The addition of oxides significantly changes the Fe2+/Fetotal ratio, thus directly affecting the polaron number density and consequently controlling DC conductivity trends for both series studied by impedance spectroscopy. Moreover, we found that short-range polaron dynamics are also under the influence of structural changes. Therefore, we have studied them in detail using model-free scaling procedures, Summerfield and Sidebottom scaling. An attempt to construct a super-master curve revealed that in addition to change in polaron number density, also the polaron hopping lengths change, and Sidebottom scaling yields a super-master curve. The spatial extent of the localized motion of polarons is correlated with polaron number density and two distinct regions are observed. A strong increase in the spatial extent of the polaron hopping jump could be related either to the structural changes due to the addition of HfO2 and B2O3 and their effects on the formation of polarons or to an inherent property of polaron transport in IP glasses with low polaron number density.In this work, we report the effect of the addition of modifiers and network formers on the polaronic transport in iron phosphate glasses (IPG) in two systems of HfO Glasses are prepared by melting homogeneous mixtures of reagent grade chemicals (HfO2–B2O3–Fe2O3–P2O5) in appropriate amounts in high-density alumina crucibles at 1200–1250 °C in the air for 1–2 h. It is known that P2O5 and B2O3 loss during melting is negligible for iron phosphate/borophosphate glasses. The experimental details of the preparation, as well as the structural characterization of these glasses, can be found in reference [The batch composition of two series of glasses labeled F and S are selected for this study. Series F glasses have the nominal composition eference .2O3 is replaced by HfO2 and B2O3, while in the S–series, HfO2 and B2O3 are simultaneously replaced by Fe2O3 and P2O5. The Fe/P ratio in the initial mixture is kept constant (0.67) for the S series. The samples are labeled according to the amount of boron and hafnium oxide in the batch. For instance, the glass F–B2Hf2 belongs to the F series and contains 2 mol% each of B2O3 and HfO2. The PXRD results of these IPG-based glasses showed the precipitation of the crystalline phase HfP2O7 in the glass matrix, but this did not affect the chemical stability of the obtained partially crystallized glasses. A similar effect, namely surface crystallization, was also observed in I(B)P glasses [N, calculated from the glass composition and density, and the fraction of ferrous ions, C = Fe2+/Fetot ratio, determined by Mössbauer spectroscopy, are given for all these glasses in In the F–series, Fe glasses , and as  glasses ,52 the e57Fe Mössbauer spectra were collected in order to characterize the valence state of iron ions and to determine relative amounts of Fe2+ and Fe3+ in studied glasses. Measurements were performed at room temperature (RT) using a constant acceleration method with a source of 57Co(Rh) having the activity of 925 MBq. For the measurement, a well-pulverized sample weighing 40 mg was homogeneously dispersed on the transparent adhesive tape in the diameter of 10 mm. The obtained spectra are presented in the R) and constant-phase element (CPE). The CPE is an empirical impedance function of the type:Electrical and dielectric properties were studied by impedance spectroscopy. Before performing measurements, annealed bars were cut into disks. For the contacts, thin gold electrodes were sputtered onto both sides of 1 mm-thick sample disks using a sputter coater SC7620, Quorum Technologies . Complex impedance was measured using an impedance analyzer  over a wide frequency and temperature range, from 0.01 Hz to 1 MHz at temperatures between 30 °C and 240 °C. The temperature was controlled to ±0.2 °C. The typical complex impedance plot consists of a single semicircle with the centers below the real axis. The equivalent circuit that represents each such depressed semicircle, with the center below the real axis is a parallel combination of a resistor  the DC conductivity is calculated, σDC = d/(R × A), and is listed in 2O7 phase in the amorphous matrix and its influence on the electrical processes are not detected by impedance spectroscopy (IS). In the case of classical glass-ceramics, more than one semicircle is usually present in the IS spectra, which is due to the formation of crystalline phases in a glassy matrix [The impedance spectra at different temperatures for F–B4Hf4 sample and their corresponding equivalent electric circuit (EEC) are shown in y matrix ,23,24. TσDC, is usually expressed by the Austin–Mott relation [σDC is DC conductivity, σ0* is the pre-exponential factor, EDC is the activation energy for the DC conductivity, kB the Boltzmann constant, and T the temperature (K). The pre-exponential factor, σ0*, contains important parameters for polaronic transport according to the relation:α is the rate of wave function decay, C is the fraction of Fe2+ ions to total iron content (Fe2+/Fetot), and R is the average hopping distance between transition metal ions (R = N^(−1/3)), phν is the phonon frequency (≈1012–1013 Hz), e is the electronic charge. For the adiabatic hopping conduction, the tunneling term exp(−2αR) in Equation (2) is approximately 1, by what αR becomes negligible. The DC conductivity variation with 1000/T for all studied compositions is shown in EDC, for each sample is calculated from the slope of log σDCT vs. 1000/T. Calculated values along with the values of DC conductivity at 30 °C and pre-exponential factor are given in −9 (Ω cm)−1 and 4.07 × 10−11 (Ω cm)−1, both of which agree well with the values for various IPG-based glasses from the literature [relation ,37,57: becomes negligible. Based on the data available, it was expected that in the case of IPG-based glasses, the non-adiabatic hopping model is more suited for describing polaron transport [σDCT) vs. activation energy, EDC, at a fixed experimental temperature T = 180 °C is plotted and presented in BT based on the DC conductivity in the adiabatic regime given by Equation (2). If the temperature obtained from the slope differs from the experimental temperature, the process is considered non-adiabatic. The temperature we obtained from the slope is equal to T = 40 °C, which is very different from the chosen temperature of 180 °C. This indicates the non-adiabatic hopping of small polaron and strong electron-phonon coupling in our studied samples and is supported by previously reported literature data on similar IP-based glasses [Stepping forward in the analysis of the electrical transport in these glass-(ceramics), we focus on the changes in the pre-exponential factor and other parameters, see ransport . To veri glasses ,53,61,622+ ions , and based on the assumption that the size distribution is random, see −1 which is in good correlation with those for similar IP-based glasses reported in the literature [The values of the tunneling factor, α, for each glass are calculated from the obtained values of pre-exponential factors, the fraction of Feterature ,60,63,642 and Fe2O3 content is shown in 2O3 or HfO2). The DC conductivity for the F–series varies from 1.00 × 10−10 to 4.07 × 10−11 (Ω cm)−1 at 30 °C with the addition of HfO2 content up to 6 mol%, and increases for sample F–B8Hf8 reaching a value of 1.12 × 10−10 (Ω cm)−1. Otherwise, the S–series shows an opposite behavior, and the DC conductivity increases by almost an order of magnitude with the addition of HfO2, from 1.00 × 10−10 to 1.35 × 10−9 (Ω cm)−1 for glasses containing 8 mol% HfO2, respectively. Interestingly, sample S–B4Hf4 shows a deviation from this trend. It can also be seen that the variation in the amount of Fe2O3 content is high in the F–series as it decreases from 40 to 24 mol%, whereas for S–series the range for iron content is narrower, between 40 and 35 mol%, because B2O3 and HfO2 are added at the expense of Fe2O3 and P2O5 in the composition. The observed trends suggest that the Fe2O3 content is not the only key parameter behind polaronic transport in these samples. As can be seen from the Mössbauer spectra in 2O3 content decreases in these iron phosphate-based glass-(ceramics), the Fe2+ concentration changes from 0.16 to 0.58. Based on this observation, we analyzed the polaron number density parameter and its changes with composition and conductivity trends in more detail.The dependence of DC conductivity for both series upon HfO2O3 but also on the fraction of Fe2+ and Fe3+ ions and is determined by the product of number density of the total iron ions and a fraction of ferrous ions (or Fe2+/Fetot ratio < 0.5). For a ratio value above 0.5, it is determined as a product of the number density of the total iron ions and the fraction of ferric ions [EDC, as a function of the number density of polarons, Nv, for both series of samples is shown in EDC decreases. This supports our conclusion that the polaronic transport in these glass-(ceramics) is not only controlled by the total iron oxide content but a crucial influence comes from the concentration of Fe2+ ions, which ultimately affects the polaron number density and governs the trend of DC conductivity. Moreover, for two compositions, namely F–B4Hf4 and S–B2Hf2, the Nv was also calculated based on Fe2+ ratio reported in reference [2+/Fetot = 0.58 (F–B2Hf2 sample) which shows a value deviating from linearity. In such a ferrous-rich condition an inhomogeneous distribution of ferrous and ferric ions could be present which means that a significant fraction of ferrous ions cannot contribute to the polaron transport [It should be noted here that the polaron number density depends not only on the total amount of Feric ions ,53. The eference  which shransport .2O3 and HfO2, an increase in Tg value was noticed, slightly higher for F–series where the wider variety of overall Fe2O3 content is present, and only Fe2O3 is exchanged with Hf and B. This behavior agrees well with a previous study of IBP glasses modified with boron oxide [2O3 increases thermal stability. It seems likely that in compositions studied, small changes in the strength and rigidity of the IP network do not affect the mechanism of polaronic transport. Nevertheless, even though from Mössbauer analysis both Fe2+ and Fe3+ ions have distorted octahedral coordination in all studied samples similar to many other iron phosphate systems [2+/Fe3+ ratio are significantly altered by the addition of B2O3 and HfO2 in both series, which again implies having a dominant effect on the polaron number density and consequently on the DC conductivity trend.From a previous structural study of these two systems , only a on oxide , where t systems ,65, theiAdditionally, we shift our attention from DC conductivity and long-range transport features to the frequency-dependent conductivity and examination of the localized motions of charge carriers.σ/(σDCT)) = F(ν/σDCT) and can be understood as mobility scaling. An indication of it is that the role of temperature is to accelerate or slow down the charge carrier dynamics without influencing the conduction mechanism. If the Summerfield procedure of scaling is valid then both axes in a double log-log plot are scaled by the same factor σDCT. This is equivalent to scaling the isotherms in such a plot along a line with slope one, which is plotted by marking the onset  frequency value of the conductivity dispersion, σ′(f0) = 2σDC, please see The conductivity spectra for F–B8Hf8 and S–B4Hf4 samples from the F– and S–series are shown in We could make important and indicative observations based on obtained slope values, see x-axis with the changes in the composition. We shifted the individual master curves along the x-axis and tried to create a super-master curve for each series, It is interesting to investigate the influence of glass composition and structure on the conductivity dispersion, the frequency-dependent part, by applying a superscaling procedure where all individual master-curves within these two series are superimposed. 2 and CeO2, similar in composition but with widely ranging concentration of ferrous ions (0.23 ≤ Fe2+/Fetot ≤ 0.58), the shift was also observed, but correspondingly in a scattered fashion.A shift in the super-master conductivity plot is reported in the literature for different ion-conducting oxide glasses and has been correlated with the alkali content ,72,73 asfshift) and the changes in the number density of charge carriers, Nv, was studied. The variation of log(σDC) and log(fshift) as a function of Nv is shown in Furthermore, the correlation between log(fshift), exhibits scattering rather than a linear trend, suggesting that its origin is not entirely related to changes in the polaron number density. This prompts us to consider the alternative scaling procedure proposed by Sidebottom [In the top part, debottom ,75, whicε* (ν) = 1/(2πνjCoZ*) can be expressed as a complex number:ε′(ν) and ε″(ν) are the real and imaginary parts of the complex permittivity. The frequency dependence of the real part of the complex permittivity, ε′(ν), at different temperatures for F–B2Hf2 and S–B6Hf6 samples, is shown in ε′∞, resulting from fast polarization processes that occur in the glasses under the applied field [ε′(ν). On the other hand, with increasing temperature and decreasing frequency, ε′(ν) increases and for all the glass-(ceramics) studied, the low-frequency plateau denoted as the value of the low-frequency static permittivity, εs, is well-developed, see ε′ = εs − ε′∞, as proposed by Sidebottom [ε′∞ is only weakly temperature-dependent, the dielectric strength for polaronic glasses can be determined directly from the temperature-dependent experimental spectra.The complex permittivity ed field . Therefodebottom ,75,77 anε′(ν) on the y-axis is scaled by the product (ε′(ν) − ε∞) T, while the frequency x-axis is scaled as the product ν/σDCT. The scaled spectra for samples F–B2Hf2 and S–B6Hf6 are shown in ε × T can be extracted, which can be related to the typical spatial extent of localized motions of polarons, <rLOC2(∞)>1/2, discussed in the following ε × T is in a range from 3910 to 5742 K and depends on the composition and level of modification of base glass structure with B2O3 and HfO2 /(σDCT)) = F((ε0Δε)/σDCν). Compared to the Summerfield procedure, Sidebottom scaling could be more challenging since for ionic conductive glasses the determination of Δε is usually hindered by the electrode polarization effect, which is not the case in polaronic glasses. Nonetheless, it can be considered truly universal since it is applicable whenever scaling is possible, i.e., when the shape of the conductivity dispersion does not change with temperature [As already mentioned above, an alternative scaling procedure for conductivity spectra is proposed by Sidebottom to account for the simultaneous change in typical charge carrier hopping distance with the change in their number density ,73. Hereperature . As far perature ,53. A peR >1/2 by the relation [B is the Boltzmann’s constant, ε0 is the permittivity of free space, and Nv is the number density of polarons. The obtained values of the extent of the localized motions of polarons are given in With regard to the length scales relevant to electrical transport, it is possible to distinguish different parameters: those calculated based on composition and density such as average distance between iron ions, R see  and pola rLrLOC2(∞)>1/2 upon the number density of polarons for studied F– and S–glass series and correlation with literature data is presented in rLOC2(∞)>1/2 for both series varies in the range between 1.87 Å to 3.24 Å. Simultaneously, the polaronic number density changes in the range between 1.35 × 10−21 cm−3 and 4.42 × 10−21 cm−3. It is hard to see the trend, but a closer look reveals useful information. The correlation between <rLOC2(∞)>1/2 and Nv for all compositions exhibits two distinct regions. Region 1 contains samples with higher Nv, above ~2.2 × 10−21 cm−3, whereas the samples with lower NV fall into Region 2. It is indicative that the transition between these regions is not only related to Nv, but also to the combination of compositions and parameters which have an impact on <rLOC2(∞)>1/2.The dependence of <2O3 and HfO2 on the Fe2+ concentration could also be observed. From 2+ concentration with an apparent minimum for compositions F–B4Hf4 and S–B4Hf4 . Additional indications of the dominant role of boron on the Fe2+ concentration could be seen in sample S–4BHf6 where the content of hafnium oxide increases, however, the Fe2+/Fetotal ratio is kept low at 18%. In the next step, with simultaneous modification of the glass network, Fe2+ concentration increases up to 38% and 36% for samples F–B8Hf8 and S–B4Hf8 and indicates that HfO2 takes control and has a positive impact, increasing the Fe2+ concentration. A shift from boron oxide to HfO2 taking control could further be seen in the fact that for the sample S–B4Hf8, the amount of boron oxide is kept at 4 mol% while HfO2 is increased to 8 mol%, which results in a jump of ferrous concentration from 17% to 36%. Also, these results could be compared with Mössbauer’s result for IPG glass with up to 20 mol% of boron oxide [2+/Fetot ratio since the fraction of ferrous ions changes in a narrow range from 0.16 to 0.22. Furthermore, considering <rLOC2(∞)>1/2 trends, it should be pointed out that for series S the overall Fe2O3 content is less under the impact of compositional changes as B2O3 and HfO2 are added at the expense of both F2O3 and P2O5 content. This results in a narrower range of iron oxide change in the S–series (40–35 mol%) in comparison to the F–series (40–24 mol%). Both aforementioned parameters affect the general result and values on Nv and <rLOC2(∞)>1/2. For example, one can see that at the shift from the first region to the second, the polaron number density is ~2.2 × 10−21 cm−3 for two extreme compositions, base glass G–B0Hf0 and F–B8Hf8. This implies that even though the Nv is similar, modification of IPG structure and addition of B2O3 and HfO2 have an impact on polaron transport. As a result, <rLOC2(∞)>1/2 increases suddenly from 2.25 to 3 Å for the sample with a high amount of boron and hafnium oxide.The influence of Bon oxide ,52, wher2+ concentration ranges from 0.16 to 0.58, shows a much broader Nv region. The increase of polaron number density from 2.2 × 10−21 cm−3 to 4.4 × 10−21 cm−3 has surprisingly no effect on the <rLOC2(∞)>1/2 which is ~3 Å. For the last sample of the F–series, F–B8Hf8, an increase in NV is observed which could be attributed to the increase in ferrous concentration due to the positive effect of high HfO2 content. At the same time, the DC conductivity trend shows a similar behavior and follows the changes of Nv, see rLOC2(∞)>1/2, one can see that, except for the sample F–B2Hf2 [Nv range (second region) with a value of ~3 Å. Once again, the polaron concentration is higher in S–series, except for S–B4Hf4, and all samples fall into the first region, which is characterized by a high Nv value. Discussing a general trend, it should be pointed out that the polaron number density increases, whereas <rLOC2(∞)>1/2 slightly decreases linearly between 1.9–2.3 Å. However, the jump to 3.2 Å for S–B4Hf4 sample  and transitions from the first region to the second one are observed.Returning to two distinct regions in  F–B2Hf2 , the sparLOC2(∞)>1/2 is close to polaron radius, rp, values calculated using the Bogomolov and Mirilin: rp = (1/2) (π/6N)1/3 relation, where N is the total number of iron ions [R parameter calculation based on composition and density shows significantly higher values (see rLOC2(∞)>1/2 are in the appropriate range when comparing different IPG systems, and a good correlation can be drawn. It is interesting to see that the compositions from this study cover both regions in Nv–higher <rLOC2(∞)>1/2 and vice versa) due to wide variation in Nv, which was not the case in our previous studies [rLOC2(∞)>1/2 with NV is characterized by a larger slope in the lower polaron number density region, see rLOC2(∞)>1/2 and NV allows clear identification of the prevalence and determination of a sharp increase in the spatial extent of the polaron hopping jump, which consequently has a direct impact on the DC conductivity trends in the studied samples. This result could be related either to the structural changes induced by the addition of modifiers or network former oxides and their effects on the formation of polarons or to the inherent property of the polaron transport in IPG with low Nv.At this point, it is useful to examine the results from ron ions . On the lues see . The obt studies ,52,53. O2O3−60P2O5 base glass (G–B0Hf0). Two different series of glasses were prepared with simultaneous additions of B2O3 and HfO2 at the expense of Fe2O3 (F–series), or both Fe2O3 and P2O5 (S–series). The observed trends in long-range DC conductivity show that the key parameter behind polaronic transport in these glasses is not only the Fe2O3 content. The addition of B2O3 and HfO2 significantly alters the Fe2+ concentration in both IP glass series, from 0.16 to 0.58, which has a dominant effect on the polaron number density and consequently on the trend of DC conductivity. As a result, a nearly linear trend is observed with an increase in the polaron number density.In this work, we use solid-state impedance spectroscopy to study in detail the electrical properties of iron phosphate glasses in which up to 8 mol% boron and hafnium oxide were added to 40FeFurthermore, we investigated the short-range polaron dynamics by applying a model-free analysis of the conductivity and permittivity spectra of the studied glassy systems and scaling procedures to gain a better insight into the polaron dynamics. Namely, Summerfield and Sidebottom scaling of the conductivity spectra confirmed the validity of the time–temperature superposition principle for all compositions. The construction of the super-master curve reveals an interesting feature. While the shape remains the same, the Summerfield scaling fails, but the Sidebottom one yields a super-master curve for both series. This indicates that in addition to change in polaron number density, also the polaron hopping lengths change. In the next step, we used experimental permittivity spectra to evaluate the spatial extent of the localized motion of polarons. Its correlation with the polaron number density reveals two distinct regions, containing samples with low and high polaron concentrations. Interestingly, the transition between these regions is not only related to the polaron number density, but also to the combination of sample compositions and parameters which have an impact on polaron-localized motions. The detailed analysis allowed clear identification of a sharp increase in the spatial extent of the polaron hopping jump, which consequently has a direct impact on the DC conductivity trends for both series. This feature could either be directly related to the structural changes induced by the addition of modifiers or network forming oxides and their effects on the formation of polarons, or to the inherent property of polaron transport in IPG with low polaron concentration.


This article has been corrected: In the new Figure 5, the authors replaced Figure 5B, where they accidently mislabeled the “E2” and “E2+inhibitor” groups and mistakenly used partially duplicated pictures. The new Figure 5B contains a new image for the “E2+inhibitor” group from the original set of experiments. The authors also provided all the original HE staining pictures for this manuscript. This correction does not change the content of the publication and does not affect the conclusion of this research.Figure 5 is presented below.New


In drug development programs, proof‐of‐concept Phase II clinical trials typically have a biomarker as a primary outcome, or an outcome that can be observed with relatively short follow‐up. Subsequently, the Phase III clinical trials aim to demonstrate the treatment effect based on a clinical outcome that often needs a longer follow‐up to be assessed. Early‐phase outcomes or biomarkers are typically associated with late‐phase outcomes and they are often included in Phase III trials. The decision to proceed to Phase III development is based on analysis of the early‐Phase II outcome data. In rare diseases, it is likely that only one Phase II trial and one Phase III trial are available. In such cases and before drug marketing authorization requests, positive results of the early‐phase outcome of Phase II trials are then likely seen as supporting (or even replicating) positive Phase III results on the late‐phase outcome, without a formal retrospective combined assessment and without accounting for between‐study differences. We used double‐regression modeling applied to the Phase II and Phase III results to numerically mimic this informal retrospective assessment. We provide an analytical solution for the bias and mean square error of the overall effect that leads to a corrected double‐regression. We further propose a flexible Bayesian double‐regression approach that minimizes the bias by accounting for between‐study differences via discounting the Phase II early‐phase outcome when they are not in line with the Phase III biomarker outcome results. We illustrate all methods with an orphan drug example for Fabry disease. For regular “large disease” drug development programs decisions to conduct a Phase III trial are based on positive Phase II trials. If these trials are only retrospectively evaluated in combination, that is, during the drug marketing authorization request, the ad hoc synthesis may induce a form of decision‐induced bias . Such a bias is not an issue if the early and late Phase trials are prospectively considered in the design phase .retrospectively be seen as supportive, even if the two clinical trials were designed/conducted independently, as typically the early‐phase outcome would be assumed to be associated with the late phase primary clinical outcome. Throughout the article the terms “retrospective (ly)” denote the retrospective combination of the available Phase II and Phase III trial after both trials are completed and their final results are available.However, it is not uncommon that in rare diseases, no more than two independent RCTs are conducted and available, one exploratory and one confirmatory.For example, Galafold  acquired marketing authorization as an orphan drug for the treatment of Fabry disease in 2016 within Europe. Fabry disease is a rare, progressive disorder with an estimated prevalence of 1:117 000 to 1:40 000.In trial 011 patients switched to migalastat 6 months postrandomization, while in trial 012 primary follow‐up was considerably longer, with switching taking place 18 months postrandomization. In the first trial, the change in average globotriaosylceramide (GL‐3) inclusions from baseline to 6 months was the primary outcome which produced nonconclusive evidence. The second trial utilized the annualized change in glomerular filtration rate  approach, which can aid in more precise treatment effect estimation, by accounting for unobserved late‐phase outcome responses via observed early‐phase outcome responses. Their method utilizes the correlation to ultimately inform the mean and variance estimates of the treatment effect on the late‐phase outcomes. For large samples their method has the potential to increase precision. However, for small sample sizes this is not necessarily true.Analysis methods that use the relation between early and late‐phase outcomes may be applied to retrospectively, but formally, synthesize the evidence on treatment efficacy across the two trials. Engel and Walstradouble‐regression (BDR) analogue can be readily constructedA Bayesian In this article, we investigate how to model and estimate the efficacy of a new treatment on the late‐phase clinical outcome, using data on early‐phase outcomes from both trials. Most literature on double‐regression focuses on design aspects such as interim analysis or seamless design of phase II/III trials, though, in the present article we propose methods that would be applied retrospectively (ad hoc) only after the Phase III trial. We propose and investigate methods that either account or do not account for the potential decision‐induced bias when combining retrospectively the Phase II and Phase III trials. We investigate the two proposed models, the bias corrected DR approach and the flexible Bayesian approach regarding their performance to estimate the treatment effect on the late‐phase outcome. We focus on two related key problems: (1) the magnitude of the type 1 error inflation when retrospectively combining data from Phase II and III and (2) how to estimate the treatment effect on the late‐phase outcome, using results from both studies and we assess this estimate in terms of bias and variance.single‐regression(SR) approach. We introduce the problem of decision‐induced bias moving from Phase II to Phase III based on the Phase II early‐phase outcome in Section 3 and then provide an approximate analytical solution. In Section 4, we propose and formulate a Bayesian two‐step solution to the estimation problem, a model that down‐weights the impact of the biomarker data via a historical power prior. This prior dynamically accounts for the bias in estimating the same treatment effect across the two trials, by accounting for additional between‐trial differences (variability) around the biomarker outcome effect. The article ends with a discussion and steps for further research.The article is organized as follows. First, we describe a bivariate linear model, we introduce its conditional form and we formalize the  retrospective pooling by utilizing DR with nonavailable Phase II late‐phase outcome data, then briefly discuss specific model variations, for example, the 2i in trial k and i in trial k, Consider a Phase II trial of total sample size 2.1σx2 and σy2 denote the true outcomes variances, ρ the true correlation between the two outcomes and i to aid readability.When all late‐phase σ02=σy2−γ2σx2, a0=ay−γax, bo=by−γbx, and γ=ρσy/σxThe above bivariate model can be conditionally expressed as2.2At the end of both trials early‐phase outcome data y Figure .σx2 are estimated via the regression of âx,b^x, sx2 and parameters a0,b0,γ, and σ02 are estimated via the regression of â0,b^0,γ^, s02, while sy2=s02+γ^2sx2, ây=â0+γ^âx, ρ^=γ^sx/sy.Based on the DR method, parameters b^y is shown in Reference z1−α3>b^y/var^(by), where z1−α3 is the (1−α3)th standard normal quantile and α3 is the alpha level of the late‐phase primary outcome of phase III trial. A direct Bayesian analogue to the conditional model ,ax,bx∼N,σx2∼IG of We can model the Phase II biomarker data . To further mimic model . The above two‐step procedure will allow for possible discounting of the Phase II trial by down‐weighing the early‐phase historical outcome data, which is further discussed in section The prior on ic model  we have ic model  we inforγ parameter, model (ρ) between the two outcomes, and it directly produces posterior Markov Chain Monte Carlo draws from Pr(by>0|Y)>ω where ω=0.95.In comparison to the direct Bayesian analogue of model , where tr, model  is more ρ∼U, then, the Phase III trial late‐phase outcome data are evaluated individually under a standard (Bayesian) linear SR model. In comparison to the SR models, the advantage of models ((Bayesian) SR models can be found in Appendix A.If we set the correlation very close to zero; that is, f models , Bayesiaf models  and m3)ρ∼U, will be truncated, where ZX2 denotes the standardized difference of the early‐phase Phase II trial outcome. If the analysis of Phase III data occurs independently from earlier Phase trial data, we expect no increase of Type I error and bias, though the power might remain low due to the limited trial sample size. In the retrospective assessment of the totality of evidence in this rare disease setting, however, positive results from both the Phase II trial and Phase III trial may well be seen as reinforcing. This informally combines evidence between trials which often results in positively biased inferences in favor of the late‐phase treatment effect b^y estimate, based on model 1−Φ(ω), ω=z1−α2−μx2σx2/n2/2, ϕ and Φ are probability density and cumulative functions of the standard normal distribution, respectively, D=w2((2σx22/n2)ζ+A2(1−w22−w32)+2A(μx2−μx)), A=(σx2/n2/2)λ, ζ=α2λ−(λ)2.The potential issues with retrospective combination of early and late phase results stem mostly from the fact that they are not independent. Usually, a Phase II decision leads to the initiation of a Phase III trial. This decision could be based on a test statistic for the early‐phase outcome and an imposed critical value; that is,  (models  and m3)z1−α2. Thon model  is givenMSE(b^y) of the double‐regression is equal to (Appendix B)An approximate value for λ parameter, (ii) the Phase II early‐phase outcome mean (μx2) and variance (2σx22/n2), (iii) the number of patients in the Phase II trial (ρ). An increase in σx2 results in an increase of MSE, while as ρ, while the power curves, in general, increase with more patients being allocated to the Phase III trial  model are estimated as (Appendix B),Based on the aforementioned bias and mean square error expressions and by replacing parameters with their estimates, the late‐phase outcome effect and variance of a (bias) corrected b^x3,b^x2 in Appendix B.1 based on the treatment arms sample sizes. For example, if the allocation ratio between arms in the Phase II trial equals to 1:2, then the Phase II early‐phase endpoint variance increases, 9σx2/2N2 and the introduced bias could be reduced by half.The above expressions hold when treatment arms within studies are equal. Nonetheless, similar analytical expressions for unequal within study allocation ratios, can be acquired by appropriately changing the variances of 4All models above, including the bias corrected model, assume that the true overall treatment effect remains common between trials, no between‐study variability on the early and late‐phase outcomes exist and therefore, all The decision‐induced bias discussed in Section η^ that represents conflict between the early‐phase outcome data of the two available trials, model  is the marginal likelihood. Ibrahim et al−2log{m(η)} for the normal outcome case. Figure η and varying levels of A number of power prior  formulations has been suggested.Pr(by>0|Y)>ω where ω=0.95.In model , similar5The main four approaches discussed are summarized in Table N patients are available on the early‐phase and data of For illustrative purposes, we assume that the two available Phase II and Phase III trials had a similar control treatment, therefore, the Phase III trial would have been designed as a placebo‐controlled trial. In this section, we assume that the decision to conduct the Phase III trial was taken on the basis of available evidence in the first Phase II trial on a single early‐phase outcome. At the end of the Phase II trial, individual data of double‐regression (BFDR) approach. Even though, they are not directly comparable  and late‐phase (τy) outcomes (see Appendix C for details). Specific alternative versions of scenarios I and II were produced by varying ρ and τy,τx.The SR, DR, DRC methods ignore any between‐study variability and therefore assume a different underlying data generating model in comparison to the Bayesian flexible le Table , we empiτy=τx=0) and null hypothesis with additional between‐trial variance (τy=τx=0.3), while the second (II) scenario describes a common alternative hypothesis on both outcomes and trials. Scenario III can occur when heterogeneous populations are selected for the Phase II and Phase III trial, while the fourth (IV) scenario describes a situation where the late‐phase outcome true effect exists but the early‐phase outcome equals to 0. All remaining settings , total sample sizes The first (I) scenario describes variations of the strict null (le Table . All sim5.1The BFDR results in treatment effects closer to the SR estimates than the DR approach under the null hypothesis simulation (Scenario I—Table 5.2τy=τx=0), DRC performs better in terms of 95% coverage whereas in scenario IIb where τy=τx= 0.3, BFDR results in coverage closest to 95%. The C(r)Is widths retained a similar behavior to the null hypothesis scenarios.In scenario II  outcomes are not predictive of a Phase III success on late‐phase clinical outcomes. If Phase II and Phase III results are then assessed  jointly to support efficacy, this retrospective (ad hoc)) assessment may be subject to decision‐induced bias and may increase uncertainty of the true primary late‐phase treatment effect. Such an informal combination of results may increase to a great extent (more than three times) the Type I error rate of null hypothesis, rendering the retrospectively combined late‐phase true treatment effect misleading. Especially in rare diseases, where the validation of early‐phase surrogate endpoints can become problematic, due to the small and often heterogeneous populations, the small sample sizes and the insufficient number of available trials, only late‐phase hard endpoints are usually appropriate to prove treatment efficacy.In this article, in addition to identifying and investigating the above issue, we explored methods that can be utilized in order for early and late Phase trial data to be combined retrospectively , while accounting for the underlying decision‐induced bias. The flexible BDR includes the borrowing of historical information, while this model downgrades the historical prior upon early‐phase outcome data conflict. The DRC method approximately corrects the biased late‐phase mean effect and variance estimate.τx is low and/or ρ is high  penalty to the Phase II effect than the Bayesian approach; which could explain its overall better performance in the simulation.In most scenarios, the DRC method better controls the Type I error and bias than the DR and BFDR methods. This is not observed in scenario Ia, where the BFDR controls better the Type I error than the DRC. This possibly happens because the BFDR approach completely downgrades the impact of Phase II trial when its early‐phase treatment effect is different than the Phase III trial early‐phase treatment effect. Therefore, on average the Bayesian approach becomes less prone to false‐positive results based on possible very positive Phase II early‐phase outcome trial effects when Both the BFDR and the DRC methods would be an attractive solution to the increased Type I error of the informal retrospective combination of two small available trials. The consideration of these methods was shown to be rather important when, (i) the preceding Phase II trial conservatively  resulted to the Phase III trial and/or (ii) the association of utilized early and late‐phase outcomes is high. An informal combination of results across Phases often happens when both of the above hold, though, when neither holds then the complexity of suggested methods may outweigh the gains of their application.Alternative versions of the BFDR model could be developed and they may perform more optimally in comparison to the current  when applied on the flexible BDR via the use of an alternative guided value.An alternative approach that controls type I error on the late‐phase outcome, while borrowing historical information, may also provide a more formal solution.In the motivating example we assumed that both trials were superiority trials, while if we had kept the initial designs, different strategies may have been more appropriate. Nonetheless, examples of two superiority trials, one Phase II and one Phase III, exist in the literature. For example, the drug development program of thalidomide for the treatment of multiple myeloma contained two randomized superiority clinical studies of similar design, a supportive (GISMM2001) and a main study (IFM 99‐06), that compared melphalan‐prednisone (control treatment) to thalidomide .Throughout the article normality was assumed, an assumption that could be challenged with rare diseases sample sizes.In this article, we performed a post hoc (retrospective) combination of available information after the conduct of the Phase II and Phase III trial. However, it may be very relevant to (prospectively) plan to pool the data from both studies and to use the early‐phase outcomes of the Phase II study to increase the precision, with which the efficacy on late‐phase outcome is estimated overall.To conclude, especially in a small population context, the often informal retrospective pooling of a single Phase II early‐phase outcome data to support the true late‐phase outcome data inference at the end of a single confirmatory Phase III trials could induce bias and it should be performed via formal numerical approaches. Such approaches should control this decision‐induced bias, in order to avoid inflating the Type I error under the null hypothesis and prevent overestimating our beliefs on the primary treatment effect. We hope that this article, except for introducing possible solutions, raises awareness of potential mishaps with post hoc combinations of trial outcome results.


Escherichia coli for originating new binding properties. As a proof-of-principle a nanobody recognizing the antigen TirM from enterohaemorrhagic E. coli (EHEC) was evolved towards the otherwise not recognized TirM antigen from enteropathogenic E. coli (EPEC). To this end, E. coli cells displaying this nanobody fused to the intimin outer membrane-bound domain were subjected to multiple rounds of mutagenic oligonucleotide recombineering targeting the complementarity determining regions (CDRs) of the cognate VHH gene sequence. Binders to the EPEC-TirM were selected upon immunomagnetic capture of bacteria bearing active variants and nanobodies identified with a new ability to strongly bind the new antigen. The results highlight the power of combining evolutionary properties of bacteria in vivo with oligonucleotide synthesis in vitro for the sake of focusing diversification to specific segments of a gene (or protein thereof) of interest.ssDNA recombineering has been exploited to hyperdiversify genomically-encoded nanobodies displayed on the surface of  Yamal Al-ramahi et al. describe an accelerated laboratory evolution method that applies ssDNA recombineering and bacterial surface display to engineer nanobody epitope recognition. As a proof-of-concept, they evolved TD4 recognition to bind the translocated intimin receptor (TirM) of EPEC pathogens, with this method bypassing the need for laborious cloning and mutagenesis procedures. While the variable (V) domain of different antibodies adopt a very similar immunoglobulin (Ig) fold structure, their antigen-binding loops  are selected out of a very large diversity pool3. Such a process occurs upon exposure of the immune system to given antigens through an intricate molecular mechanism of clonal selection, amplification and affinity/specificity maturation5. One remarkable feature of this course is that in vivo diversification of the CDR sequences for optimally binding a molecular target is virtually limited to 2–4 residues of the primary protein sequence5. The single V domain of the naturally occurring heavy chain camel antibodies—also called VHHs or nanobodies (Nbs) when expressed in a recombinant form—represent an extreme case of this state of affairs6. While the structures of the Ig V-domain scaffold of nanobodies are virtually identical, their CDRs  or ribosome display9 systems enables the selection of variants after diversification. Variability can be introduced either in vivo with a limited targeting of the VHH sequence10 or in vitro with degenerate codons at CDRs using PCR with oligonucleotide primers11 or DNA synthesis13. Yet, while such in vitro methods for creating targeted variability in the CDRs push the potential number of antibody variants close to those of the natural immune system (>1013)14 they are limited to in vitro selection methods (e.g. ribosome display) that avoid cloning and cell transformation steps needed for more robust and versatile in vivo selection approaches 15.Given the structural simplicity of camel-derived Nbs it comes as little surprise that they have emerged as a favourite platform for recreating in the laboratory some of the mechanisms involved in their natural diversification and thus breed large libraries which can later be panned for selecting best binders to specific targets. In particular, the possibility to express VHHs in multiple surface display platforms10 variants/library) with single nucleotide precision in living cells. Furthermore, as such oligo recombineering can be iterated and coupled to continuous selection, it allows the continuous generation and selection of evolved variants with beneficial mutations16. We thus envisioned a combination of surface-display of single-copy genome-encoded Nbs on E. coli cells17 with the cyclic and DNA-segment directed mutagenesis method called DIvERGE18. This in vivo ssDNA recombineering strategy progressively hyper-mutagenizes specified segments of any gene of interest (e.g. the CDRs of a given VHH) without off-target mutagenesis in adjacent DNA. This could then be followed by capturing by physical means the bacteria displaying the best nanobody variants on their surface19.On this background we wondered whether we could boost a targeted diversification of the CDRs of VHHs by combining the evolutionary power of bacteria in vivo with the ease of synthesizing mutagenic oligonucleotides in vitro. Compared to other methods, oligonucleotide-mediated in vivo mutagenesis (i.e. oligo recombineering) can generate extreme diversity  of enterohaemorrhagic E. coli (EHEC) strains towards the otherwise non-recognized but functionally homologous TirM domain of enteropathogenic E. coli (EPEC) strains21. The resulting, evolved nanobodies recognizing the new TirMEPEC antigen were then secreted and easily purified from bacterial culture media by adopting the E. coli hemolysin protein secretion system22 integrated in the experimental pipeline. In this way, the work below results in nanobodies potentially useful for diagnostics and passive immunotherapies against EPEC and EHEC infections causing human diarrhoeal diseases23. Furthermore, the data below provides an example of how an all-prokaryotic platform can be designed to artificially embody key features of the immune system for a large number of applications—including diagnostics, therapeutics and environmental needs e.g. detection of biodegradative enzymes as pollution markers24.In this article we show how combination of ssDNA recombineeringD < 5 nM) and specificity for TirMEHEC21, a protein domain required for intimate attachment of the pathogen to intestinal epithelial cells with actin pedestal formation during infection by EHEC26. However, TD4 Nb did not show any significant binding for the equivalent TirM antigen of EPEC strains, despite the fact that the 24-amino acid epitope recognized by TD4 is 71% identical in both variants21. On this basis, we picked TD4 as a good candidate for sampling whether one given Nb could be evolved with DIvERGE technology18 towards new binding properties. To this end, we adopted E. coli strain EcM1luxSAtir17  locus of E. coli K12 genome28. Furthermore, EcM1luxSAtir strain was deleted of the fimA-H operon, encoding type 1 fimbriae29 and has an insertion of the luxCDABE operon of Photorhabdus luminescens30 into the matBCDEF operon, encoding the E. coli common pilus31. These genetic modifications inactivate the expression of cognate endogenous E. coli cell surface structures that could interfere with binding of the displayed Nb to target antigens17. This makes E. coli EcM1luxSAtir an optimal strain for displaying the Nb moiety of the fusion in an active form for interacting with TirMEHEC, whether in solution or bound to a surface17. As sketched in Fig. E. coli carrier. This in turn facilitates application of ssDNA recombineering for targeted diversification of such DNA segments, eventually resulting in variability of the primary sequences of the Nb’s CDRs.The Nb called TD4 has high affinity  that transiently supresses mismatch repair—thereby increasing occurrence in vivo recombineering events in at least four orders of magnitude34. The pORTMAGE3-bearing strain was then cyclically transformed with a cocktail of target-specific mutagenic 90-mer oligonucleotides. The whole of synthetic ssDNA used in the experiment included 3 distinct pools, each targeting one of the segments of the VHH gene that encoded domains CDR1, CDR2 and CDR3, respectively. These oligos were designed34 to introduce 1.5 mutations on average per 100 nucleotides of the targeted sequence to thus generate an average of single amino acid changes within the 3 CDRs of the VHH protein. Following ten consecutive rounds of DIvERGE mutagenesis the DNA sequences corresponding to the VHH genes borne by E. coli were amplified from the merged genomic DNA extracted from the culture with primers TirDSF and TirDSR  and to the parental cells incubated with the original antigen TirMEHEC (positive control). Colony forming units (CFU) stemming from bacteria in the washout eluate (unbound fraction) and in the antigen-bound fractions were determined before and after MACS cycles for all samples  , washed and stained with streptavidin-phycoerythrin . Samples were then inspected in a fluorescence flow cytometer along with controls of the non-diversified strain (E. coli EcM1luxSAtir) and the input library (diversified but unselected) treated under identical conditions. The results shown in Fig. EPEC. Additional rounds of selection enhanced the TirMEPEC binding signal on bacterial cells ca. 3-fold, which remained constant or showing a small reduction in the fourth and fifth cycles. Taken together, these data settled the efficacy of the workflow sketched in Fig. HH variants with potentially new specificities.In order to enrich the library of Vgen Fig. , washed cle Fig. . The samEHEC Nb enabling binding to TirMEPEC, the genomic regions of the VHH were amplified as above in each of the MACS cycles. The distribution of amino acid replacements was then determined through DNA sequencing of the resulting PCR products with PacBio CCS technology (see Methods). Figure HH gene variants were subsequently amplified by PCR and separately sequenced. By matching the resulting discrete sequences with those of the pooled data we could calculate the occurrence and evolution of each variant relative to the overall mutant frequency after each round of MACS sorting  and magnetic beads36. Hence, clones with wild type VHH were excluded from further analysis. Eventually, mutant clones were identified that were gradually enriched along the MACS cycles, the most abundant being those with VHH encoding Nbs with single mutations in residues H107Y, T108R and D116G in their primary sequence  were tested for their ability to bind the original antigen TirMEHEC in comparison with the one used for selection (TirMEPEC). For this, cells from cultures expressing the intimin-VHH fusions were incubated with either 250 nM of biotinylated TirMEPEC or 50 nM biotinylated TirMEHEC, followed by staining with streptavidin-allophycocyanin . Bacteria were then subjected to fluorescence flow cytometry analysis and the results are shown in Fig. EPEC, the counterparts expressing TD4H107Y, TD4T108R and TD4D116G exhibited a considerable binding to the new antigen −1, where Y is the fluorescence intensity, Bmax is the fluorescence signal extrapolated to very high concentrations of antigen, X is the concentration of the antigen and Kd is the equilibrium dissociation constant . Based on best fit estimations, the apparent equilibrium dissociation constant (Kd) for the new antigen TirMEPEC exposed apparent binding affinities of H107Y, T108R and D116G mutants in a similar range, between 17 to 33 nM  and the resulting plasmid constructs were then placed in a dedicated E. coli strain that expresses HlyB and HlyD components of the hemolysin protein secretion system (see Methods Section22). The strains bearing these fusions were grown in LB, induced with IPTG, and the secreted nanobodies purified from the culture media . Affinities were determined by coating ELISA plates with purified His-tagged versions of TirMEHEC and TirMEPEC and the values estimated as shown in Fig. EHEC is best recognized by the non-evolved TD4 (Kd = 5.5 ± 1.19 nM) in a fashion about one order of magnitude better than TD4H107Y (Kd = 47.74 ± 12.38 nM) and >20-fold in respect to TD4T108R and TD4D116G. The new antigen TirMEPEC was not recognized by the parental TD4 Nb at any concentration tested but, the evolved counterparts manifest their newly gained interaction to the new target with apparent Kd values of 20.8 ± 2.31 nM (TD4H107Y), 176.3 ± 47.11 nM (TD4T108R) and 397 ± 140.9 nM (TD4D116G), respectively. Divergence of the apparent Kd values determined with the monovalent Nbs with those estimated with E. coli bacteria are likely reflecting avidity effects due to the distinct physics of the Nb-antigen interaction in monovalent (purified) versus multivalent (cell-displayed) formats36. In this respect, purified TirMEPEC showed a dimeric behaviour in solution as determined by size exclusion chromatography . The resulting shape of the original TD4 model was then overlapped pair-wise with each of the other 3 evolved variants, with the outcome shown in Fig. EPEC structure that were not amenable to binding by the non-evolved TD4 Nb21. In contrast, an acidic aspartate at position 116 was replaced with a neutral and smaller glycine, suggesting that in this case the change likely eliminated an inhibitory repulsion between a charged amino acid and an incompatible site of TirMEPEC. These replacements are also predicted to change the orientation of the new amino acid in respect to the original one, although due to the size of the side chains, this was more pronounced in the case of H107Y and T108R. In any case, replacement of a single amino acid in TD4 was sufficient gain of a new binding specificity while their former capacity to bind TirMEHEC was not abolished. This was noticed in the three purified nanobodies, but the tradeoffs were different in each case. TD4H107Y was the one with higher TirMEPEC-binding affinity and the mildest loss affinity for TirMEHEC. In contrast, acquiring a new binding towards TirMEPEC for TD4T108R and TD4D116G was at a higher cost for their interaction with the former antigen.For the sake of gaining insight into the changes that endowed TD418 of the VHH gene integrated as single copy in the bacterial chromosome and fused to intimin outer membrane anchor domain for bacterial display, and the resulting library directly screened and characterized for binding to a new antigen target of choice. However, the apparent affinities for the evolved Nbs on the bacterial surface do not exactly match with their binding affinities observed as purified monovalent proteins, likely reflecting complex avidity effects of the multimeric format of the Nb displayed on the bacterial surface and the dimeric nature of the purified antigen (TirMEPEC). Nonetheless, the hereby proposed experimental pipeline delivers a suite of Nbs endowed with a range of affinities for the target that cover different needs. In the specific instance addressed in this work we pursued to expand the therapeutic potential of a specific anti-TirMEHEC Nb that inhibits the attachment of EHEC bacteria to human intestinal tissue towards EPEC strains, which are a major cause of acute and chronic diarrhoea in infants23. But one can envision a number of scenarios which make the workflow of Figs. 39. This can be specially useful when thereby selected Nbs are subsequently bulk-produced in bacteria with the hemolysin export system—also adopted in our experimental roadmap above. While alternative microbial platforms for in vivo generation of Nb repertoires do exist to the same end41 we believe that the presented pipeline is simple and contributes to ongoing efforts towards point-of-care technologies for diagnostics and other medical and environmental needs44.The results above showcase the power of merging surface display of Nbs with genomic site-specific diversification elicited by DIvERGE technology in combination with MACS-based cyclic selection. The CDRs of the encoded Nb can be easily targeted by ssDNA mutagenic recombineeringE. coli strains used in this work can be found in Supplementary Table 45. Purification of TirMEHEC and TirMEPEC was done as described in refs. 22. Protein concentrations were estimated using Protein assay reagent (BioRad) or BCA protein assay (Thermo Fisher Scientific). Whenever required, proteins were analyzed in denaturing polyacrylamide gels.HH sequence. For this, 3 randomized regions of the mutagenic oligonucleotides (Supplementary Table 34. Each mutagenic oligo pool was synthesized on an ABI3900 DNA synthesizer (Applied Biosystems) following a modified protocol based on phosphoramidite chemistry. The synthesis cycles were applied on a controlled pore glass as solid support and were carried out as follows. First, deprotection was done with trichloroacetic acid 3% (w/v) in dichloromethane. Second, incoming phosphoramidite was dissolved in 55 × 10−3 M anhydrous acetonitrile, then mixed with the other three amidites in and subsequently coupled by activation with 5-ethylthio-1H-tetrazole. Third, capping was done with acetic anhydride 10% (v/v) in anhydrous tetrahydrofuran and N-methyl-imidazole 16% (v/v) and pyridine 10% (v/v) in an anhydrous tetrahydrofuran solution. Finally, the oxidation was done with a solution of 5 g of iodine per litre of a mixture of pyridine:water:tetrahydrofuran in the ratios 0.5:2:97.5, respectively. Cycles were repeated until the 90th nucleotide and then, the DNA strands were cleaved from the solid support using concentrated ammonia. Crude oligos were purified by high-performance liquid chromatography (HPLC) on a Shimadzu ADVP-10 HPLC system. HPLC fractions were concentrated and dimethoxytrityl (5 ´-DMTr) protecting groups were removed using a PolyPak column (Glen Research) following the manufacturer’s protocol. Oligonucleotides (Integrated DNA Technologies) were eluted, lyophilized and suspended in TE buffer (1X) pH 8.0. Their sequences are listed in Supplementary Table ssDNAs for recombineering and library diversification were designed to minimize secondary structure, to target the lagging strand of the DNA replication fork and to incorporate 1-2 mutations per CDR1, CDR2 and CDR3 segments of the VE.coli EcM1 bearing pORTMAGE3 (Addgene Plasmid ID #72678) taken from LBa,Km was inoculated into 2 mL LBKm and incubated overnight at 30 °C, 170 rpm. This cell culture was used to prepare a 1:100 dilution in 10 mL LBKm that was then incubated at 30 °C and 170 rpm until reaching 0.4 OD600nm. Next, this was incubated 15 min in a water bath at 42 °C with shaking at 250 rpm and then the flask containing the cells was kept 10 min in ice. The grown cells were harvested by centrifugation at 4000 rpm for 8 min at 4 °C and supernatant discarded. Sedimented cells were washed by resuspension in 10 mL ice-cold water, centrifugation at 4000 rpm for 8 min at 4 °C and supernatant discarded. The sediment of cells from the last wash was suspended in 160 µL of ice cold dH2O and kept on ice. A mix of 3 µL with primers pools tirM1, tirM2 and tirM3  was dissolved in DMSO and immediately added to these protein solutions in a biotin:protein molar ratio of 20:1 and the mix was incubated for 30 min at RT with gentle agitation. Then 1 M Tris-HCl, pH7.5 was added for a final concentration of 50 mM and the mixture was incubated for 1 h on ice. The sample was concentrated to a final volume of 500 µL in a 3 kDa Centricon (Merck) by centrifugation at 5000 g. Finally, the efficiency of biotinylation was checked in a Western blot revealed with POD-anti-biotin .Fractions of the antigens EHEC TirM or EPEC TirM were diluted in 1.5 mL PBS to 1 mg ml8 bacteria were washed by centrifugation  and supernatant discarded and then were suspended in 2 mL of BSA 0.5% in PBS 1X. The same process was repeated again from the beginning. Then, the cells were washed again and suspended in 200 µL of 250 nM of the biotinylated antigen TirMEPEC in a solution of BSA 0.5% in PBS 1X, mixed gently and incubated 1 h at 4 °C. These cells were washed three times, suspended in 80 µL of a solution of BSA 0.5% in PBS 1X, mixed gently with 20 µL of anti-biotin microbeads and incubated for 30 min at 4 °C. Then, the cells were washed three times and suspended in 500 µL of BSA 0.5% in PBS 1X. A column was placed on the magnet and rinsed with 1.5 mL of PBS + BSA 0.5%. Then, the cell suspension was applied onto the column and the flow-through collected. The column was washed three times with 500 µL of PBS + BSA 0.5%, and the three flow through were collected together in the same tube. The column was removed from the separator and placed on a suitable collection tube. Next, 2 mL of PBS + BSA 0.5% were added onto the column. The magnetically labelled cells were immediately flushed out by firmly pushing the plunger into the column.An amount of 10600nm was centrifuged at 4000 g for 2 min at 4 °C followed by 20,000 g for 2 min at 4 °C. The supernatant was discarded and the sediment was washed three times with 1 ml of pre-chilled and filter-sterilized PBS and centrifugation at 4 °C and 4000 g for 2 min followed by 20,000 g for 2 min at 4 °C and supernatant discarding. Then, cells were harvested by centrifugation  and the sediment suspended in 500 µL of ether, 250 nM of biotinylated TirMEPEC in PBS or 250 nM of biotinylated BSA in PBS. Cells were incubated for 90 min at 4 °C, washed three times with PBS as before and labelled with streptavidin-phycoerythrin. The cells were washed with PBS and fluorescence was quantified in a flow cytometer .Cells were grown overnight in 20 mL cultures at 30 °C and 170 rpm. A volume of cells equivalent to 0.25 ODkm and EcM1LuxSATir in a flask with 25 mL of LB. These four flasks were subsequently incubated overnight at 30 °C and 170 rpm. From each culture, a volume corresponding to 1 OD600nm was centrifuged at 4000 g for 2 min at 4 °C followed by 20,000 g for 2 min at 4 °C, the supernatant was discarded and the sediment was washed three times with 1 ml of pre-chilled and filter sterilized PBS and centrifugation at 4 °C and 4000 g for 2 min followed by 20,000 g for 2 min at 4 °C. Then, the cells were harvested by centrifugation  and the sediment suspended in 500 µL of PBS. Next, 9 fractions with 25 µL of this cell suspension were centrifuged at 20,000 g and 4 °C for 2 min, the supernatants discarded and the cells incubated 90 min at 4 °C with a fixed amount of biotinylated antigen and increasing volumes of PBS from 0.25 to 1.6 mL to attain a final concentration range between 0 nM and 250 nM TirMEPEC or 0 nM and 50 nM TirMEHEC, as indicated. After incubation, cells were centrifuged at 4000 g and 4 °C for 2 min followed by 20,000 g at 4 °C for 2 min, then washed twice with 1 mL of filtered and sterilized PBS at 4 °C and labelled with Streptavidin-allophycocyanin. After a final washing step with PBS, the median fluorescence intensity of APC was quantified in a cytometer . Data of median fluorescence intensity relative to maximum median fluorescence intensity obtained from the cytometer were plotted against the concentration of biotinylated antigen to estimate the dissociation constant (KD). Curve was fitted according to non-linear least squares regression method and one site-specific binding saturation kinetics model using the data analysis tool in Prism software (GraphPad).The strains EcM1T108R, EcM1H107Y and EcM1D116G were inoculated in separate flasks with 25 mL of LB® Microbial DNA Isolation Kit . Fractions of the isolated genomes were used as templates for PCR with the primers tirDSF and tirDSR and the obtained amplicons were purified and sent for sequencing with Illumina MiSeq set-up. Isolated genomic DNA from the libraries rMAGE5, rMAGE10, MACS1EHEC, MACS1CR, MACS1EPEC, MACS2EPEC, MACS3EPEC, MACS4EPEC and MACS5EPEC was used as template for a PCR with the primers tirDSF and tirDSR and the resulting amplicons were cleaned NucleoSpin® Gel and PCR Clean-up (Macherey-Nagel) and sent for PacBio circular consensus sequencing (CCS). For screening and for sequence confirmation, colony PCR products of around 1060 bp were obtained with the oligonucleotides CheckSATir_F and CheckSATir_Rev using Q5DNApolymerase (New England Biolabs) and the program as follows: 30 s at 98 °C, 30 cycles  and for final extension 120 s at 72 °C. PCR products were cleaned with the kit NucleoSpin® Gel and PCR Clean-up (Macherey-Nagel) and sent to Macrogen for sequencing with the oligonucleotide CDRI_Fwd. Results were used to estimate the frequency of mutations in the libraries.Bacterial DNA was isolated from fresh overnight cultures of the libraries rMAGE1, rMAGE2, rMAGE3, rMAGE4, rMAGE5 and rMAGE5 control with the kit UltraCleanE. coli CC118. Afterwards, each of the modified plasmids was isolated and confirmed by sequencing using the oligos 7 and 8  and glucose (20%) were sequenced with the oligonucleotides 9 and 10 and, once verified were named pEHLYA_H107Y, pEHLYA_T108R and pEHLYA_D116G, respectively.Changes found in the DNA of the evolved VHH variants were first recreated in the frame of the wild-type TD4 sequence. To this end pCANTAB6_VhhTD4 plasmid  and pEHLYA-derivatives with Vamy, TD4, and mutant Nbs fused to the C-terminal end of hemolysin A, were plated on LB agar with glucose and the proper antibiotics (Ampicillin and Chloramphenicol). One isolated colony of each strain was grown in aerobic conditions overnight at 30 °C in 25 mL of LB with glucose 2% and antibiotics. The resulting culture was used to prepare 200 mL of at 0.05 OD600nm in LB with antibiotics that was grown until 0.4 OD600nm. Following that, IPTG to 10−3 M final concentration was added and the culture was further incubated for 6 h at 30 °C and 100 rpm. Next, the culture was centrifuged 4000 g for 15 min, the supernatant was passed to a new tube and centrifuged 4000 g for 15 min, the supernatant was passed again to a new tube and centrifuged 4000 g for 15 min, supernatant was collected in a new tube and supplemented with PMSF to 10−3 M final concentration. Each volume was supplemented with PBS 10x to a final PBS 1X concentration. Samples containing the thereby secreted [His]6-tagged VHH-HlyA fusions were next passed through Talon CellThru resin (Clontech). For this, 2 ml of this metalloaffinity matrix, was placed in a 15 ml tube and it was subsequently washed by centrifugation 2 min at 700 g, supernatant discarding and suspension in 10 mL of PBS 1X. The resin was washed again, then it was centrifuged 2 min at 700 g, the supernatant was discarded and the resin was added to the suspension containing the secreted Nbs in 50 mL tubes. The mix was incubated in a spinning wheel and, after 30 min, it was added gradually to a 0.7 × 5 cm chromatography column (Bio-Rad). Once packed, this resin was washed in the column by passing gradually up to 30 mL of 5 × 10−3 M imidazole in PBS (1X) through the packed resin. Then, the protein was eluted in 1 mL fractions of 0.15 M imidazole in PBS (1X). Fractions were stored at 4 °C until needed.Expression and purification of his-tagged TirM EPEC and EHEC antigens was conducted exactly as described previously22. Briefly, Maxisorp 96-well immunoplates (Nunc) were coated by adding 50 μL/well of a solution with the antigen EPEC TirM or EHEC TirM at 10 μg mL−1 in PBS (1X) and incubating overnight at 4 °C. Then, blocking was done by discarding the liquid from the plate and adding 200 μL/well of PBS (1X) with skimmed milk (3%) and incubation for 60 min at 23 °C. Next, the liquid in the plate was discarded and the plate washed three times. Each wash consisted in adding 200 μL/well of a solution made of Tween 20 (0.05 %) in PBS (1X), incubation for 5 min at 23 °C and liquid discarding. Then, 50 μL with a Nb in a solution of skimmed milk (3%) in PBS (1X) was added per well in the range of 0–100 nM in the case of TD4 and 0 to 400 nM in the case of H107Y, T108R or H116G. Following incubation was 90 min at 23 °C. Next, the liquid was discarded and the plate washed three times, each consisting in adding 200 μL/well of Tween 20 (0.05 %) in PBS (1X), incubation for 5 min at 23 °C and liquid discarding. Mouse antibody anti-Etag diluted 1:2000 in a solution of skimmed milk (3%) in PBS (1X) was added and the plate was incubated for 60 min at 23 °C. Then, the plate was washed three times. Following that, the anti-mouse-POD antibody was diluted 1:1000 in the solution with skimmed milk (3%) in PBS (1X), the mix was added to the wells and the filled plate was incubated for 60 min at 23 °C. Then, the liquid was discarded and the plate was washed three times. Finally, results were developed as follows. One o-phenylenediamine tablet  and 10 μL hydrogen peroxide (30 % w/v) were solved in 20 mL phosphate-citrate buffer (pH 5) and 80 μL/well was added from the mix. The plate was incubated for 20 min at 23 °C, protected from light and then, the reaction was stopped with 20 μL HCl (3 M)/well. Finally, OD490nm was determined using a microplate reader .ELISA conditions were based on a modified version of a previously described protocolhttps://zhanglab.ccmb.med.umich.edu/I-TASSER/) to obtain the output which included each structural model file. The models of the structural alignments with the parental Nb TD4 were obtained with the function CEalign in PyMOL software. Alignment images were edited in PyMOL and exported as files with the extension png. Next, image cuts to show specific details were done with the software Fiji (https://imagej.net/Fiji).The amino acid sequences of the Nbs (table IV) were introduced as input in FASTA format in i-TASSER online server , the median fluorescence intensity values by cytometry and the absorbance values by ELISA were analyzed to obtain best-fit values, standard errors and 95% confidence intervals (Table In Fig. obodies x0−9 M, thFurther information on research design is available in the Peer Review FileSupplemental InformationDescription of Additional Supplementary FilesSupplementary Data 1Reporting Summary


We have corrected this error.Following publication of the original article , it was The corrected versions of Figs. 2B and 3B are included in this erratum.Figure 2B:Figure 3B:


However, conventional trial‐and‐error strategy significantly slows down the development of SACs in Li–S batteries. Here, the Li2S oxidation processes over MN4@G catalysts are fully explored and energy barrier of Li2S decomposition (Eb) is identified to correlate strongly with three parameters of energy difference between initial and final states of Li2S decomposition, reaction energy of Li2S oxidation and Li—S bond strength. These three parameters can serve as efficient descriptors by which two excellent SACs of MoN4@G and WN4@G are screened which give rise to Eb values of 0.58 and 0.55 eV, respectively, outperforming other analogues in adsorbing LiPs and accelerating the redox kinetics of Li2S. This method can be extended to a wider range of SACs by coupling MN4 moiety with heterostructures and heteroatoms beyond N where WN4@G/TiS2 heterointerface is predicted to exhibit enhanced catalytic performance for Li2S decomposition with Eb of 0.40 eV. This work will help accelerate the process of designing a wider range of efficient catalysts in Li–S batteries and even beyond, e.g. alkali‐ion‐Chalcogen batteries.The sulfur redox kinetics critically matters to superior lithium–sulfur (Li–S) batteries, for which single atom catalysts (SACs) take effect on promoting Li 4@G and WN4@G with the best redox performance are screened. This descriptors‐guided design strategy can be extended to a wider range of SACs, e.g., interface and more complex coordination environments of metal center.Three out of nine parameters identified in this work can serve as efficient descriptors by which two excellent single atom catalysts (SACs) of MoN Despite all the mentioned merits, these cells were found to be restricted to a sequence of obstacles that result in poor battery performance during cycling. Among them, the practical challenge named “shuttle effect” is a primary reason for the recession of Li–S batteries. During the discharge process, soluble long‐chain lithium polysulfide (LiPs), Li2Sn , can diffuse through the separator to the lithium metal anodes and be reduced to the short‐chain LiPs, Li2Sn . These reduction products proliferate back to the cathode and are reoxidized to long‐chain LiPs, which leads to the irreversible capacity fading as well as low Coulombic efficiency.Rechargeable lithium–sulfur (Li–S) batteries are regarded as the ideal next generation energy storage devices for their highenergy‐density and low cost.1, 2 Although carbon materials have been integrated into cathode for hosting sulfur or LiPs, it was demonstrated that the nonpolar pristine carbon manifest weak adsorption strength to polar LiPs, indicating that physical confinement is far from meeting the requirement to suppress the shuttle effect. Therefore, various of materials including metal oxides, sulfides, carbonaceous materials, have been investigated, which exhibit stronger chemical adsorption for LiPs. However, only strengthening the affinity between anchoring materials and LiPs cannot significantly suppress the shuttle effect because of the sluggish kinetics of the conversion of immobilized LiPs and Li2S, thereby rendering the cathode surface unexpected accumulation of LiPs and increasing the chances of LiPs diffusion toward anode. Moreover, the LiPs accumulation would cause unrestrained deposition of Li2S which is difficult to be fully oxidized during charge process due to high energy barrier (Eb) for decomposition, which will lead to the decrease of rate capability. Therefore, increasing research attention has recently been paid to the acceleration of conversion kinetics, in which the transformation from insoluble of Li2S to long‐chain LiPs was highlighted.To address these issues and, particularly, avoid the deleterious shuttle effect, significant research efforts have been dedicated to searching for anchoring materials which are efficient in the immobilization of soluble intermediates. There are two main methodologies, namely physical confinement and chemical adsorption.12 Alt4@G is the most considered model of SACs with metal coordinated by four N atoms embedded in graphene, which also inspired recent research efforts on their applications as sulfur hosts of Li–S batteries. The employment of SACs as anchoring materials can not only immobilize soluble intermediates but also accelerate redox kinetics of Li2S deposition. However, the mechanism of Li2S decomposition catalyzed by SACs is far less understood and thus the key parameters affecting the catalytic performance of SACs remain unclear. Therefore, the design of SACs for Li–S batteries relies solely on the conventional trial‐and‐error method. Given numerous degrees of freedom for adjusting the local atomic environments, the catalytic properties of SACs can be tuned accordingly in a wide range, which necessitate more advanced and efficient design strategy. High throughput calculations have been widely implemented to search for high efficiency SACs for various electrochemical reactions such as oxygen reduction reaction (ORR), oxygen evolution reaction (OER), hydrogen evolution reaction (HER), and N2 reduction reaction (NRR). Therefore, this could be an optimal strategy as well to screen SACs as high performance of anchoring materials for Li2S decomposition in Li–S batteries. To this regard, suitable descriptors are critical for screening high performance SACs for Li2S decomposition which enables exploring efficiently the configurational phase space of SACs. Although Eb of Li2S decomposition could be utilized as a descriptor, direct calculation of Eb consumes a large number of computational resources using the climbing‐image nudged elastic band (CI‐NEB) method, which hinders significantly the implementation of high throughput calculations. Therefore, to search for simple and easily obtained descriptors correlating well with Eb of Li2S dissociation is essential but challenging.Single atom catalysts (SACs) have been proved efficient for catalyzing various of electrochemical reactions, which accordingly attracted much research attention on their applications in Li–S batteries. Up to now, MNatteries.26 The2S oxidation and accordingly identify the key parameters strongly correlated with the Eb of Li2S decomposition. The relationship of Eb and nine key parameters were extensively investigated, and we identified three of them possessing the strongest correlation with Eb which can serve as descriptors to screen SACs with excellent catalytic performance for accelerating Li2S oxidation from 30 systems. This descriptor‐based strategy was also extended to other catalysts containing structural features other than MN4 moiety.Herein, a series of graphene‐supported SACs as anchoring materials for Li–S batteries were systematically investigated, using density functional theory (DFT) simulations, to deepen the understanding of the mechanism underlying Li2Figure 4@G were modeled by depositing transition metal (TM) atoms onto the nitrogen doped graphene. The average bond lengths of TM—N bonds are listed in Table As shown in 2S on MN4@G were considered and only the most stable adsorption patterns are shown in Figure 4@G and TaN4@G as examples for illustrating the Li2S adsorption the Li atoms are adsorbed at the hollow sites and the S atom tends to be adsorbed near the metal atom. As presented in Figure 2S are lower than −2.0 eV in most of the cases, exhibiting a stronger affinity to Li2S than that on graphene and consistent with the previous works. The strongest binding strength toward Li2S was found on TaN4@G, with the Eads value of −5.15 eV and the weakest interaction took place on AuN4@G (Eads = −1.23 eV). Group IIIB, IVB, VB, and VIB elements exhibit ultrastrong adsorption strength to Li2S , while VIII and IB elements show poor adsorption properties. The Eads of Li2S on VN4@G, MnN4@G, FeN4@G, CoN4@G, NiN4@G, and ZnN4@G are −3.93, −2.61, −2.81, −2.53, −1.47, and −2.53 eV, respectively, which is in good agreement with the previous report. The substrates of AgN4@G with Li2S adsorption and HgN4@G show significant structural distortions , calculated by the corresponding energy integral below Fermi level (Ef)  of 0.91. The ICOHP values of Li–S interaction on VN4@G, NbN4@G, and TaN4@G surfaces are −3.71, −3.30, and −3.36, which are larger than those for their analogues demonstrating weaker chemical bonds of Li–S and lower Eb values in the former three SACs. Note that, the data point for VN4@G is a statistical outlier because it possesses lower ICOHP than that for NbN4@G and TaN4@G but gives rise to smaller Eb. This discrepancy could be attributed to the additional substrate affinity to Li atoms  Figure , were usEb. We considered the energy difference of initial and final states of Li2S decomposition (ΔE) essential for calculating Eb based on CI‐NEB method, which can be defined as: ΔE = E(*Li+*LiS) − E(*Li2S), where E(*Li+*LiS) and E(*Li2S) represent the energies of final and initial states as illustrated in Figure Eb = 1.07 × ΔE + 0.16 (ΔE > 0 eV), with the remarkable R2 of 0.98, signaling that there is an extremely strong positive correlation between ΔE and Eb. To this end, ΔE could be used as an effective adsorption descriptor, with which we can precisely estimate the Eb values for evaluating the catalytic performance of SACs toward Li2S oxidation.We next investigated several simple parameters and explored their correlations with 2S decomposition on the surface of the catalyst. As the first step of charging process in Li–S batteries, decomposition of Li2S could also be described by *Li2S → *LiS + Li+ + e− where Li+ is isolated in vacuum instead of being adsorbed on SACs, which is different from the process aforementioned for calculating Eb based on CI‐NEB method. To calculate reaction energy is a prevailing method to evaluate the performance of catalysts for electrochemical reactions. Thereby we here introduced reaction energy ΔE(*LiS) = E(*LiS) + E(Li)bcc − E(*Li2S) as a possible adsorption descriptor for Li2S oxidation, where E(*LiS), E(Li)bcc, E(*Li2S) refer to the total energy of adsorbed LiS cluster, energy of single Li atom in bcc bulk phase and total energy of adsorbed Li2S, respectively. Using the scaling relationships, Eb as a function of ΔE(*LiS) was plotted in Figure Eb = 0.87 × ΔE(*LiS) + 0.38 with R2 = 0.92.To a significant degree the catalytic performance of a metal is determined by the affinity strength of the reaction intermediates of Licatalyst.41 As  and CO2 reduction, we explored the possibility of adsorption energy of LiS (Eads (LiS)) serving as an additional descriptor for Li2S oxidation. Using the scaling relations, Eb as a function of Eads (LiS) was investigated , it still resonates roughly with the results from ICOHP, ΔE, and E(*LiS)  Figure  in partiφ having all parameters related to intrinsic properties of catalytic center, which can be expressed as φ = θ × (χM + αnNχN)/χS, containing the information of valence electrons in the occupied d orbit of TM atom (θ) and the electronegativity of its ligand atoms (χ) has been proposed to design electrocatalysts toward ORR, OER, and HER. We set up φ in the same way, where a is set to 1, and nN, χN, and χS are the number of nearest‐neighbor nitrogen atoms, electronegativity of nitrogen and sulfur, respectively. Its correlation with Eb was explored. As can be seen from Figure φ and Eb is not so remarkable as represented by a low R2 value of 0.78. This might be caused by the fact that φ does not precisely describe the intrinsic properties of SACs. As we found that φ is greatly determined by the periodic law, reflected by θ and χM, and the electronegativity of N has a very limited modification to the goodness of fit of φ‐Eb relationship . These results generally indicate that VN4@G, TaN4@G, and NbN4@G substrates could weaken the bonding of Li and S atoms, which is known as adsorption activation. In contrast, Li2S adhere to NiN4@G, PdN4@G, PtN4@G, and AuN4@G can maintain the structure similar to the pristine one indicating their worse catalytic performance than TaN4@G and NbN4@G for the Li2S decomposition. More details of bond length and bond angles of adsorbed Li2S are shown in Table  was performed to investigate the adsorption induced charge variations of Li2S  and Li atom , respectively. In particular, the adsorption induced charge difference of Li in Li2S varies roughly linearly with respect to Eb, indicating that the activation of Li atom is more beneficial to the decomposition of Li2S.We also explored the correlation of 2S decomposition and extracted the relationships of Eb and nine parameters , and ICOHP can better correlate with Eb than Eads (LiS), φ, and other parameters evidencing by their better goodness of fit, which could serve as descriptors for fast screening.Overall, based on the 14 randomly selected systems of SACs we gained in‐depth understanding the mechanism of LiE, ΔE(*LiS), and ICOHP to explore their predictive capability for estimating Eb and further to screen high performance SACs for Li2S decomposition. Before that we first validated the prediction capability of ΔE using the available data from the literature about the decomposition barriers of Li2S over other SACs anchored on N doped graphene and pristine C2N. As shown in Figure E−Eb linear relationship. This gives us confidence that this descriptor of ΔE can provide efficient theoretical guidance for fast screening active SACs for Li–S batteries. Thus, the descriptor of ΔE was utilized to evaluate the catalytic performance of all SACs systems by calculating Eb values based on the equation of Eb = 1.07 × ΔE + 0.16 (ΔE > 0) as shown in Figure We focus mainly on three descriptors of ΔEb < 0.85 eV) are found to be WN4@G, MoN4@G, NbN4@G, TaN4@G, VN4@G, and ReN4@G which comprise middle transition metal atoms having both partially occupied d orbitals and enough d electrons. In particular, SACs of Mo and W on N4@G are predicted to be the potential catalysts with the highest catalytic activity, which even have lower Eb than VN4@G benchmark. This means that among 30 SACs MoN4@G and WN4@G could catalyze Li2S oxidation with the fastest kinetics. Their low Eb values could also be reflected by the variation in bond length and the Bader charge of Li atom upon adsorption where the most significant change occurs for Li2S on WN4@G and MoN4@G, as shown in Figure Eb for WN4@G, MoN4@G directly calculated using CI‐NEB method  and ICOHP could reproduce the trend of Eb, as shown in Figure E indicating they are also good descriptors for screening SACs although the Eb values predicted by ICOHP and ΔE(*LiS) have bigger discrepancies than by ΔE with respect to those given by CI‐NEB method as listed in Table 4@G predicted by ICOHP, which has been aforementioned that this big discrepancy is assigned to the relatively strong interaction between Li atoms and substrate. Otherwise, the predictive results of Eads(LiS), φ and other parameters related to adsorbed Li2S are significantly different from those predicted by ΔE, ΔE(*LiS), and ICOHP as illustrated in Figure Eb.ΔEb = 1.07 × ΔE + 0.16 (ΔE > 0), Eb = 0.87 × ΔE(*LiS) + 0.38, and Eb = −1.30 × ICOHP −3.49, one can expect SACs with MN4 moiety possessing better catalytic performance than WN4@G and MoN4@G for catalyzing Li2S oxidation should have ΔE < 0.36 eV, ΔE(*LiS) < 2.22 eV, or ICOHP > −3.11. This needs further investigations.Based on the equations of 4@G and WN4@G as anchoring materials in Li–S batteries, we then explored their stability to clarify the experimental feasibility of synthesis. We found that these three SACs are more energetically stable embedded in the center of N4@G moiety than anchored on the nearby graphene sites, which is shown in Figure 4@G, WN4@G, and VN4@G were investigated in Figure 4@G was set as a benchmark in this study, which was treated as the optimum catalyst in previous work.Before further investigating the performances of MoNous work.35Eads of LiPs on MoN4@G, WN4@G, and VN4@G are almost less than −3.0 eV, showing strong adsorption strength. Taking the adsorption of Li2S6 as an example, the Eads are −3.14, −4.29, and −2.96 eV on MoN4@G, WN4@G, and VN4@G, respectively. Previous work demonstrates that strong affinity strength to LiPs can effectively mitigate the shuttle effect. It can be seen that the Eads of all intermediates on MoN4@G and WN4@G are lower than that on VN4@G benchmark, exhibiting a stronger adsorption strength, which suggests MoN4@G and WN4@G are potential optimum anchoring materials for entrapping LiPs.DFT calculations were then performed to examine the adsorption performance of those catalysts for all of the reaction intermediates during the redox of sulfur. As demonstrated in Figure 8 to Li2S were calculated and exhibited in Figure 2S8 to Li2S is accompanied by the production of isolated state Li2S2 (Equation (3)–(8) in the Supporting Information). The overall free energy changes are −0.41, −0.33, and −0.82 eV for MoN4@G, WN4@G, and VN4@G, respectively, suggesting that the overall reaction on those substrates are exothermic. Whereas, for individual lithiation step, the conversion from S8 to Li2S4 are exothermic and the subsequent two steps are endothermic. Obviously, the reduction from Li2S2 to Li2S possess the largest positive change of Gibbs free energy, indicating that the conversion from Li2S2 to Li2S is the rate‐determining step (RDS) in the entire lithiation process. The reduction kinetics can be characterized by the change of Gibbs free energy of RDS (ΔGRDS). As can be seen from Figure 4@G (ΔGRDS = 4.25 eV) and WN4@G (ΔGRDS = 4.36 eV) exhibiting similar efficiency as that on VN4@G benchmark (ΔGRDS = 4.24 eV). Given the excellent catalytic performance for Li2S oxidation, WN4@G and MoN4@G exhibit remarkable performance of adsorbing LiPs, accelerating reduction kinetics of Li2S, which will significantly suppress the shuttle effect and improve the rate performance of Li–S batteries and in particular will be expected to perform better than ever reported VN4@G. This deserves further experimental investigations.To gain further insights into the kinetics of lithiation process, the Gibbs free energy profiles for discharge reactions from Srmation).49 The2S is critical for suppressing shuttle effect and thus a superior catalyst should keep balance between both oxidation and reduction processes. To confirm the overall catalytic performances of those screened VN4@G, MoN4@G, and WN4@G, we further explored the redox processes of other MN4@G  which possess ΔE < 1.0 eV. As shown in Figures 2S2 → Li2S, consistent with previous work. When ΔE < 1.0 eV, we now have 10 SACs whose performances are compared based on the values of ΔGRDS and Eb summarized in Table Eb undergoes significant change by 108% from WN4@G (Eb = 0.55 eV) to TcN4@G (Eb = 1.15 eV) indicating a pronounced diversity of catalytic performances of those SACs, while ΔGRDS varies from ZrN4@G (ΔGRDS = 3.83 eV) to TaN4@G (ΔGRDS = 5.36 eV) by 40% maximum. Except for Ta, other SACs exhibit comparable capability for reduction process due to similar values of ΔGRDS. We found SACs (Mo/W/Re/NbN4@G) are as active as VN4@G for both reduction and oxidation of Li2S. Among them, MoN4@G and WN4@G with superior activity for Li2S oxidization deserve further investigation in experiment.Conversion of LiPs and Lious work.50 WheE, ΔE(*LiS), and ICOHP can serve as distinguished descriptors for rationally designing potential catalysts of MN4@G catalyzing Li2S decomposition and next we extended this strategy to predict the catalytic performance of various types of 2D materials with dispersed atomic metals for Li2S decomposition. Recently, various strategies have been proposed to enhance the catalytic performances of SACs toward electrochemical reactions, which involve heterostructures and heteroatoms dopants other than N to tailor the local atomic environments. Therefore, we used ΔE, ΔE(*LiS), and ICOHP to search for enhanced catalytic performance compared to WN4@G and MoN4@G by introducing interface effect and S dopant to modulate electronic properties of active metal center. We constructed three typical models as representatives shown in Figure 4@G/TiS2 and WN4@G/G) and one MN4@G with tailored local environment (MoN4S2@G). WN4@G/TiS2 and WN4@G/G are built by introducing monolayer of TiS2 and graphene, respectively, while MoN4S2@G is realized by doping two sulfur atoms into the second coordination sphere of TM atom. Based on the fitted equation of Eb = 1.07 × ΔE + 0.16, the estimated Eb values of Li2S for WN4@G/TiS2, WN4@G/G, and MoN4S2@G are 0.43, 0.72, and 0.90 eV, respectively. Amazingly, additional CI‐NEB calculations give rise to Eb of 0.40, 0.83, and 0.91 eV in the given order, confirming the prediction of descriptor ΔE, as shown in Figure Eb values (Table E(*LiS) one can obtain the largest deviation with those from CI‐NEB method in particular for the case of WN4@G/TiS2. Moreover, the trend of Eb given by CI‐NEB method could not be reproduced by ΔE(*LiS). Therefore, ΔE(*LiS)‐Eb relationship needs to be modified for specific ligand environment which needs further investigation.We revealed that Δtructures53 and4@G/TiS2 possesses the merits of its individual components which will behave better than TiS2 or WN4@G in suppressing shuttle effect since WN4@G/TiS2 shows superior catalytic performance outperforming WN4@G monolayer. All these results not only evidence that the incorporation of monolayer TiS2 could further reduce the energy barrier of Li2S decomposition over WN4@G but also confirm the rationality that ΔE and ICOHP can be extended to wider range of catalysts for Li2S oxidization. Generally, these two parameters could guide further investigations for rationally designing SACs with heterostructures and/or tailored MNx moiety. Any other catalysts having configurations like MN4@G/TiS2 and MN4S2@G are expected to have better catalytic performance for Li2S decomposition when their ΔE < 0.36 eV or ICOHP > −3.11. In addition, we believe this screening strategy could also be extended to screen catalyst in alkali‐ion‐Chalcogen batteries.Heterostructure of WN32S oxidation catalyzed by graphene supported SACs and identified three out of nine parameters correlating strongly with the energy barrier (Eb) of Li2S dissociation, which are bond strength , energy difference of initial and final states of Li2S decomposition (ΔE) and reaction energy of Li2S decomposition (E(*LiS)). The equations describing the relationships of Eb, ΔE, and ICOHP are Eb = 1.07 × ΔE + 0.16 (ΔE > 0), Eb = 0.87 × ΔE(*LiS) + 0.38, and Eb = −1.30 × ICOHP −3.49, respectively. Under the guidance of descriptors ΔE, ΔE(*LiS), and ICOHP, MoN4@G, and WN4@G were screened out as optimal catalysts for catalyzing Li2S oxidation with the Eb values of 0.58 and 0.55 eV, as well as competitive trapping capability and reduction activity, indicating they can serve as superior performance anchoring materials in Li–S batteries. More importantly, the efficient descriptors ΔE and ICOHP could be extended to predict the decomposition barrier of Li2S on various types of catalysts containing dispersed TM atom centers, such as heterointerface and SACs of MN4@G with more complex local environment. Heterointerface of WN4@G/TiS2 exhibit promising catalytic performance for Li2S decomposition outperforming WN4@G by giving rise to Eb of 0.40 eV. We believe that these efficient descriptors of ΔE, E(*LiS), and ICOHP could help in fast screening and designing wider range of electrochemical catalysts in Li–S batteries and even other alkali‐ion‐Chalcogen batteries.In summary, we systematically investigated the mechanism of LiThe authors declare no conflict of interest.Supporting informationClick here for additional data file.