Spatial confinement is widely employed by nature to attain unique efficiency in controlling chemical reactions. Notable examples are enzymes, which selectively bind reactants and exquisitely regulate their conversion into products. In an attempt to mimic natural catalytic systems, supramolecular metal–organic cages capable of encapsulating guests in their cavity and of controlling/accelerating chemical reactions under confinement are attracting increasing interest. However, the complex nature of these systems, where reactants/products continuously exchange in-and-out of the host, makes it often difficult to elucidate the factors controlling the reactivity in dynamic regimes. As a case study, here we focus on a coordination cage that can encapsulate amide guests and enhance their hydrolysis by favoring their mechanical twisting towards reactive molecular configurations under confinement. We designed an advanced multiscale simulation approach that allows us to reconstruct the reactivity in such host–guest systems in dynamic regimes. In this way, we can characterize amide encapsulation/expulsion in/out of the cage cavity (thermodynamics and kinetics), coupling such host–guest dynamic equilibrium with characteristic hydrolysis reaction constants. All computed kinetic/thermodynamic data are then combined, obtaining a statistical estimation of reaction acceleration in the host–guest system that is found in optimal agreement with the available experimental trends. This shows how, to understand the key factors controlling accelerations/variations in the reaction under confinement, it is necessary to take into account all dynamic processes that occur as intimately entangled in such host–guest systems. This also provides us with a flexible computational framework, useful to build structure–dynamics–property relationships for a variety of reactive host–guest systems. Encapsulation of guests in metal–organic cages allows control over chemical reactivity. Focusing on the hydrolysis of amides, here we show an effective molecular simulation approach to reconstruct reactivity in host–guest systems in dynamic regimes. In particular, the molecular crowding in the cavity of the cage was found to favor the mechanical twisting of the amides towards reactive (cis isomers) configurations.10 The rational design of similar synthetic molecular systems requires a detailed comprehension of (i) the molecular and chemical–physical factors that control their reactivity in space and time, and (ii) how to master them in order to produce new classes of catalytic systems. However, in such supramolecular host–guest systems, the reactivity is coupled with a dynamic equilibrium where the guests exchange in/out of the cage hosts, which makes them difficult to rationalize.16,18,19In an attempt to mimic natural catalytic systems, chemists have designed synthetic cavities capable of encapsulating reactants with high selectivity.10,16,20In general, the reactivity in such systems is controlled by several factors, such as guest binding/unbinding, the solubility of the guests in the outer solution, the characteristic timescale for the reactions, possible entrapment in metastable states, and molecular concentrations in the system. In the presence of coordination-cages, a major role in reactivity is played by non-covalent cavity-guest interactions.9,15,20–22 reaching a detailed understanding in terms of the molecular factors that control the reactivity in systems in which the molecular species are in continuous exchange is typically difficult at the experimental level. Computer simulations are extremely useful to this end. Quantum mechanical (QM) approaches23 or semi-empirical approaches such as density functional tight binding (DF-TB) methods 24 have been employed to study chemical reactivity and reactive pathways with notable precision. Ab initio molecular dynamics (MD)25–27 and metadynamics28–30 simulations have been also widely adopted to study chemical reactions of reactants in reactive configurations, in cases where the reactions require crossing free-energy barriers. However, the study of such systems on timescales that also allows accounting for their supramolecular host–guest dynamics  is not trivial. For this reason, a comprehensive description of such complex dynamic systems has been difficult to attain until now. Recent computational approaches based on enhanced sampling methods, such as, e.g., metadynamics (MetaD), hold great potential in this sense.16,31–33 Such recently employed approaches, e.g., to study the isomerization of photochromic switches (azobenzenes) encapsulated in coordination cages, allowed the demonstration of the tight interplay between isomerization dynamics, molecular crowding, and host–guest exchange dynamics.16 This approach is versatile and holds high potential for studying chemical reactions in host–guest systems in dynamic regimes in general.Despite notable efforts,10 where the encapsulation of amide guests within a coordination cage was found to considerably enhance the amide hydrolysis. This is an interesting test case not only because one amide functional group (the peptide bond) is the structural foundation of proteins, but also because the hydrolysis of amides is a classic example of a spontaneous reaction hindered by very high kinetic barriers, which can be lowered by physical means.34 X-ray crystallography and NMR measurements were used in this system to characterize the encapsulation of a number of electron-rich diaryl amides into different octahedral coordination cages, differing in their metal corners and having electron-deficient walls. In particular, here we focus on the host–guest system shown in vs. blue curves). This has been imputed to the fact that reactive conformations of the amide guest (cis-twisted) are stabilized within the cage host.10 Such evidence is found to be consistent with single-crystal X-ray diffraction analyses, showing that the amide can twist when encapsulated within the cage cavity.10As a case study, here we focus on the neat host–guest system recently reported by the group of Fujita,ab initio MetaD simulations to estimate the relative reactivity of the various conformations that the amide guest can assume . Altogether, (ii) + (iii) provide statistical information on the relative probability of stabilizing reactive guest conformers in the cage cavity. Then, in phase (iv), we conduct AA MetaD to study the host–guest equilibrium and to reconstruct the probability of effectively having the guest encapsulated in the cage cavity, the guest encapsulation residence time, etc. Altogether, (ii) + (iii) + (iv) provide information on the probability of effectively having reactive amide conformers encapsulated in the cage cavity. Finally, in phase (v), all data are then combined in order to obtain a probabilistic estimation of reaction acceleration (a) in the host–guest systems, which essentially compares the probability of having reactive amide conformers within the cage vs. in solution. We obtain results that are found in excellent agreement with the available experimental trends, as well as a flexible computational framework, which can be used, in principle, to study a variety of dynamically reactive host–guest systems and to draw structure–dynamics–reactivity relationships useful for rational design.To obtain a submolecular-resolution insight into the behavior of such a system in dynamic regimes, here we devised a comprehensive multiscale computational strategy. This includes the following key steps: (i) all-atom (AA) MD simulations to characterize the  features of the host–guest system under equilibrium conditions; (ii) AA metadynamics (MetaD) simulations to investigate which guest conformers are more/less favored in the cage cavity or in solution; (iii) 22.1et al. to host the hydrolysis of encapsulated amide guests.10 This is an octahedral coordination cage composed of four self-assembled electron-deficient panel ligands -1,3,5-triazine) and six metal-based corners (cis-endcapped Pd(ii) complexes).35 As a first step, starting from the X-ray crystal structure reported in the literature,10 we built an AA model for cage 1 which was then preliminarily minimized and equilibrated in explicit water solvent and under standard (room) conditions of temperature and pressure via a classical MD simulation. In particular, all the AA models used herein have been parametrized based on the general AMBER force field (GAFF)36 and all simulations have been run with the GROMACS-2020.2 software37 patched with Plumed-2.7 38  of the atomic positions and of two variables (d1 and d2) estimating respectively the height and equatorial width of the octahedral cage revealed that the cage structure is rather rigid under experimentally relevant conditions , with a rather persistent hydrophobic cavity that allows encapsulating one or multiple guest molecules in its interior (the electron-acceptor π planes of the triazine-based ligands of 1 are particularly apt at interacting with electron rich guests).10,35 As the main guest, here we focus on guest 2, N-thiophene-2-carboxamide, an electron-rich diaryl amide .39 Experimentally obtained crystal structures show that cage 1 can encapsulate at the same time up to two 2 molecules in its internal cavity.10 An AA model was developed for guest 2, paying particular attention to the accuracy in the force field parametrization of the central amide bond dihedral, which defines the trans and cis conformers of the amide, how much one is energetically favored with respect to the other, and the related transition barrier. To achieve this, we optimized the original amide ω dihedral force field parameters of the GAFF, in order to obtain a trans-to-cis isomerization free energy profile consistent with the experimental data available for N-metyl-acetamide  in cage 1.10 In particular, X-ray and NMR measurements showed signals corresponding to a cis : trans 1 : 1 2 dimer in the 22 ⊂ 1 complex, and to a cis-twisted conformation in the 2·3 ⊂ 1 complex. For the purpose of our investigation, here we decided to model all possible combinations of conformers, mainly aiming to explore the presence of any correlation between confinement, crowding and the rotation of the amide bond. We also parametrized an AA model for co-guest 3  conditions of temperature and pressure for at least 1 μs for each system, which was found to guarantee sufficient sampling for all variables reported in Aromatic amides, like 2, exist mainly in a -guest 3 . Startintrans conformer showing in general a larger shift compared to 2cis , while this trend is even increased when co-guest 3 is also present within 1 (higher crowding). We will discuss in the next section how this affects which conformer (2transvs.2cis) is more favored within the cage in different complexes.When in the cage, 2 tends to stay shifted from the geometrical center of the cavity to maximize interactions with the walls, with the 2 to 2cis . This iscis conformer has a smaller SASA, and is more compact than 2trans in all cases. These SASA and contact data provide information on different packings of the 2cis and 2trans conformers within the cage under increasing crowding conditions. These are consistent with the free-energy surfaces (FESs) reconstructed from the histograms extracted from the MD shown in cis allows for a tighter packing within the cage (bottom). At the same time, the 2cis conformer is found to be less mobile within 1, as it is demonstrated by a narrower FES dark minimum (minimum energy configuration) compared to that of the 2trans conformer (top).The solvent accessible surface area (SASA) of the guests shows that 2 is less exposed to the solvent when this is encapsulated within 1, and even less when this is co-encapsulated together with 3 . In gene−, if the reaction occurs as catalyzed under basic conditions) on the carbonyl carbon.34,41 While the cavity of cage 1 is markedly apolar and the encapsulation of the guests in the cavity is essentially driven by hydrophobic effects, some accessibility by the solvent is therefore still required for the reaction to take place. We calculated the number of contacts between the solvent molecules and the carbonyl group of the 2 guest in various cases. In general, we can observe that the amide of 2trans is slightly more accessible to the solvent compared to that of 2cis when the guests are free in solution (dotted distributions), and a similar depletion is observed upon mono-guest confinement (dashed). However, the situation is surprisingly switched in the tightly packed 2·3 ⊂ 1 systems (cis is found to be more exposed to the solvent within cage 1 compared to that of 2trans (red vs. blue solid curves). This suggests that, while on the one hand the cis conformer of 2 is more tightly packed within the cage, the equilibrium configuration of 2cis in the 2·3 ⊂ 1 complex may allow, at the same time, for an increased propensity to react.As a first and generally rate-limiting step, the amide hydrolysis reaction requires the nucleophilic attack of water  in or out of the cage cavity.We explore at a deeper level the difference in terms of reactivity between the trans configurations with a torsion angle ω close to π, with a sparse population in cis conformation (ω ∼ 0). The degree of steric conflict of the two residues flanking the amide bond is typically larger in cis amides, resulting, for example, in only ∼5–6% occurrence of cis peptide bonds in protein structures.42,43 To estimate the relative probabilities for finding different conformers under different conditions, we used WT-MetaD simulations, providing free energy profiles of the ω isomerization.Under realistic conditions, amide bonds are usually found in trans-to-cis isomerization of ω angles consists of a local conformation change that is often compensated by local variations of the backbone angles ϕ and ψ of the residues flanking the amide.42 In order to assess how the free energy profile for the isomerization of ω is affected by the torsion of ψ and ϕ, we selected these 3 dihedral angles as our CVs and ran WT-MetaD simulations activating/biasing the trans-to-cis transition of 2  under different conditions  and estimate the free energy barriers  (The nditions . Prelimidetails) .cis conformer in pink), (ii) 2 is encapsulated in cage 1 , (iii) the isomerizing 2 is co-encapsulated in cage 1 with another trans2 guest , and (iv) 2 is co-encapsulated in cage 1 with co-guest 3 . The results show that the stability of the conformers of 2 is significantly affected by confinement. The free energy differences between the cis and trans conformers  of 2 with respect to the trans conformer in all simulated complexes. We move from a cis : trans ratio of ∼10−7 : 1 for one 2 free in solution to ∼10−5 : 1 in the mono-encapsulated system (2 ⊂ 1), ∼10−4 : 1 when two 2 guests are co-encapsulated in the cage (22 ⊂ 1), to ∼10−2 : 1 in the 2·3 ⊂ 1 system. The 2 dimer encapsulation (22 ⊂ 1) system, where one of the two guests is kept fixed in a trans configuration in accordance with experiments,10 falls in between the 2 ⊂ 1 and 2·3 ⊂ 1 cases. The strongly twisted conformation at ω = π/4 remains extremely unlikely in all systems, despite a similar thousand-fold stabilization via confinement. The most crowded case, 2·3 ⊂ 1, shows a ∼10 000× increase in the probability of finding the more reactive cis conformer with respect to the case where 2 is free in solution. This is remarkable, considering that experimentally this case is the one showing the strongest acceleration in the hydrolysis reaction.10 These simulations have allowed us to identify the confinement in the cage cavity as a key factor in determining the relative probabilities of the different conformers of guest 2. As a next step, we investigated the propensity of the accessible guest conformers to react.The results shown in nformers  indicateetc.), here we modeled the hydrolysis reaction when the amide is free in solution vs. when it is encapsulated in the cavity of cage 1. Under moderate pH conditions (6 < pH < 13) and in the absence of other catalysts, the hydrolysis of amides occurs via hydroxide attack, forming a tetrahedral intermediate (TI), followed by a second step consisting of the C–N bond rupture.34 The first step of the reaction is typically considered the rate determining step of the process41 .34 For this reason, for the study of the hydrolysis of amide 2, here we focused our investigation only on the formation of the TI via nucleophilic attack by a solvated hydroxide  simulations44 to study the reactivity of 2 free in solution vs. confined in the cage cavity. Considering the computational cost of these simulations and the complexity of our systems, we employed a semi-empirical density-functional tight-binding (DF-TB) method,45,46 in its self-consistent charge corrected variant SCC-DFTB.47 Recently shown to provide comparable accuracy to DFT with large basis sets in terms of prediction of barrier heights and reaction energies for organic molecules,48–50 SCC-DFTB has indeed been already successfully used to investigate hydrolysis reactions in biological systems.51,52 Despite its approximations, this method guarantees satisfactory accuracy in our case at an affordable computational cost,24 a point particularly relevant given that our method of choice, i.e. infrequent WT-MetaD, requires running multiple replica simulations in parallel. Noteworthily, SCC-DFTB has been shown to be more accurate than other semi-empirical approaches, e.g. AM1 and PM3, for biological applications at a similar computational cost .53,54 We simplified our models by studying a system with an OH− near the amide (ω) of amide 2 to representative values, in order to simulate attack on different conformers. We compared 2 conformers with ω equal to 0 (cis conformer), π (trans conformer), and π/4 (a twisted cis conformer). We employed replica infrequent ab initio WT-MetaD simulations27 to obtain information on the reaction coefficients (rate of hydroxide attack and TI formation) for the various conformers of 2 , we reconstructed the unbiased kinetics for the transition events and could estimate the characteristic transition times, τoff and τon. The kinetic constant for hydroxyde release can be calculated as koff = 1/τoff. The kinetic constant for hydroxyde attack (kon) can be obtained in a similar way from the τon accounting for the OH− concentration in the system. In our simplified setup, the constraint on the OH− is such that the OH− starts in close proximity to the amide  and it is not allowed to leave and jump into solution (and to have one water molecule replace its generated void). This is consistent with having an entire solution of OH− ions and having an ion always at a solvation distance from the amide. In other words, this is like having a solvation box where all water molecules are replaced by OH− ions. Such an approximation essentially removes the diffusion of the ion at the reaction site from the study of the reactivity. This is sensible for our purpose, as such diffusion, i.e., the probability of the ion reaching the amide from bulk solution, is directly dependent on the ion concentration, i.e., on the pH of the solution. In terms of the probability of finding an OH− close to the amide, such a constraint effectively produces an effect consistent with having an OH− concentration equal to that of water molecules in pure water (concentration of 55.6 M). Thus, the approximation introduced by the constraint on the amide-OH− distance, in our case is traduced into fixing an OH− concentration of 55.6 M in the system (or of perfectly basic pH = 14).Previously used, he amide , and conω values). The results in ω dihedral. As expected, the lowest reactivity against hydroxyde attack was observed for the trans conformer, while the cis and the twisted cis amide conformers were found to be more reactive. In particular, the latter is the only conformer with a Kreac = kon/koff >1  useful to compare the reactivity between the different amide conformers. The χω scores in y axis) clearly show how, compared to the twisted cis amide conformer (ω = π/4), the cis amide (ω = 0) is ∼1000 times less reactive, while the reactivity of the trans amide conformer (π) is basically negligible (∼109 times less reactive than the π/4 conformer).This simulation approach allowed us to obtain thermodynamic and dynamic characterization of the reaction as a function of the amide conformation  The most reactive twisted conformers  are, at the same time, highly improbable, even with increased molecular crowding. (iii) The 2cis conformer, moderately reactive (but sensibly more reactive than the 2trans one), is unfavored in solution against 2trans, but it becomes more and more relatively favored as the crowding increases upon confinement, emerging as the prominent reactive species in the cage. Our results then converge on the idea that the reaction acceleration by confinement is strongly governed by other dynamic processes of the system, particularly the stabilizing effect of confinement on the more reactive conformers.Altogether, the results shown in cis–2trans dimer in the 22 ⊂ 1 case and a 2cis-twisted conformation in the 2·3 ⊂ 1 complex. This seems to indicate that in these complexes a 2cis conformer is more favored than 2trans. While this may seem to contradict the simulation results discussed above, it is worth noting that all the results collected up to this phase are valid only under the assumption that the encapsulated guests always remain within the cage cavity. Nonetheless, these are host–guest systems, in which the probability of finding the guests within the cage obeys a well-defined supramolecular equilibrium. Estimating the effective probability of finding the guests within the cage requires also studying the dynamics of guest encapsulation/exchange in-and-out of the cavity. As it will be demonstrated in the next section, accounting also for the intrinsic supramolecular dynamics of these host–guest systems provides results that are globally in very good agreement with all available experimental pieces of evidence.The available experimental X-ray structures for these complexes show a 22.3vs. when it is encapsulated into the cavity of cage 1. However, to obtain a complete picture, one last point is missing, namely, the effective probability of having such reactive guest conformers in the cage cavity. In fact, such host–guest systems obey a supramolecular equilibrium, where the affinity between the host and the guest dictates the propensity for the guest to stay encapsulated within the cage cavity or to be expelled in solution. Based on our strategy, characterization of the thermodynamics and kinetics of the guest encapsulation/expulsion in-and-out of the cage cavity is thus an additional necessary step. Under realistic conditions, the encapsulation/expulsion of guests as, e.g., 2 or 3 in/out of the cavity of cage 1 may require crossing considerably high free energy barriers,16,32 which makes them rare events in the timescales accessible via classical atomistic MD simulations. As recently carried out for other host–guest16 and dynamic supramolecular systems,33,55 we thus reconstructed the thermodynamics and kinetics for the processes of encapsulation/expulsion of amide 2 in/out of the cavity of cage 1 by means of a well-suited WT-MetaD56 simulation protocol  WT-MetaD simulations has been reported previously for other systems,57,58 including dynamic supramolecular ones,16,33,55 given the complexity of the systems studied herein, this stands out as a very efficient, robust and powerful approach. The extracted data are collected and shown in Up to this point, we have probabilistically compared the reactivity of the amide guest when it is in solution trans), we could observe that in general the encapsulation of 2cis is more favored than that of the 2trans isomer in all studied cases . On the other hand, the dynamics (and stability) of the 2cis-complexes is more impacted by the presence of co-guests in the cage cavity, which are found to stabilize the 2 encapsulation in the cage cavity by ∼2–4 orders of magnitude in the presence of 3 (lower koff: more improbable/slower 2 expulsion out of the cavity). As shown in koff for the expulsion of the 2cis guest out of the cage cavity drops from ∼7.1 × 101 s−1, when only one 2cis is present in the cavity of 1, to ∼1 s−1 or ∼5.9 × 10−2 s−1, when 2cis is co-encapsulated in the cavity of 1 with a 2trans or with a 3 co-guest respectively. The kon, on the other hand, is found to be globally similar in all simulated cases  for all the host–guest systems  and the effective host cavity, thought of as that accessible for the guest considering the cage and the presence of eventual encapsulated co-guests. From the calculated Δ systems . It is wagnitude . This meKenc values indicate that in all the cases simulated herein the amide guest can be, in good approximation, assumed as always encapsulated in the cage cavity. In fact, from the thermodynamic data we can extrapolate the partition probability Pin = Kenc/(1 + Kenc). Pin indicates the relative probability of having 2 in vs. out of the cage cavity. In particular, Pin tends to be 1 when the K is high and the guest has a high probability of being encapsulated in the system, as it is the case in general for all host–guest complexes explored herein .Nonetheless, the high 2.4ω dihedral in solution and in confinement; (iii) the reaction itself. For each of them, we have devised a probability metric and observed how it varies across the different investigated cases. Our last step is thus to put these processes (and probabilities) together, in order to capture the full dynamic complexity of the system.We have thus far investigated, separately, the three main processes that concur to define the global dynamic picture of amide hydrolysis under confinement : (i) enca, as the ratio between the observed reactivity with or without the presence of a cage in the system – i.e., when the reactant, guest 2, is encapsulated within the cage cavity (K( ⊂ 1)) or when it is free in solution (K(sol)):To this end, we can define a reaction acceleration index, a will thus depend on the likelihood that the hydrolysis of 2 occurs in vs. out of the cage. From our simulations we have seen that the conformational free energy landscape of the amide guest may change upon encapsulation (changing the relative free energy difference between cis and trans conformers). As a consequence, the probability of crossing the rotational barrier around the amide bond also changes. In particular, we could observe that the more reactive 2cis conformer is more and more stabilized as the crowding increases in the cavity of cage 1 . In general, the reaction acceleration f cage 1 . The simion time . Altogeti.e., the relative population ratio between having 2 in the cage vs. in solution at the equilibrium), and the solvent molecule accessibility to the amide  upon encapsulation. Noteworthily, all these parameters can be extracted from our simulations.The reactivity in the system depends on the propensity of the visited 2 conformers to react, their relative population in the different complexes, their probability of encapsulation (K(⊂ 1), as the sum of the reaction constants (Kω(⊂ 1)) for all amide conformers (ω) visited by the guest reactant 2 in the cage cavity:χω is the hydrolysis reaction constant associated with the possible amide conformers ω (see Pconfω(⊂ 1) is the relative statistical weight for all different conformers ω in the cage cavity (Pinω(⊂ 1) is the probability of effectively having each specific conformer ω in the reactive environment – in this case, inside the cage cavity (see ω(⊂ 1) is the average number of contacts between solvent molecules (water or OH−: key reactants for amide hydrolysis) and the amide's carbonyl , can be defined as:Pconfω(sol) and NWω(sol) refer respectively to the relative probabilities for 2, when alone in the solvent, to assume the different conformers ω, and the corresponding number of amide carbonyl-solvent molecules contacts. In this case, in the absence of the cage in the system, guest 2 is by definition always out of the cage, and Pinω(sol) = 1. Thus, Accordingly, the global reaction constant in the absence of the cage in the system , Transforming the formula for acceleration into:G values in Pin(⊂ 1) term tends to be ∼1 (vide supra):Moreover, it is worth noting that, given the high ΔReducing the formula for the acceleration to:Pin(⊂ 1) ∼ 1), in this specific case the reaction acceleration in the system is found to be little dependent on the guest encapsulation/expulsion equilibrium. On the other hand, the reactivity turns out to be rather controlled by the fact that the guest is more favored to assume reactive conformations inside the cage cavity (compared to the case when this is free in solution). This is in full agreement with the available experimental evidence on these systems.10This means that, given the high propensity for guest encapsulation  is very low. These are distorted, very unstable conformers, with a survival lifetime which tends to be 0, and for which the product χωPconfω ∼ 0. The unique conformers with survival life and Pconfω ≠ 0 are 2cis (ω = 0) and 2trans (ω = π). The latter, however, is substantially non-reactive (χπ ∼ 0), so that also in this case χπPconfπ ∼ 0. Based on these observations, in our case the reactivity of the system seems thus to be largely related to (i) how much over-stabilized the reactive 2cis conformer is and (ii) how accessible the amide is to the co-reactant solvent molecules in such conformation in the cage cavity vs. in solution.Finally, it is worth noting that while the summations in reactive : χπ ∼ 0,a for the various host–guest complexes reported in 2 ⊂ 1 system. A dramatic a ∼ 150 is obtained for the 2·3 ⊂ 1 complex. While such estimated a values may differ quantitatively from those obtained from the experiments , the trends can still be safely compared. e.g., 22 ⊂ 1 and 2·3 ⊂ 1) provides an additional case useful for comparison. In particular, the limited computed acceleration seen in this case supports the evidence that molecular crowding within the cage cavity is a key player in the reactivity in the host–guest system.By combining these data, we estimate the reaction acceleration a parameters against some of their key constitutive terms.In order to obtain an insight into the key molecular determinants controlling the reaction acceleration in these host–guest systems, as shown in cis and trans conformers among the different systems is the main factor determining the final reaction acceleration using a and the relative probability of finding the 2cis conformer with respect to the 2trans conformer in solution and in the different encapsulation complexes  for 2cis in all systems . This affinity, as evidenced by a qualitative investigation of how the different components interact with each other, both in our simulations and in the available experimental X-ray structures,10 is mainly driven by π–π stacking between the four triazine ligands of cage 1 and the aromatic rings of amide 2, particularly favorable in the cis conformation and by residual solvophobic interactions between such ring groups. A similar trend can also be observed when looking at the weighted number of contacts between the host and the guest (i.e. a proxy for the host–guest interaction energy). If we consider the interaction between 2 and cage 1 to be consistent among all the investigated systems, we can trace the trend back to the interaction between the guest and co-guest (or the absence thereof in the 2 ⊂ 1 case), with 3 showing a greater stabilizing effect for 2cis within the cage cavity compared to another 2 co-guest. Noteworthily, as revealed by the obtained trends shown in a. The remarkable stabilizing effect of co-guest 3 is driven by a balance between having an aromatic structure able to participate in π–π stacking with guest 2, its considerable solvophobicity that makes its encapsulation within the cage cavity very stable, and a size (volume = 172 Å3)39 that allows the amide to reside comfortably in the cavity in a cis (reactive) conformation. Another 2 co-guest, while still being able to engage in favorable interactions (3)39 which constrains the first guest in less favorable configurations and produces a less stable complex.Noteworthily, a quasi-exponential trend is observed between the computed reaction acceleration  systems . In thescis conformer when it is encapsulated in the cavity of the various complexes vs. when it is alone in solution.16,55,59 While a correlation with the computed reaction acceleration is observed (see 2 ⊂ 1 and 2·3 ⊂ 1). However, the differences in acceleration between the various systems do not correlate in a neat manner with the ΔSASA calculated for the various cases. This reveals that (i) non-specific hydrophobic effects alone are not sufficient to grasp the complexity of these reactive systems and suggests that (ii) like in most receptor-ligand complexes in nature, specific molecular interactions are probably relevant in controlling host–guest affinity.To obtain information on how much of the host–guest interaction is due to solvophobic effects, we calculated the reduction in the solvent accessible surface area (SASA) of the 2rved see , the tre310 By combining a multi scale modeling scheme with metadynamics simulations, we couple the study of the intrinsic dynamics of the host–guest system with that of the amide hydrolysis reaction. The approach allows us (i) to characterize the barriers to the hydrolysis reaction as a function of the conformation assumed by the amide, (ii) to study the conformations that the amide guest can assume and estimate their relative probabilities in vs. out of the cage cavity, and (iii) to characterize the dynamic processes of guest encapsulation/expulsion in/out of the cage. Altogether, we demonstrate how this allows us to reconstruct the reactivity in such dynamic host–guest systems by calculating the relative reactivity of the various molecular conformers/arrangements that the amide guest can assume and to what extent the reactive conformations are more/less favored in vs. out of the cage, by re-weighting the results considering the probability of effectively having the guest encapsulated inside the cage cavity. This allows us to estimate a reaction acceleration score, which estimates to what extent the reaction is accelerated/hindered in the host–guest system and allows comparing between system variants.Understanding reactivity in dynamic regimes and in systems in which reactants and products are in continuous exchange is a non-trivial task. Here we report a computational approach that allows us to reconstruct the reactivity in dynamic host–guest systems and to study in atomistic detail the key (molecular and dynamic) factors that control it. As a case study, we focus on the hydrolysis of amide guests encapsulated in the cavity of a coordination cage, for which experimental evidence for reaction acceleration has been recently reported.2 ⊂ 1 and 2·3 ⊂ 1). We characterized all of them, and we estimated their hydrolysis acceleration factor. The results show unambiguously that the encapsulation of the amide in the cage cavity tends to stabilize the reactive conformers of the amide guest. This is key in such specific systems, considering the high encapsulation constants obtained for all studied cases . We clearly observe how, when a co-guest is also co-encapsulated with the amide guest in the cage, the crowding in the cavity augments and the reactive cis conformer of the amide guest is more and more stabilized. The acceleration scores estimated from our computations are found to have remarkable trends with the reaction acceleration observed experimentally. Overall, our computational results are found in optimal agreement with the experimental results by Takezawa et al.,10 while it is worth noting that in our approach these emerge bottom-up, from a comprehensive study of the molecular and supramolecular dynamics of the host–guest system and of its key molecular equilibria. This provides a general character to this approach, as confinement-induced reaction acceleration (or deceleration) in such supramolecular  host–guest systems can only be explained by taking into account all the dynamic processes that occur within them. The comprehensive picture of hydrolysis and of how this may be modulated under confinement that we obtain here provides a general high-resolution framework for building structure–property relationships. This approach constitutes a useful general-purpose platform, useful to explore strategies towards the rational design of host–guest systems with a molecular-level control of chemical reactivity. Based on the results that we show herein, this approach offers a useful platform to explore, e.g., the effect of tuning the properties/features of the guest  or of the cage cavity , as recently shown in other host–guest reactive systems.16 Its versatility and generality may also be useful to explore ways to gain control over system reactivity by tuning host–guest dynamics by using, e.g., guest mixing and guest–guest encapsulation competitions,10 which could allow tuning the residence time and fractions of encapsulated guests within the cage cavity and, consequently, the reaction in the system. The approach described herein appears to be well suited for the case studied, and it is general enough that it can be useful to also study other types of dynamically reactive systems. While very effective in our case, the reconstruction of the dynamic reactivity by correcting this with weights that account for effective (probabilistic) reactant availability and their propensity to react has some intrinsic limitations. For example, in our case it does not explicitly consider the possible active roles of intermediate states that can promote/hinder the reactivity. Also, in this specific case in the calculations of reactivity we do not explicitly account for the effect of the cage on the hydrolysis reaction. In our case, the effect of the cage is accounted for implicitly: this enters into play as the encapsulation of the guest within the cavity favors reactive amide conformers or it allows for fewer solvent molecules  to reach the proximity of the amide compared to that when the amide is in solution. Clearly, this is a valid assumption as long as the structure of the cage does not have a catalytic role in the reaction (as it is assumed to be the case in such systems).10 Nonetheless, it is worth noting that the versatility of our approach makes it well suited for methodological extensions. For instance, when needed, it is in principle possible to include explicitly the cage in the study via, e.g., a hybrid QM/MM approach where the reactive sites are modeled explicitly at the quantum level while the surroundings are treated with classical atomistic force fields. Another methodological limit is related to the complexity of the reaction and of the system as a whole. One practical limit of MetaD simulations lies, in fact, in the possibility to choose a reduced number of appropriate collective variables that can effectively and correctly model all the phases from (ii) to (iv) of this study. While in this case MetaD simulations are effective, it is worth noting that the approach must not necessarily be built on MetaD simulations. Any enhanced sampling method that allows the probabilistic factors of i.e., combining the study of reactivity with that of the intrinsic dynamics of the system) but not on a specific method provides flexibility and robustness to the approach. Furthermore, the results we report herein demonstrate how, despite the relatively simple concepts on which our approach is built, this is effective, and it offers concrete opportunities to reconstruct reactivity in dynamical host–guest supramolecular systems. It is also worth underlining how, in achieving such goals, this approach offers a first attempt to link the typical computational chemistry study of chemical reactions to that of a complex dynamic reaction process, more typical of chemical engineering.In the specific case study used here, we obtain clear evidence that the reaction acceleration is controlled by the crowding effects accompanying the guest encapsulation. We compared four cases, where the amide guest 2 is alone in solution, when it is encapsulated in the cage (2 ⊂ 1), or when it is co-encapsulated together with other co-guests in the cage cavity (2In general, all these considerations underline how gaining control over the dynamics of these systems is key to controlling reactivity within them and indicate how approaches such as those described herein can offer relevant support towards the development of new types of reactive supramolecular systems.https://doi.org/10.5281/zenodo.7064016.Complete computational materials and data pertaining to the study conducted herein are available at: M. D. P., L. P. and M. C. created the molecular models and performed the simulations. All authors contributed to the analysis of the results and to the writing of the manuscript. G. M. P. conceived the research and supervised the work.There are no conflicts to declare.SC-013-D2SC02000A-s001


The damming materials will be deteriorated during the long-term service of concrete arch dam by weathering, seepage dissolution, freeze–thaw and other factors which will put dams at potential risk. However, there are few researches on the reliability assessment of arch dams considering the effect of material deterioration. In this paper, a prediction model of concrete mechanical properties under freeze–thaw cycles is established using regression analysis. Considering various uncertain factors, a time-varying structural reliability analysis model is established using finite element equivalent stress method and an optimized response surface method with improved sampling strategy and convergence criterion. A case study is later carried out. The results show that, at the early stage of freeze–thaw, the structural failure probability will increase steadily in a small range. However, as the tensile strength of concrete continues to decrease under long-time freeze–thaw action, the reliability of the arch dam structure will suddenly fall off a cliff. The proposed deterioration model and reliability analysis probability model are reasonable and practical and could be serve as a technical support for similar engineering projects. Most of them were built between the 1950s and 1970s which are close to the standard service life. With the increase of service time, the damming materials will be deteriorated by weathering, seepage dissolution, freeze–thaw and other factors which will put dams at potential risk.Arch dam is a very economical high order statically indeterminate shell structure which could make full use of arch effect to maximize load bearing. China has a large number of arch dams, accounting for 40% of the world4. Trubic5 provided an analysis of the air voids parameters in fresh concrete and the proving of the concrete resistance to the freezing and thawing effects already in fresh concrete. Hong6 studied the degradation law of mechanical properties of concrete subjected to freeze–thaw cycles. When damage degree is less than 40%, the compressive strength of freeze–thaw damaged concrete is not lower than 70% of its original strength. Paul7 presented an assessment of the freeze–thaw resistance of preformed foam cellular concrete. The results showed that compressive strength, depth of initial penetration, absorption and absorption rate are the important variables in producing cellular concrete that is resistant to cycles of freezing and thawing. Density and permeability were shown not to be significant variables. However, there are few researches on the reliability of arch dams considering the effect of material deterioration. David et al. conducted an investigation of a 40-years old concrete arch dam and studied the concrete deterioration characteristics through coring test, sonic test and petrographic examination8. Many other concrete structures in long-term service have also been studied11. The concrete deterioration characteristics are focused in the above researches. Relevant material deterioration models and structural safety and reliability evaluation models of arch dams are not studied and established. Wang et al. introduced a concrete degradation model with aging because of chemo-mechanical damage and studied the seismic response of an arch dam with aging effects12. Pan et al. proposed a unified approach for long-term behavior and seismic response analysis of AAR  -affected concrete dams by combining AAR kinetics, effects of creep and plastic-damage model in the finite element method13. Joshi et al. studied the effect of alkali silica reaction on the deflection of an arch dam14. The currently proposed degradation models did not take into account many uncertain factors, such as external load uncertainty, structural mechanical uncertainty and so on. The structural reliability research is relatively scarce. Yilmaz et al. proposed a stochastic deterioration modelling approach to evaluate the performance deterioration of corroded concrete structures caused by reinforcement corrosion15. However, it does not reflect the effect of concrete deterioration on arch dam. In general, the reliability of the arch dam structure under freeze–thaw effect, considering various uncertainty factors, still needs further study.At present, the deterioration of concrete materials under freeze–thaw conditions has been studied intensivelyIn this paper, a freezing–thawing deterioration material model of concrete is proposed through regression analysis. The time-varying strength and reliability of an arch dam under freezing–thawing action are analyzed by using a finite element equivalent stress method and response surface method. The research results provide a method and basis for the safety evaluation of arch dam structure considering material deterioration. Other material deterioration progress, such as weathering, seepage dissolution, et al., could also be similarly analyzed by the proposed method.ED gradually decreases with the increase of freeze–thaw times, and the change rule is basically the same. In order to study the time-varying reliability process of arch dam under static condition, the dynamic elastic modulus ED needs to be transformed into static elastic modulus ES. According to “Standard for seismic design of hydraulic structures” (GB 51247-2018)16, under seismic conditions, the dynamic elastic modulus of concrete increases by 50% on the basis of static elastic modulus. It could be assumed that the dynamic elastic modulus of concrete under freeze–thaw action remains the same as 1.5 times of static elastic modulus, namely:freeze–thaw action is generally considered to be the internal microstructure changing process of cement mortar due to repeated freezing and melting of internal moisture. The freeze–thaw action mainly affects the strength and elastic modulus of concrete. Under freeze–thaw effect, the dynamic elastic modulus of concrete 23, the test data of plain concrete are extracted. And the static elastic modulus of concrete under freeze–thaw action could be reverse analyzed and obatined which is shown in Fig. ES,0, ED,0 are the initial static and dynamic elastic modulus of concrete without freeze–thaw action, respectively; ES,n, ED,n are the static and dynamic elastic modulus of concrete after n times of freeze–thaw action.Based on the exponential fitting model regression analysis of the dynamic elastic modulus test data in relevant literatures22 also proposed a deterioration model which is shown in Eq.  and and26 anIt could be noted that the deterioration curve proposed in this paper lies in the middle of other models. Zhao's test data did not include cases where the relative residual dynamic elastic modulus is below 0.9 which makes the regressed deterioration curve much steeper when the relative residual dynamic elastic modulus decreases further. Hong thinks when damage degree is less than 40%, the compressive strength of freeze–thaw damaged concrete is not lower than 70% of its original strength. Therefore, the conservative estimate of the residual compressive strength of freeze–thaw damaged concrete is equivalent to the residual dynamic elastic modulus. As more experimental data are used in this paper, a relative compromised compressive strength deterioration characteristic curve is obtained.27, the tensile strength of concrete under freeze–thaw action could be determined as follows:fc,n, ft,n are the compressive strength and tensile strength of concrete after n times of freeze–thaw action, respectively; fck, ftk are the standard compressive strength and tensile strength of concrete, respectively.Assuming that the proportional relationship between compressive strength and tensile strength of concrete under freeze–thaw action is the same as the standard value of concrete strengthEquations –4) defi defi4) d28. In order to reduce stress concentration, the finite element internal force analysis results will be treated by equivalent stress method in the post-processing stage29.An input–output model is needed to evaluate the reliability of arch dam structure. In the input–output model defined by finite element method, stress concentration could occur at the corner edge of the dam resulting in the decrease of the reliability indexσij′ in the global coordinate system  is first calculated by finite element method. Then, a local coordinate system  with radial and tangential axes of the arch is established at node k of the horizontal arch circle, where the X-axis is parallel to the tangential direction of the arch center line, the Y-axis is parallel to the radius direction, and the Z-axis is the vertical direction. The coordinate’s origin is on the arch center line. The stress σij′ in the global coordinate system could be converted into the stress in the local coordinate system σij = T by means of the coordinate transformation matrix:T] is the coordinate transformation matrix.li, mi, ni) is the direction cosine of coordinates x, y and z.The stress at each node of arch dam y could be 1 + y/r. r is the curvature radius of the arch center line. The internal force of the beam could be obtained by integrating the stress and its moment along the thickness direction:T is the thickness of arch circle, y0 is the centroid coordinates of the beam section.On the horizontal section of the beam, the unit width on the center line of the arch is taken. The width at point The radial section width of arch ring per unit height is defined as 1. The internal force of an arch could be obtained by integrating the arch stress and its moment along the thickness direction:σz, σx, τzx are distributed linearly from upstream to downstream, the stress of dam body could be calculated by the method of material mechanics according to the internal forces of arch and beam. The remaining stress components need to be obtained by taking the microelements from the upstream and downstream edges and solving the equilibrium equations of the microelements.Since the shear stresses are paired, the vertical shear and torque of the arch could be determined without calculation. Assuming that The vertical normal stress of the cantilever beam on the horizontal plane could be calculated as follows:The horizontal normal stress of the arch in the radial plane could be calculated as follows:The tangential horizontal shear force of the horizontal cantilever beam could be calculated as follows:φd is the angle between the dam surface and the vertical direction in the radial vertical plane; ηd is the angle between the dam surface and the tangent line of the arch center line in the horizontal plane.The other three stress components could be calculated as follows:The three stresses parallel to the dam surface could be calculated as follows:t is the arch ring thickness; Ab = Aa = t; Ru = r + t/2; Rd = r − t/2; Pd is the downstream dam surface pressure.The equivalent principal stress of the downstream dam surface could be calculated as follows:Pu is the upstream dam surface pressure.The upstream dam surface stress calculation is the same as the downstream. The main calculation formulas are as follows:30. In response surface method, a fitted response surface is adopted to replace the complex functional equation, which is essentially an improvement of Monte Carlo method and could greatly improve the efficiency of reliability evaluation31. However, the calculation accuracy of response surface method depends on the fitting degree of the selected sample points near the limit state surface32. It is necessary to obtain sufficient samples near the limit state surface. But too much sampling will affect the computational efficiency, and has little effect on the accuracy improvement. Therefore, an improved sampling strategy and convergence criterion are proposed to achieve a balance between computational accuracy and efficiency.When the probability distribution of input variables is determined according to the test data or specification parameters, the response surface method could be adopted to analyze the structural reliabilityX =  as the random input variable. The response surface function without cross terms could be defined as follows:The quadratic polynomial is usually selected as the response surface function form. Set n + 1 samples will be needed to fit the response surface function containing n random variables. However, 2n + 1 samples may not be close enough to the limit state surface, so the fitting effect could not be determined. The optimal response surface could be determined by stepwise trial calculation, which is expressed as follows:As the Latin hypercube sampling algorithm is selected for sampling, at least 2ith step trial is defined to fit k response surfaces. The number of response sampling points used to fit each response surface is respectively in + n + 1, (i + 1)n + n + 1, (i + 2)n + n + 1, …, (i + k-1)n + n + 1. The target probability of each response face is solved separately as Pf,i, Pf,i+1, Pf,i+2, …, Pf,i+k-1. If the following formula is true:Pf), min(Pf), E(Pf) are respectively the maximum value, minimum value and mean value of the target probability in the set of stepped response surfaces, ε is the allowable error.The ith step trial calculation has achieved a suitable fitting effect near the limit state surface. The response surface whose target probability is closest to E(Pf) will be taken as the representative fitting response surface. Otherwise, i + 1 step trial calculation will be conducted, and n sample points will be newly added on the basis of the original sample points for trial test until Formula . Rigid constraints are applied to the bottom surface of the foundation, and linkage constraints are applied on the lateral surface along the direction of water flow. The overall model mesh is shown in Fig. 33, the working conditions of the dam strength reliability analysis could be determined and shown in Table According to the actual operation of the arch dam and “Design specification for concrete arch dams SL282-2018)”82-2018”334. Recent studies show that the arch sealing temperature and residual temperature are the main sources of temperature load uncertainty35. In this paper, the arch sealing temperatures are treated as random variables to characterize the uncertainty of temperature load. The arch sealing temperatures of eight layers of arch rings from arch bottom to arch top were separately considered as independent random variables. The coefficient of variation is set as 0.05. Temperature load is calculated according to “Design specification for concrete arch dams (SL282-2018)”33 considering average body temperature Tm and equivalent linearly-distributed temperature difference Td. The temperature load when the arch sealing temperature is 16.0 °C is shown in Table −6.Temperature load is one of the main loads of arch damAccording to laboratory test, the elastic modulus of dam body varies from 16.8 to 22.6GPa. The standard deviation is 2.28GPa. Thus, the median value of 19GPa is adopted and variation coefficient is taken as 2.28/19 = 0.12. Other material property parameters are generally taken in the same way. The Initial tensile strength of concrete defined according to Eq. .3. The initial tensile strength is 2.74 MPa.The dam body is made of C15 concrete. According to the laboratory test, the recommended initial elastic modulus is 19GPa. The Poisson's ratio is 0.20. The density is 2350 kg/m3.The foundation is argillaceous limestone. According to the laboratory test, the recommended elastic modulus is 6 GPa. The Poisson's ratio is 0.25. The density is 2600 kg/mThe arch sealing temperature is usually taken as the temperature that is a little higher than the average annual temperature at the dam site, which is 15.9 °C. Thus, the arch sealing temperature is taken as 16.0 °C.Random variables and their statistic parameters are shown in Table Because the compressive strength of arch dam is generally rich, and arch dams are mostly damaged by tension, the failure mode is selected as tensile failure in the structural reliability analysis.The initial mean value of each parameter is adopted to simulate the initial dam stress state. As shown in Fig. The nephogram of principal stress shows that the finite element simulation produces local stress concentration. Thus, equivalent stress treatment is adopted and the results are shown in Table k of stepped response surfaces expressed above, the larger the aggregation number k is, the harsher the convergence condition will be. Generally, the calculation results will be more secure, meanwhile the corresponding amount of calculation will be increased. In fact, a good convergence could be achieved when k ≥ 3. Considering the calculation accuracy and efficiency comprehensively, the aggregation number of stepped response surfaces k is set to be 4, and the allowable error ε is set to be 0.5.The response surface method proposed in this paper is adopted to evaluate the structural reliability. According to the definition of aggregation number In order to directly reflect the cumulative probability distribution of structural stress, the maximum principal tensile stress is taken as the objective function. The initial concrete tensile strength is taken as a constant to solve the structural failure probability at initial state (without freeze–thaw effect). The stepwise response surface convergence process is shown in Table P(σ)i is the strength failure probability of each working condition, P(C)i is the annual occurrence probability of each working condition which could be replaced by statistical frequency.Considering the annual occurrence of each working condition as mutually independent events, the annual structural failure probability could be estimated by the following formula:It is assumed that a freeze–thaw cycle is completed when the daily minimum temperature is below − 1 °C for over five consecutive days followed by five consecutive days above -1 °C. According to the statistics of the reservoir operating temperature shown in Fig. Compared with the dead water level working condition, the maximum principal tensile stress cumulative probability curve in the condition with higher water level ((1), (2), (5)) has lower slope and longer span, which indicates that the possible stress fluctuation range of arch dam body is larger and the arch structure is in a relatively unstable state under high water pressure load.The calculation results of normal water level & temperature rise condition (working condition (2)) are close to that of dead water level & temperature drop condition. However, since the former happens almost every year after impoundment, it is the actual control condition of the failure probability of arch dam. The other working conditions occur less frequently in one year and have little influence on the annual failure probability. Therefore, the failure probability of arch dam is approximately equal to the failure probability of normal water level & temperature rise condition (working condition (2)).With the increase of freeze-thawing times, the failure probability of arch dam structure generally increases and the reliability decreases. At early stage of freeze–thaw action (approximately within 50 times of freeze–thaw cycles), though the tensile strength of concrete is decreasing, the failure probability will increase steadily in a small range, and the reliability of the arch dam structure will not decrease significantly. Because the stress level of dam body is also relieved when the elastic modulus is reduced. After that, as the tensile strength of concrete continue to decrease, the structural failure probability will suddenly increase. After 80 times of freeze-thawing cycles, the annual failure probability of the arch dam structure will reach about 1% and the arch dam structure will be unreliable.Although the residual strength decreases slowly due to freeze–thaw effect, it has a great influence on the reliability of the structural strength. Because the residual tensile strength slowly approaches the maximum tensile stress of the structure. As the dam has freeze–thaw action 29 times during its 50 years of service, it is estimated that the dam will reach a 1% probability of failure in about 88 years. The problem of freeze–thaw damage should be paid attention to and corresponding pre-strengthening measures should be taken.According to the calculation results, it could be known that:The relationships between freeze–thaw cycles, concrete dynamic elastic modulus, static elastic modulus and compressive and tensile strength are analyzed. A freeze–thaw deterioration model of concrete is established through regression analysis to simulate the deterioration process of mechanical properties which could provide a simple and practical analysis method for engineering application.In view of the contradiction between the computation effect and computation amount of response surface method, a convergence criterion and strategy for sampling is proposed so as to determine the relatively optimal response surface. The case study results show that the response surface fitted after the fifth step trial is basically stable, and the fluctuation is not obvious. To some extent, the proposed convergence criterion could guarantee the effectiveness of response surface calculation, avoid excessive sampling quantity, and reduce the amount of calculation. It is feasible to use the convergence criterion proposed in this paper to determine the relative optimal response surface.The simulation results of time varying structural reliability of arch dam show that, at early stage of freeze–thaw, though the tensile strength of concrete is decreasing, the reliability of the arch dam structure will not decrease significantly, because the stress level of dam body is also relieved when the elastic modulus is reduced. However, as the tensile strength continues to decrease under freeze–thaw action, the failure probability will increase rapidly. After 50 times of freeze-thawing cycles, the structural reliability will suddenly fall off a cliff. As the dam has freeze–thaw action 29 times during its 50 years of service, it is estimated that the dam will reach a 1% probability of failure in about 88 years. The problem of freeze–thaw damage should be paid attention to and corresponding pre-strengthening measures should be taken. Learning from the stress distribution, the stress level of the arch end of both sides is high, and the weak parts such as the arch end of both sides are suggested being monitored and maintained in daily operation.The material of concrete arch dam is easy to deteriorate under freeze–thaw action which affects its structural reliability and service life. In this paper, a freeze–thaw deterioration model of mechanical properties of concrete materials is established based on measured data. A response surface method with improved convergence criterion is proposed. The time-varying reliability process of arch dam structure under freeze–thaw action is studied. The main conclusions could be made as follows:


The patellofemoral joint involves a combination of bony structures and soft tissues to maintain stability. Patella instability is a disabling condition, and the cause is multifactorial. The main risk factors include patella alta, trochlea dysplasia, excessive tibial tuberosity to trochlea grove (TT–TG) distance, and excessive lateral patella tilt. In this case report, we highlight the thinking process of diagnosis and method for selecting the optimal treatment in accordance with the guidelines by Dejour et al. when we are presented with a patient with patella instability. A 20-year-old Asian woman without underlying medical conditions, presented with recurrent (>3 episodes) right patella dislocation for 7 years. Investigations revealed a type D trochlea dysplasia, increased TT–TG distance, and excessive lateral tilt angle. She underwent trochlea sulcus deepening, sulcus lateralization and lateral facet elevation, lateral retinacular release, and medial quadriceps tendon–femoral ligament (MQTFL) reconstruction. Due to the complexity behind the anatomy and biomechanics of patella instability, an easy-to-follow treatment algorithm is essential for the treating surgeon to provide effective and efficient treatment. MQTFL reconstruction is recommended for recurrent patella dislocation due to satisfactory clinical and patient reported outcomes and a reduced risk of iatrogenic patella fracture. Controversies for surgical indication in lateral retinacular release, and whether the sulcus angle is an accurate parameter for diagnosis of trochlea dysplasia, remain, and further research is required. Patella instability is a disabling condition commonly affecting the female population in their second and third decades of life ,2. The pThe dislocation rate of the patella for the first time is 17–20% and as high as 44–70% with consecutive dislocations [Throughout the years, the literature has continuously been presented with innovative advances and controversies for the diagnosis and management of patella instability. In 2021, Dejour et al. published a treatment guideline that highlights and proposes treatment recommendations for each of the four main risk factors . Due to In this case report, we highlight the thinking process of diagnosis and method for selecting the optimal treatment in accordance with the guidelines when we are presented with a patient with patella instability.A 20-year-old Asian woman without underlying medical conditions, presented with recurrent (>3 episodes) right patella dislocation for 7 years. Physical examination revealed patellofemoral arthrosis, positive apprehensive test, positive J-sign  and highComputed tomography (CT) and magnetic resonance imaging (MRI) confirmed suspicion of trochlea dysplasia. Patellotrochlear index (PTI)  on MRI wShe underwent trochlea sulcus deepening, sulcus lateralization and lateral facet elevation, lateral retinacular release and medial quadriceps tendon–femoral ligament (MQTFL) reconstruction. Postoperative recovery was smooth and without complication. Dejour et al. published an updated treatment guideline  for pateWith consideration to the above guidelines, surgical treatment is recommended for our patient because she had more than three dislocations.Characteristic signs of trochlea dysplasia are best seen on a lateral standing plain radiograph . The croThe Dejour Classification classifies trochlea dysplasia into four types based on the three characteristic signs ,2 Figur. Type A Our patient had type D trochlea dysplasia as evidenced on imaging with the presence of a crossing and double contour sign, supratrochlear spur and a vertical ‘cliff’ between a hypoplastic medial facet and convex lateral facet. Sulcus deepening trochleoplasty is recommended for type D trochlea dysplasia . In our Patella alta is a patella that rides abnormally high in relation to the trochlea groove, with reduced patellofemoral articular contact, and increases the extent of knee flexion before the patella can engage the trochlea groove ,8,12,13.Patella height can be determined by various methods, though, to date, there is no consensus on the most reliable measurement method . The CatThe cartilage of the patellofemoral joint is evident on a sagittal MRI, which allows patella height to be assessed using the patella–trochlea cartilage overlap ,15. ThisUsing the above methods and parameters for calculation, our patient does not have patella alta (Caton–Deschamps method (PTG:PG) = 33.62:32.49 = 1.03; Insall–Salvati method (LT:LP) = 42:38.95 = 1.07; PTI value was 0.36), and therefore tibia tubercle distalisation was not performed.TT–TG distance is used to determine the degree of lateralisation of the tibia tubercle in relation to the deepest part of the trochlea groove, that is, the alignment of the extensor mechanism ,11. It iThere is no consensus on which image modality (CT versus MRI) is considered the gold standard to measure this parameter. Whilst excellent interobserver reliability is demonstrated on both CT and MRI , the values derived from these two tests may not be interchangeable  . PatientFor our study, TT–TG distance was measured on MRI imaging. Whilst a TT–TG distance of 22.07 mm in our patient suggests that tibial tubercle medialization should be performed, it was not enacted in our case because we believed the main contributor to the increased TT–TG distance was a proximal malalignment; therefore, both lateralization, in addition to deepening of the new sulcus, was performed to correct proximal alignment and hence TT–TG distance.Lateral patella tilt is confirmed if the angle between the posterior condylar axis and mid-patella line is greater than 20° and suggests trochlea dysplasia . In DejoIn our patient, lateral patella tilt measured on CT was 28.7° and intraoperatively, excessive lateral tightness was examined so lateral release was performed in conjunction with the other procedures.A systematic review demonstrated that MPFL reconstruction is recommended particularly for patients with recurrent dislocations, as the MPFL can contribute up to 60% of resistance to lateral displacement of the patella ,24. The In contrast to MPFL reconstruction, no transosseous patella drilling or bone tunnels are required for suture anchor placement for MQTFL reconstruction. Transosseous drilling and bone tunnels in the patella increase the risk for development of iatrogenic patella fractures. In Spang et al.’s comparative study of the patellofemoral kinematics and stability between four conditions: an intact MPFL, transected MPFL, reconstructed MPFL and reconstructed MQTFL , they foWe performed a MQTFL reconstruction in our patient due to satisfactory clinical and patient reported outcomes in the literature, and the reduced risk of iatrogenic patella fracture.The sulcus angle measured on merchant view of the knee is a commonly used parameter for the diagnosis of trochlea dysplasia ,4,28. ItHowever, in our case study we have noted that the sulcus angle on a knee merchant view may not be a reliable criterion for Dejour types B–D trochlea dysplasia as widely reported in the literature. This is because the knee is flexed at 45° on a merchant view, well below the position of the trochlea. This could explain why our patient who presented with Dejour type D trochlea dysplasia had a sulcus angle of 138°, despite the literature pointing out abnormalities exist when the angle is greater than 145°.The main limitation of this case report is the inclusion of one single case that is typical for patellofemoral instability. However, a typical case allows readers to generalise the results when they are presented with patients of similar clinical conditions. A follow-up period longer than 6 months would be more satisfactory.Due to the complexity behind the anatomy and biomechanics of patella instability, an easy-to-follow treatment algorithm is essential for the treating surgeon to provide effective and efficient treatment without fear of missing essential components. MQTFL, as an alternative to MPFL reconstruction, is recommended for recurrent patella dislocation due to satisfactory clinical and patient reported outcomes and a reduced risk of iatrogenic patella fracture. Controversies for surgical indication in lateral retinacular release, and whether the sulcus angle is an accurate parameter for diagnosis of trochlea dysplasia, remain, and further research is required.


Not diagnosed or mistreated scapholunate ligament (SL) tears represent a frequent cause of degenerative wrist arthritis. A newly developed deep learning (DL)-based automated assessment of the SL distance on radiographs may support clinicians in initial image interpretation.n = 201) for the automated assessment of the SL distance. Afterwards the DL algorithm was evaluated  and correlated with results of an experienced human reader and with arthroscopic findings.A pre-trained DL algorithm was specifically fine-tuned on static and dynamic dorsopalmar wrist radiography (training data set P < 0.01).The evaluation data set comprised arthroscopically diagnosed SL insufficiency according to Geissler’s stages 0–4 . Diagnostic accuracy of the DL algorithm on dorsopalmar radiography regarding SL integrity was close to that of the human reader (e.g. differentiation of Geissler’s stages ≤ 2 versus > 2 with a sensitivity of 74% and a specificity of 78% compared to 77% and 80%) with a correlation coefficient of 0.81 (A DL algorithm like this might become a valuable tool supporting clinicians’ initial decision making on radiography regarding SL integrity and consequential triage for further patient management. Artificial intelligence (AI) is an umbrella term for mathematical algorithms making intelligent decisions without human intervention . RecentlThe scapholunate ligament (SL) is an important component of wrist stability. SL tears and consecutive insufficiency might lead to scapholunate advanced collapse (SLAC), which represents the most common cause of posttraumatic wrist degeneration . To prevThis manuscript makes use of recent advances in the field of DL to develop an algorithm for wrist radiography in order to detect SL dissociation as a surrogate parameter for SL tears and the evaluation of this algorithm in comparison to the radiological reports and arthroscopic findings of the same patients.The institutional review board approved the study design (review board Application Number 600/2021BO2). The study was conducted in accordance with the Declaration of Helsinki (as revised in 2013).n = 201; 50.7% right hand, 49.3% left hand, 31.8% ulnar abduction, 30.8% radial abduction, 37.3% neutral hand position). These were randomly selected out of the institutional picture archiving and communication system (PACS) in July 2021. There were no specific exclusion criteria  of wrist radiographs of patients with clinically diagnosed wrist pain and suspected SL tear and corresponding radiographs of the contralateral unaffected hand. Inclusion criteria was the presence of a timely associated arthroscopic correlation within a maximal time interval of three months. There were again no specific exclusion criteria. The radiographs were out of the study period October of 2010 till December of 2017.Thereafter the DL algorithm analyzed an evaluation data set  with dose relevant parameters of the standard wrist protocol set to 52 kV and 2.5 mAs. Clinic’s patients’ radiographs from external institutions acquired on different X-ray scanners from different vendors and with different study protocols, which had been transferred to the institutional PACS radiographs for diagnostic and therapeutic reasons were also included.A DL algorithm was developed for automated assessment of scapholunate distance in wrist radiographs using the Mask RCNN Resnet 50 FPN model . Mask RCn = 201 wrist radiographs with varying degrees of SL dissociation was used. The model was trained for accurate segmentation of the scaphoid and lunate. To avoid overfitting and ensure robust performance, the average precision (AP) was tracked on a validation set of n = 80 wrist radiographs, and we evaluated this performance every 250 epochs. Our rationale behind this spacing was to prevent any premature conclusions based on isolated outcomes on the relatively small validation set for single epochs that might not correctly represent the model´s performance on an unseen test set. The base learning rate was set to 0.00–025, the weight decay to 0.0001, and the model was trained for 2000 optimization steps, after which no further improvements were observed on the validation set. A batch size of 2 was used to balance the trade-off between memory usage and computational efficiency. We implemented the DL algorithm using the Python package detectron2 v0.6 and used default preprocessing steps of this package. All training and inference code is publicly available at www.github.com/blinded-for-review.The pretrained model was then fine-tuned to segment the contours of the scaphoid and the lunate on the radiographs. For this purpose, the peripheral bone contour of scaphoid and lunate was given in the training data set, regardless of overlays of cortical and medullary bone or of different bones. The smallest distance between the two segmented bones was defined as the SL distance. For fine-tuning a data set of According to the training the DL algorithm segmented the scaphoid and the lunate and measured the smallest SL distance (mm) between the two bones. The human reporter (five years of experience in dedicated wrist radiography) measured the SL distance (mm) according to the recognized cortical contour of the scaphoid and the lunate. Since there is no international consensus for the interpretation of a normal SL distance, results were analyzed according to recent literature and most common practice with cut-offs at both ≤ 2 mm versus > 2 mm and at ≤ 3 mm versus > 3 mm , 18. In The arthroscopic SL integrity was analyzed based on surgical reports on the Geissler’s classification .Statistical analysis was performed using the software package scipy (version 1.10.0). The performance of the algorithm was evaluated and is presented by receiver operating characteristic (ROC) curve. Additionally, we determined the optimal threshold for the DL algorithm when predicting the SL integrity using Youden's J statistic . We calcAs a proxy measurement to filter out observations of misclassified scaphoid or lunate, we additionally analyzed the subgroups of radiographs that had a predicted SL distance below 5 mm and 7 mm, respectively.We compared the DL algorithm's classifications to those of the human reporter and arthroscopy, which was considered the gold standard for diagnosing the SL integrity with the Spearman's rank correlation coefficient. The SL distances from the human reporter was compared to the SL distances from the DL algorithm using the Pearson's correlation coefficient.n = 267; 73.4%) compared to female patients , mean age was 47.5 [± 12.8] years with a total range of 10–78 years. Out of the 1604 images in the evaluation data set, 791 are from the left hand, 813 from the right hand, 532 in radial abduction, 534 in ulnar abduction, and 538 in neutral hand position.The evaluation data set comprised a male predominance . Table The spearman’s rank correlation coefficient between the measured SL distance of the human reporter and the Geissler’s classification was 0.58 P < 0.0. Table 1There were no radiographs that could not be assessed by the DL algorithm (no output failure). The average SL distance measured by the DL algorithm was 2.3 mm, with a range of 0.0 mm to 15.9 mm. As shown in Fig. P < 0.01). When removing measurements of the DL algorithm above 7 mm and 5 mm, the Pearson's correlation coefficient between the measured SL distance by the DL algorithm and the human expert was 0.80 (P < 0.01) and 0.81 (P < 0.01), respectively.Pearson's correlation coefficient between the measured SL distance by the DL algorithm and the human expert was 0.60 (The receiver operating characteristic (ROC) curve shows robust DL algorithm performance see Fig. . The areP < 0.01). Table n = 9 radiographs excluded, which would correspond to an output failure of 0.56%), the sensitivity and specificity with optimal cut-off of ≤ 1.96 mm versus > 1.96 mm were 0.70 and 0.84 for Geissler’s classification of 0 versus > 0, 0.71 and 0.83 for ≤ 1 versus > 1, and 0.72 and 0.79 for ≤ 2 versus > 2. When removing predicted values above 5 mm (n = 24 radiographs excluded), the sensitivity and specificity were 0.67/0.86 (Geissler 0 vs > 0), 0.68/0.85 (Geissler ≤ 1 vs > 1), and 0.70/0.81 (Geissler ≤ 2 vs > 2).The Spearman's order correlation coefficient between the DL measured SL distance and the Geissler’s classification was 0.52 (SL tears are known as the most common cause of posttraumatic degenerative wrist arthritis . As for P < 0.01) and reached a comparable diagnostic accuracy, e.g. for the discrimination of Geissler’s stages ≤ 2 and > 2 the sensitivity was 74% and the specificity was 78%—without an output failure . Hence, according to Geissler’s suggestions [Although the newly developed DL algorithm measured SL distance in a slightly different way—measuring the absolute shortest distance between scaphoid and lunate on each radiograph, whereas the experienced human reporter is assumed to have been able to detect overlap and thus measure the shortest distance between the cortical boundaries of the two bones—the newly developed DL algorithm correlated strongly positive with the human reporter , thereby 3 T MRI is reported to be superior over 1.5 T MRI , 23. TheTo the knowledge of the authors there are no similar studies on a DL algorithm for automated interpretation of the SL distance as a surrogate parameter for SL injury published so far. Studies on automated image interpretation in the field of MSK imaging had rather focused on fracture detection or bone age , 24–28. n = 201, that has been used for fine-tuning of the DL algorithm compared to a number of mostly several thousand that have been used in others of the cited studies. Nevertheless, the algorithm worked surprisingly robust, unharmed by for example metal implants. It should also be mentioned that the relatively small number of evaluation cases (n = 364 patients) may limit the generalizability of the findings to other populations. However, it is worth noting that the evaluation data set included a large number of radiographs (n = 1604), including radiographs from external institutions, which strengthens the robustness of the DL algorithm. Of course, the algorithm may be further improved due to more training and a prospective evaluation of the algorithm in clinical routine could strengthen the results. Finally, only a single DL algorithm was evaluated in this study. The field of DL is rapidly evolving, and new, better-performing models are constantly being developed that could potentially further improve the reported results. Therefore, it is possible that newer DL models may outperform the algorithm used in this study.As a limitation of the reported results, it has to be mentioned that like in other studies the newly developed DL algorithm for the automated assessment of the SL integrity is not error-free. Nevertheless, the authors believe that algorithms like this may become an important tool for medics as they might suggest an image interpretation and medics will keep the authority to accept or overrule this suggestion. Also, an algorithm may direct the attention of unexperienced medics to relevant pathologies and support or even improve their image interpretation, while experienced readers may save time not having to measure simple distances or angles. Further limitations are the retrospective study design and the relatively small data-set of In regard of the potential and the common use of radiography after an acute wrist trauma aiming to exclude fractures and to deliver an orientation as for triaging patients for further diagnostics, e. g. MRI clarification of suspected SL tears, the newly developed DL algorithm showed high diagnostic accuracy for the assessment of SL integrity compared to arthroscopic correlation and it might become a supporting tool for interpreting wrist surgeons and radiologists.