provide appropriate boundary conditions for a protein (for example), in which certain residues are exposed to the aqueous solvent (ε = 78) while others are buried in the protein's hydrophobic interior. For the latter environment, values ε ≈ 4 are often used in classical biomolecular electrostatics calculations, e.g., to compute pK a values. In some cases, larger values have been used for the nonpolar dielectric constant, up to ε = 10-20. The HetPCM approach has an appealing simplicity and would be easy to combine with fragment-based methods that can be used to extend the reach of quantum chemistry to proteins. To the best of our knowledge, however, this model is not derivable starting from a well-defined permittivity model ε(r), in contrast to the original (homogeneous or isotropic) PCM. Thus, HetPCM has been introduced as an ad hoc modification of the original model, which has not been rigorously tested against exact continuum electrostatics theory. We do so in the present work, using a generalized Poisson equation solver (PEqS) to provide a benchmark result for the solute-continuum polarization energy associated with any model permittivity function. This function, ε(r), is defined pointwise in three-dimensional space, allowing different spatial domains to have different permittivities. Crucial to this testing regiment is the construction of model systems for which the function ε(r) unambiguously places each continuum-exposed atomic sphere into a region where the value of ε is approximately constant, such that the model function ε(r) can be used to assign a permittivity to each atomic sphere. We accomplish this via a construction of model functions ε(r) based on Voronoi cells, that can then be mapped onto atomic dielectric constants.
6753b4db7be152b1d031afe9	3	Fig. (a) Schematic depiction of the CF3(CF2)3SO3H molecule in a heterogeneous solvation environment, such that the acidic -SO3H group and the perfluorocarbon tail are embedding in media with different dielectric constants, ε1 and ε2, respectively. A solute cavity consisting of atom-centered spheres is indicated, whose whose boundary is denoted by Γ. Note that ε = 1 within the solute cavity (r ∈ Ω). (b) Depiction of the surface quadrature grid points used for PCM calculations. The size of each discretization point si is an indication of its contribution ai to the cavity surface area, as defined in eq. 10.
6753b4db7be152b1d031afe9	4	in which ρ sol (r) is the charge density of the atomistic solute, obtained herein from a quantum chemistry calculation, and ϕ tot (r) is the total electrostatic potential. The latter includes the potential generated by ρ sol (r) but also contributions from polarizing the continuum. Gaussian electrostatic units are used in eq. 2, such that 4πε 0 = 1. The original PCM solves the model problem defined by by eq. 2 and the sharp dielectric interface in eq. 1, where r ∈ Ω in eq. 1 indicates the interior of the molecular cavity (see Fig. ) and ε s is the (static) dielectric constant of the solvent. In the special case where ρ sol (r) vanishes for r /
6753b4db7be152b1d031afe9	5	∈ Ω, the model defined by eqs. 1 and 2 can be mapped onto an equivalent boundary-element or apparent surface charge (ASC) problem, defined at the cavity surface Γ that is indicated in Fig. . That remapping defines the PCM. The most fundamental version of this remapping has been called the integral equation formulation (IEF). For classical solutes where there is no charge leakage into the continuum, IEF-PCM is an exact reformulation of isotropic Poisson boundary conditions, as can be demonstrated numerically. For QM charge densities, the tails of ρ sol (r) do penetrate into the medium, to the tune of ∼ 0.1-0.2 electrons for small molecules. However, an alternative derivation of the IEF-PCM equation demonstrates that this approach implicitly (albeit approximately) accounts for the volume polarization due to this escaped charge. Therefore, IEF-PCM is an accurate reformulation of Poisson boundary conditions even in the case of a QM solute. In matrix form, the IEF-PCM equation is 3
6753b4db7be152b1d031afe9	6	The input is a vector v containing the solute's electrostatic potential, evaluated at a set of discretization points on the cavity surface (Fig. ), while the output is a vector of surfaces charges q at the same points. These charges, {q i }, describe the polarizing effect of the medium. Several other ASC-PCM methods can be cast in the form of eq. 3, 3,10,67 and IEF-PCM is defined by a particular choice of the matrices K ε and Y ε :
6753b4db7be152b1d031afe9	7	The matrices S and D are discretized forms of the socalled single-and double-layer operators, Ŝ and D. The former generates the surface electrostatic potential, and its matrix representation S consists of the Coulomb interaction between surface elements. The operator D † generates the normal electric field at the cavity surface. The IEF-PCM version of Poisson's equation provides a theoretical basis for several other approaches to implicit solvation including Generalized Born models and Debye-Hückel theory. Replacing DAS in eq. 4 with (DAS + SAD † )/2 affords the surface and simulation of volume polarization for electrostatics [SS(V)PE] method, which is formally equivalent to IEF-PCM at the level of integral equations. However, the SS(V)PE form is more sensitive to the quality of the surface discretization and may exhibit numerical artifacts at "crevices" between atomic spheres. These artifacts generally disappear for larger atomic spheres, which afford smoother cavities, but for this reason we will use the IEF-PCM form exclusively.
6753b4db7be152b1d031afe9	8	The conventional choice for C-PCM is ζ = 0 in eq. 9, corresponding to the conductor limit of IEF-PCM. Other choices (chiefly ζ = 1/2) are sometimes encountered but will not be used here. For ε s 30, C-PCM is numerically indistinguishable from IEF-PCM. Even for smaller values of ε s the differences are modest, perhaps 1-2 kcal/mol in the electrostatic solvation energies of small molecules. 2.2. HetPCM. The IEF-PCM and C-PCM methods are defined by eq. 3 and the dependence on ε s is contained wholly within the factors f ε and fε (ζ) in eqs. 6 and 9, respectively. The HetPCM approach to be tested here, which was introduced in Ref. 41 for IEF-PCM and in Ref. 42 for C-PCM, consists in modifying this factor to use a different value of ε s for each atomic sphere. To fully specify the model, however, it is necessary to consider how we discretize the solute cavity surface.
6753b4db7be152b1d031afe9	9	We employ the switching/Gaussian ("SwiG") discretization algorithm, which uses atomcentered Lebedev quadrature grids for each atomic sphere. Each surface point s i is assigned a Lebedev quadrature weight w Leb i and a switching amplitude F sw i with 0 ≤ F sw i ≤ 1, as shown in Fig. . The nature of the switching functions that is detailed elsewhere. The size of each discretization point s i in Fig. corresponds to the surface area a i that is assigned to that point, and which is the diagonal entry of the matrix A that was introduced in Section 2 .1. For a discretization point s i on the surface of atom B whose atomic radius is R vdW,B , we set
6753b4db7be152b1d031afe9	10	where ε i = ε(s i ) is a dielectric constant for the surface element s i . For C-PCM, the factor fε (ζ) in eq. 9 could be modified analogously. In the examples considered herein, we use no more than two different values for the quantities ε i , namely, ε solv = 78 for solvent-exposed parts of the cavity surface and something smaller (ε nonp = 2-10) for hydrophobic portions of the surface. These are trivial modifications, in any implementation of C-PCM or IEF-PCM.
6753b4db7be152b1d031afe9	11	The HetPCM approach is a simple but ad hoc modification of the ASC-PCM formalism, which we intend to test against rigorous Poisson boundary conditions that can describe an anisotropic continuum environment in a general way, by specifying a permittivity function ε(r). This flexibility facilitates the use of a heterogeneous (anisotropic) dielectric environment. The function ε(r) represents a model that can be used, for example, to describe the air/water interface, or different regions of a protein 81-83 or other complex system. Given a model ε(r), the corresponding Poisson boundary conditions are implemented in an exact way, up to controllable discretization errors. Such methods have a long history in classical biomolecular electrostatics calculations, and a variety of numerical solvers have been developed, but implementations of Poisson boundary conditions for electronic structure calculations have also been reported. The finite-difference algorithms that are typically used to solve eq. 2 require that the permittivity function ε(r) to be smoothly varying, else the induced charge may vary wildly in space and the discretization problem becomes severe.
6753b4db7be152b1d031afe9	12	In the present work, eq. 2 is solved for densities ρ sol (r) from Hartree-Fock (HF) or density-functional theory (DFT) calculations, using the PEqS algorithm described previously. We partition the electrostatic potential ϕ tot into a contribution arising directly from the solute's charge density (ϕ sol ) and a polarization potential (ϕ pol ) arising from the induced charge in the continuum:

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