Withania somnifera Dunal (ashwagandha) and AYUSH-64 have been used for the prevention and management of COVID-19 in India. The present study explores the effect of Ashwagandha and AYUSH-64 on important human CYP enzymes  to assess their interaction with remdesivir, a drug used for COVID-19 management during the second wave. The study also implies possible herb−drug interactions as ashwagandha and AYUSH-64 are being used for managing various pathological conditions. Aqueous extracts of ashwagandha and AYUSH-64 were characterized using LC-MS/MS. A total of 11 and 24 phytoconstituents were identified putatively from ashwagandha and AYUSH-64 extracts, respectively. In addition, in silico studies revealed good ADME properties of most of the phytoconstituents of these herbal drugs and suggested that some of these might possess CYP-450 inhibitory activity. In vitro CYP-450 studies with human liver microsomes showed moderate inhibition of CYP3A4, 2C8, and 2D6 by remdesivir, while ashwagandha had no inhibitory effect alone or in combination with remdesivir. AYUSH-64 also exhibited a similar trend; however, a moderate inhibitory effect on CYP2C8 was noticed. Thus, ashwagandha seems to be safe to co-administer with the substrates of CYP3A4, CYP2C8, and CYP2D6. However, caution is warranted in prescribing AYUSH-64 along with CYP2C8 substrate drugs. Furthermore, preclinical and clinical PK studies would be helpful for their effective and safer use in the management of various ailments along with other drugs.Ayurvedic medicines   The modern system of medicine has emerged as the primary choice for the treatment of most of the health-related issues . On the The World Health Organization (WHO) data and existing literature show that more than 70–80% world population use CAM for their healthcare needs. Particularly, in Western countries, CAM has become increasingly popular over the last few decades . The heaAshwagandha is a well-known Rasayana (∼rejuvenator) also referred to as “Indian ginseng” and is well known for its immunomodulatory, adaptogenic, anti-cancer, anti-diabetic, and anti-COVID-19 activities , Katuki roots (Picrorhiza kurroa Royle ex. Benth), Kiratatikta whole-plant (Swertia Chirata Pexbex. Karst), and Kuberaksha seed . AYUSH-64 formulation has been known for its anti-malarial activity since 1994  parameters were assessed with the known phytoconstituents from both the herbal formulations. In addition, we also used remdesivir  as a representative example for HDI studies. Remdesivir, a ProTide (a prodrug of nucleotide), is diffused into the cells and is converted to a stable metabolite GS-441524 via an intermediate GS704277  and AYUSH-64 (Batch No.: 89–1/2020-CCRAS/Admn) were prepared in a GMP-certified facility and were provided by the National Medicinal Plants Board (NMPB), Ministry of AYUSH , Government of India. The aqueous extract of ashwagandha was prepared as per the Ayurvedic procedure given in classical texts .Testosterone, ketoconazole, 6β-hydroxy testosterone, paclitaxel, rosiglitazone, remdesivir, NADPH, 6-hydroxy paclitaxel, dextromethorphan HBr, dextrorphan, quinidine, NADPH, and mixed gender HLM  were used in this study. Water, methanol, and acetonitrile of LC-MS grade were obtained from Merck Life Science, Pvt. Ltd, India. The aqueous extracts of al texts  and the ashwagandha and AYUSH-64 were reconstituted (10 mg/ml) in 100% methanol followed by sonication for 30 min. The sample was filtered through a 0.2-mm filter and analyzed with UHPLC Ultimate 3000 coupled with an Orbitrap mass analyzer. The HSS-T3 C18 column  was used for chromatographic separations, and the column oven temperature was maintained at 40°C. Mobile phases A and B contain water with 0.1% formic acid and acetonitrile with 0.1% formic acid, respectively. The gradient elution started with 1% B to 95% B over 14 min , paclitaxel (5 µM), and dextromethorphan (5 µM) are used as probe substrates for CYP3A4, 2C8, and 2D6, respectively. The marker metabolites 6β-hydroxy testosterone, 6-hydroxy paclitaxel, and dextrorphan were used for measuring the enzyme activity. The incubation mixture was prepared by gentle mixing of PBS , HLM (20 mg/ml), and DMSO. The amount of PBS, HLM, and DMSO in the incubation mixture was 98.57%, 1.2%, and 0.23% for CYP3A4 and CYP2D6, while for CYP2C8 it was 98.87%, 0.9%, and 0.23%, respectively. For each isozyme, an aliquot of 99.5 µ of the incubation mixture (containing HLM and the substrate) was spiked with 0.5 µl of investigational herbal drug/positive control/remdesivir working solution in a microcentrifuge tube and mixed by swirling and mild shaking. Thereafter, 10 µl of 10 mM NADPH was added to initiate the reaction and incubated additionally for 10 min at 37°C. The reactions were stopped with the addition of 400 µl of quenching solution (%100 methanol) . The samashwagandha and AYUSH-64) were prepared in seven different concentrations: 1, 10, 20, 50, 100, 1,000, and 2,000 μg/ml. The solutions were centrifuged at 10,000 rpm for 30 min, and the supernatant collected was used for further analysis. The concentration of positive controls ranged from 0.0005 to 5.00 µM, 1.00 to 0.01 µM, and 5.00 to 0.0005 µM for ketoconazole (CYP3A4) (The stock solution (500 mg/ml) of the investigational herbal drugs was prepared in water and kept overnight in a mechanical shaker at a speed of 200 rpm at 37°C. The investigational herbal drugs ((CYP3A4) , rosigli(CYP3A4) , and qui(CYP3A4) , respect(CYP3A4) .The marker metabolites  were formed as a result of the metabolism of substrates  by CYP3A4, CYP2D6, and CYP2C8, respectively, and measured using LC-MS/MS. Water with 0.1% formic acid was used as the mobile phase A and methanol and 0.1% formic acid as mobile phase B. The run time of 14 min with a flow rate of 0.300 ml/min was used. The chromatography separation method employed was similar to the conditions mentioned in A Thermo Scientific Xcalibur system was used for data processing and data analysis. The standard metabolite retention time and MS/MS fragmentation were matched with samples. The respective metabolite concentration was monitored accordingly under different experimental conditions .ashwagandha and AYUSH-64) were selected on the basis of the daily maximum human dose when diluted in 1 L of gastrointestinal fluid followed by their distribution in 56 L of total body fluid   activity in HLM was calculated graphically by the nonlinear regression analysis of logarithmic inhibitor concentration (log conc.) versus % of the inhibitory activity plot using GraphPad Prism 5. The data were expressed as mean ± standard deviation. The ratio of I/Ki was used to predict plausible clinical interactions, and >0.1 was considered for the possible clinical interaction. [I] is the mean maximum surrogate plasma concentration (Cmax) at the steady state after administration of the highest clinical dose of inhibitor in humans as per published reports. The inhibitory constant (Ki) was calculated using the following equation:ICm] are substrate concentration and Michaelis constant, respectively  are helpful in deciding to use the test drugs and their putative PK-PD approaches  equals to Km (for IC50 determinations) and spanned Km (for Ki determinations); 2) the substrate incubated for a short time (5 min) to minimize metabolism-dependent inhibition and inhibitor depletion; and 3) the concentration of HLM was low (0.1 mg/ml or less) to maximize the unbound fraction of the inhibitor. Under these conditions, predicted Ki values, based on IC50/2, correlated with the experimentally determined Ki (b) It is further supported by the recent publication on the reliability of estimating K0 values . It was mined Ki .where [S] and [Kectively . The intics data . Insteadble HDIs . The ratashwagandha and AYUSH-64  have shown that the aqueous extracts prepared as per the procedure given in Ayurvedic texts did not show inhibitory effects  absorption  were quantified (S1 File). In the present study, the I/Ki ratio of ketoconazole (CYP3A4 inhibitor) was 28.73, while for remdesivir it was 6.87, which showed moderate/weak inhibition of CYP3A4 as compared to ketoconazole. The results were consistent with the previous reports  on the CYP2C8 isoenzyme system, while AYUSH-64 displayed moderate to weak inhibition as IC50 was <100 μg/ml. In addition, AYUSH-64 in combination with remdesivir (IC50 = 85.21 ± 18.3) also showed a similar trend  also referred to as “Indian ginseng,” is well established for its immunomodulatory, adaptogenic, anti-cancer, anti-diabetic, and anti-COVID-19 activities . However, in vitro experiments with human liver microsomes exhibited IC50 values above 100 μg/ml, rendering that the extract is safe for use. The present study thus predicts the safety of the tested herbal extracts and their use with other conventional drugs for the management of COVID-19. We, however, did not examine the time-dependent change in the CYP activity, and thus we do not completely rule out their interaction with CYP protein or modulation in their expression. Future in vitro studies with individual phytoconstituents, in vivo studies, and clinical HDI studies might therefore be helpful in assessing their safety.The COVID-19 . Moreovei ratio of remdesivir for CYP3A4, 2D6, and 2C8 was 38.07, 27.04, and 16.62, respectively, which is similar to the existing reports . Thus, both AYUSH-64 and remdesivir had weak or moderate interaction with CYP2C8, and the IC50 of the combination was 85.21 ± 18.3. Indeed, it is possible that these two together might compete with each other and exhibit moderate inhibition of CYP2C8 activity. Interestingly, there was no mention of any significant interaction or adverse reaction during the clinical studies conducted with ashwagandha and AYUSH-64  alone and in combination with remdesivir. The results obtained signify that AYUSH-64 had no inhibitory interaction with CYP3A4 and CYP2D6 as ICAYUSH-64 . The HDIashwagandha have been in human use for a long period, and so far, no serious adverse effects have been recorded. The bioactive compounds of ashwagandha and AYUSH-64 are known to elicit immunomodulatory activity and possess plausible anti-viral properties. However, to accrue the potential benefit of using the integrative approach for managing complex diseases, it is pertinent to have a proper scientific assessment of the herb−drug interactions. Limited in vitro CYP inhibition studies in the present investigation with ashwagandha and remdesivir demonstrated probable safety of the combination. Moreover, AYUSH-64 did not exhibit an inhibitory effect on CYP3A4 and 2D6, while a moderate inhibition of CYP2C8 was observed, warranting a degree of caution for integrative pharmacotherapeutic management. Well-designed experimental and clinical studies would be more helpful to further demonstrate their safety.Overall, the use of Ayurvedic formulations as an adjuvant with conventional therapy seems to have additional benefits in the management of chronic diseases and COVID-19. AYUSH-64 and


In the study of deformation prediction and disaster warning during karst slope construction, the influencing factors and deformation law should be comprehensively considered. The layout of the deformation monitoring points of karst slope is affected by the thickness of karst overburden soil, dissolution and fragmentation degree, karst development degree, slope cracking degree, fault or weak interlayer and other factors. In this paper, the author aimed at the problem of construction deformation prediction and disaster warning of karst slope, proposed an improved model of cusp mutation by applying and optimizing the cusp mutation model, analysed the deformation trend and sudden change type of the slope, and obtained the critical control early warning value of slope deformation. Therefore, it is feasible to analyse the deformation and mutation characteristics of karstified slope by using a virtual reality-mutation model. In addition, based on the empirical formula of the slope sliding limit deformation rate and grey prediction model, the critical control warning value of slope deformation is obtained, which provides a basis to quantify the deformation index of risk evaluation. This method provides a new idea to predict karst slope construction deformation and catastrophic deformation warning and has a reference value for similar engineering examples. Karst is a general term for the geological processes where water acts on soluble rocks  mainly via chemical dissolution. This phenomenon is supplemented by mechanical processes such as erosion, latent erosion and collapse of flowing water, and the phenomena produced by these processes. In recent years, construction deformation of karst slope has a concern. On one hand, during the construction period of the cutting slope, the construction disturbance destroys the original balance state of the mountain, makes the slope produce excessive deformation and even induces the slope to become unstable. On the other hand, the mountains in karst areas have been corroded for a long time, and the internal structure is mostly damaged, which results in low strength and poor stability of the rock mass structure. This makes the deformation prediction of karst slope during construction more complicated and catastrophic deformation warnings more difficult. At this time, the deformation of karst slope during construction is particularly important, especially the deformation prediction and catastrophic deformation warning of karst slope during construction.6 to promote theoretical development and provide technical guidance for engineering. Studies on this issue are mainly divided into three categories: karst slope catastrophe mechanism15, karst slope stability evaluation20, and early warning research based on slope monitoring data23.To comprehensively study the deformation prediction and catastrophic deformation warning of karst slope during construction, scholars at home and abroad have performed many studies7 analysed the deformation and failure mechanism of three typical types of mixed soil–rock slopes, revealed the typical disaster modes, and proposed disaster control measures in the entire process. Liu et al.8 used the geological historical process mechanism analysis method and numerical simulation method to study the deformation and failure mechanism of a layered gently inclined hornstone slope in the karst area of Guizhou, and four types of deformation and failure mechanisms were summarized. Xiong et al.9 studied the engineering geological conditions of the steep cliff of Shenxiu Cave and used FLAC3D for numerical simulation to study the layered rock foundation pit slope. Aiming at a high fill project at the southern end of Chongqing Wulong Civil Airport, Xiong et al.10 analysed the displacement distribution characteristics and deformation development law of fillers, dam bodies and dissolved chambers and selected the optimal karst treatment scheme by comparison. Shi11 systematically summarized the karst distribution law, development characteristics and instability mechanism of the roadbed section of the Shanghai-Kunming High-speed Railway (Guizhou section). Liang Tang took the drink horse pond landslide as the main research object and analysed its deformation failure mode. The use of 3DEC simulation reproduced the landslide deformation and failure and movement process12. Based on the research and data collection, Hua Huang thoroughly studied the North Slope of the Xianfeng opencast coal mine deformation destruction mechanisms and stability evaluation13. Baiping Song concluded with geological properties, potential dangers, hazard causes and analysed the geological inducing and damages in geologically complicated Yunnan. Meanwhile, the inducing mechanism and inducing chains for highway engineering geological environment are provided in the Paper14. He15 took the Wangxia unstable rocks on the WU Gorge bank slope as an example and studied the formation and failure mechanism, lasting effect and hazard assessment.Research on karst slope disaster mechanism mainly focuses on the displacement change rule, deformation failure mode, deformation failure mechanism, instability mechanism, etc. Xu et al.16 took the Changsha Ice & Snow World mine project as an example to study the dynamic stability of steep rocky high slope by combining on-site measurement and indoor numerical analysis. Based on the background of a building foundation pit slope engineering project in Chenzhou city, Dajian Li established the model of experimental analysis based on the two typical karst cave sections in the slope. Through model tests and numerical simulations, the deformation and stability characteristics of building slope containing karst caves have been studied17. Based on the background of karst development in Guangxi, Zhao18 analysed the karst problems that affected the stability of foundation pit slope. Based on the concrete project, Chen19 analysed the stability of the high reinforcement slope, pile anchor slope and karst foundation using theoretical, FlAC3D numerical and field monitoring.Research on karst slope stability evaluation mainly focuses on influencing factors, stability analysis, stability evaluation, dynamic stability, etc. Liu et al.23 mainly focuses on numerical simulation calculations, prediction models, the selection of calculation parameters, etc. Gao et al.21 addressed the problem of the fuzzy selection basis of dual-structure slope simulation calculation parameters and the complicated force and deformation of the slope body in the supporting process. Gao et al.22 studied the problem of the loss of anchor cable prestress over time in a soil–rock dual-structure slope. Gao et al.23 proposed and used the deep displacement monitoring data and the p value test method to check the simulation parameters.Early warning research based on slope monitoring dataThe above studies involve the catastrophe mechanism of karst slopes and stability evaluation of karst slopes, but relatively few studies have both. In addition, most of the above studies involve monitoring data processing, model testing and numerical simulation, but there is less theoretical research.23, prove that the proposed virtual reality-mutation model in this paper and the critical control warning value of catastrophic deformation are scientific, reasonable and effective. This method provides a new idea to predict karst slope construction deformation and catastrophic deformation warning and has a certain reference value for similar engineering examples.Based on the application and optimization improvement of mutation theory and the cusp mutation model in slope deformation, the virtual reality grey trend-cusp mutation model  is proposed in this paper. This model is applied to the Damao slope, and the study shows that the mutation effect can be transformed between two deformation trends or states, including convergence–divergency mutation type (C–D), divergence–convergence mutation type (D–C), convergence–convergence mutation type (C–C), and divergence–divergency mutation type (D–D). Thus, the main deformation and mutation forms of the Damao slope are the C–D mutation type, D–C mutation type and C–C mutation type. Among them, the C–D mutation type greatly affects the deformation state of the Damao slope and easily causes its cracking and failure. Therefore, it is feasible to analyse the deformation and mutation characteristics of karstified slope using the virtual reality-mutation model. In addition, based on the empirical formula of the slope sliding limit deformation rate and grey prediction model, the critical control warning value of slope deformation is obtained, which provides a basis to quantify the deformation index of risk evaluation. Theoretical analysis, field data processing, and early warning research based on slope monitoring dataThe Damao slope is located in guiweng (Guiyang to Weng'an) highway in central Guizhou, which belongs to the middle and low mountain karst landform, with limestone as the main formation lithology and 310°∠16° occurrence. The original mountain natural slope is 13°–32° with large fluctuations. The surface of the mountain is mostly covered by a karstic overburden soil layer with uneven thickness. Vegetation grows at the top of the mountain, and there are small terraced fields on both sides of the mountain. A small amount of rock mass is exposed on the local surface of the mountain, and most of them show corrosion characteristics, as shown in Fig. The layout of the deformation monitoring points of karst slope is affected by the thickness of karst overburden soil, dissolution and fragmentation degree, karst development degree, slope cracking degree, fault or weak interlayer and other factors. In the study of deformation prediction and disaster warning during karst slope construction, the influencing factors and deformation law should be comprehensively considered.The selection and arrangement of monitoring points for karst slope should be representative and can accurately reflect the deformation of karst slope. The author determines the monitoring point location according to the engineering analogy method and expert experience method. Moreover, in areas with strong karst development and poor geological and hydrological conditions, monitoring points should be arranged based on the optimal placement method of slope monitoring points based on the principle of error analysis.Stable state F of a karst slope is mainly affected by the internal gestate disaster factor (IGDF) and external gestate disaster factor (EGDF). Its function expression is,The karst slope is divided into m (m → ∞) sections for stable state analysis, and the sections with the most serious influence of internal and external gestate disaster factors  = Max  are selected to determine the slope state. When the stable state of the karstified slope is Fi, the corresponding internal and external gestate disaster factors of the karst slope are , but it is actually .According to Taylor's formula, Eq.  is expanCombining Equation entclass1pt{minimaI and ΔFE are the difference in slope state caused by the judgement deviation of internal and external disaster gestate factors respectively.We assign a value, i from Ei + ΔE is small, ΔE tends to 0, and ΔFE tends to 0. However, the internal disaster gestate factors are concealed in the interior of the slope, such as lithologic characteristics, geological and hydrological conditions, and karst development, and are difficult to determine, and the spatial difference is large, which results in the serious deviation of Ii from Ii + ΔI; i.e., ΔI is large. Therefore, Eq. , dissolution, fragmentation, cave development, slope cracking, fault and weak interlayer.According to the layout theory of monitoring points, the deformation monitoring points of the Damao slope were arranged in the form of "dense at the top and sparse at the bottom", as shown in Fig. In addition, to avoid mutual interference between slope monitoring and construction, monitoring points were arranged step by step with the progress of slope construction, as shown in Table The slope construction monitoring data from April 26, 2014, to November 15, 2014, were selected for analysis, and the slope deformation data were transformed into equal time interval series with “week” as the unit, as shown in Figs. Figures In the direction of height, the cumulative deformation of the Damao slope gradually decreased from top to bottom. Among them, the level-3 slope and level-4 slope had more obvious deformation. On June 28, 2014, the cumulative deformation of the level-4 slope was 217.0–278.4 mm. On November 15, 2014, the accumulated deformation of the level-3 slope was 60.1–86.6 mm, and the tilt measurement data showed that there might be a sliding zone at 6–9 m of the lower part of the third-grade slope. However, by November 15, 2014, the cumulative deformation of the level-1 slope and level-2 slope was only 15.2–21.9 mm and 29.9–37.9 mm, respectively. The tilt measurement data also showed that the level-1 slope and level-2 slope had no obvious sliding zone and were relatively stable overall.In the horizontal direction, Table Spatial difference characteristicsTime difference characteristicsFigures Mutation theory is a mathematical theory that solves discontinuity problems based on mutation and leap phenomena. Mutation theory is the theory of singularities. Using topology as a tool and based on the theory of structural stability, the mutation model of related systems is established, as shown in Table 26:Dual mode. When the system is under the action of the same control parameter, two different states may appear, i.e., jump from one mode to another mode.Unreachable. It is impossible for the system to achieve stable equilibrium in the real sense on some state variables, i.e., the system has unstable equilibrium positions, which are not differentiable and transient.Abrupt. A small change in control parameters will cause a huge change in state variables, which will make the system jump from one critical point of local minimum to another critical point of local minimum.Divergence. For continuous smooth changes, small perturbations of control parameters only cause small increments of state variables. However, in the neighbourhood of the degradation point, a small change in parameters will cause a large change in state variables.Hysteresis. Hysteresis occurs when a physical system cannot repeat a process of change in a strict reversed-phase.In engineering applications, three types of abrupt change models are commonly used: the cusp type, folding type and dovetail type, among which the cusp model is the most widely used because it can easily construct a critical surface and has an intuitive geometry. The cusp mutation model has five typical characteristics29, i.e.,To apply the cusp mutation model, the general form of the potential function in the cusp mutation model should be constructed firstAccording to Eq. , the equThe projection of the singularity set (S) on the O-UV plane is called the bifurcation set (B), which can be obtained from the discriminant of Eq. . EquatioAccording to Eq. , when \dx in Eq.  has a rex in Eq.  has threx in Eq.  has threWhen 30. Then, the cumulative sequence is expanded into a power series form by Taylor’s formula,The slope deformation data are discrete data, and the slope deformation data are usually processed by accumulative smoothing as the initial data of the slope deformation mutation analysisAfter taking the first five terms of Eq. , we can According to Eq. , the chaThen, the matrix form of Eq.  is13\docAccording to the least square method, the following equation can be obtained,Based on cusp mutation theory, the accuracy of the model can be guaranteed by taking the first four terms of the Taylor series. Thus, the derivative of Eq.  can be oBy assigningEquation  can be rIn addition, to eliminate the third term in Eq. , we can Substituting Eq.  into Eq.Equation  can be rWe obtain:Variable substitution is made for Eq. , i.e.,22b0 is the parameter , and ut and vt are control variables. Meanwhile, from Then, Eq.  can be wAccording to Eq. , the disSubstituting Eqs. , 18) an an18) anΔt = 0.Δt < 0, and According to the cusp mutation theory, there are two possible mutations of the slope state:ut = 0, vt = 0, and Eq. ,  and Eq.  has trip< 0, Eq.  has threWhen the control variable crosses the bifurcation set, the state variable T jumps, as shown in Fig. Thus, the time difference between the critical state of slope deformation and the abrupt instability state of slope deformation ist < 0 and t and the duration of the critical state, i.e.,Under condition (2), when ΔAccording to the slope deformation cusp mutation model, whether the slope deformation state has mutation are directly related to the slope deformation data. However, the measured slope deformation data is very discrete, and it is difficult to eliminate it by one-time accumulation processing, which directly affects the establishment of the potential function of the cusp mutation model and causes the inaccurate prediction of the mutation model. In addition, the mutation model cannot determine the direction of slope mutation, i.e., it can only estimate whether the slope is unstable. Because the slope deformation sequence is based on the zero of deformation, and the zero of deformation is considered the stable point, the mutation point must be the unstable point. Therefore, it is unreasonable to determine whether the slope is unstable only by using the Δ value.Based on these deficiencies, the virtual reality grey trend-cusp mutation model  was proposed. Based on the idea of the relative change in the slope deformation trend, the relative potential function of the slope deformation virtual reality grey trend was established; then, the slope deformation mutation problem was solved.From the data of x(1)(i), slope deformation sequence Perform grey modelling for the two sequences and obtain the fitting and prediction sequences of the entire sequence. The fitting and prediction sequences of The fitting and prediction sequences of If a reality trend grey sequence is assigned as follows, If a virtual trend grey sequence is assigned as follows, Thus, the difference value sequence of the virtual reality grey trend is,Select a group of difference value sequences with length 6 from the neighbourhood of the ith data of the sequence , divergence–convergence mutation type (D–C), convergence–convergence mutation type (C–C), and divergence–divergence mutation type (D–D). Thus, the deformation and mutation forms of the Damao slope are mainly the C–D mutation type, D–C mutation type and C–C mutation type, as shown in Table Table 30C is the final slip velocity of the slope deformation failure section (divergent section); v0 is the initial deformation velocity of the slope deformation failure section (divergent section); K is a constant and if generally 4.6–10.4.In engineering, the warning control of slope deformation mainly includes deformation rate control and cumulative deformation control. In practical applications, slope deformation is generally controlled by combining the deformation rate with the accumulated deformation. In this paper, according to the empirical formula of the slope sliding limit deformation rate proposed by BroadbentAccording to the deformation data at each monitoring point on the Damao slope, the initial velocities of monitoring points D1 and D2 at their failure sections are 5.6 mm/week and 6.4 mm/week, respectively. According to Eq. , when thBased on the grey upper and lower limit segmentation prediction model, the failure sections of monitoring points D1 and D2 were predicted, and the deformation velocity prediction models at the failure sections of monitoring points D1 and D2 were obtained as follows:D1 and vD2 be equal to vc; then, the total time of slope sliding at monitoring points D1 and D2 isLet vThus, the sliding time of the Damao slope at monitoring point D1 is 10.4–14.0 weeks, which is 2.0–5.6 weeks later than the cracking time. The sliding time of the Damao slope at monitoring point D2 is 9.8–12.8 weeks, which is 1.7–4.7 weeks later than its cracking time. Therefore, the process of catastrophic instability of the Damao slope can be inferred, which mainly includes slope excavation disturbance, slope deformation start, slope cracking failure, slope sliding and instability.The cumulative deformation prediction formulas of monitoring points D1 and D2 at the failure sections were obtained based on the grey upper and lower limit segmentation prediction models respectivelyBy substituting Eqs.  and 49)49) into Combined with the deformation characteristics of the Damao slope itself, the lower limit of the average deformation velocity and average cumulative deformation of monitoring points D1 and D2 at sliding failure were taken as the critical control warning values of its catastrophic deformation, i.e., the critical deformation velocity of the Damao slope is 127.0 mm/week, and the critical cumulative deformation is 254.1 mm.Based on the principle of error analysis, the deformation monitoring points of karst slope are arranged in areas with thick karst overburden, dissolution and fragmentation, cave development, slope cracking and faults or weak interlayers. In addition, according to the karst geological characteristics of the Damao slope, the deformation monitoring points adopt the layout form of “dense at the top and sparse at the bottom”.Mutation analysis of slope deformation data is performed by the mutation model. Because the mutation model is often inconsistent with the actual slope deformation, an improved virtual reality mutation model is proposed, considering that mutation can be converted between two deformation trends or states, such as convergence–divergency mutation type (C–D), divergence–convergence mutation type (D–C), convergence–convergence mutation type (C–C), divergence–divergency mutation type (D–D). The application in the deformation and mutation of the Damao slope, shows that the deformation and mutation forms of the Damao slope are mainly the C–D mutation type, D–C mutation type and C–C mutation type. Among them, the C–D mutation type greatly affects the deformation state of the Damao slope and easily causes the cracking and failure of the Damao slope.Based on the empirical formula of slope sliding limit deformation velocity and combined with the grey prediction model, the sliding time of the Damao slope at monitoring point D1 is 10.4–14.0 weeks, which is 2.0–5.6 weeks later than the cracking time. The sliding time of the Damao slope at monitoring point D2 is 9.8–12.8 weeks, which is 1.7–4.7 weeks later than its cracking time. In addition, the lower limit of the average deformation rate of 127.0 mm/week and the lower limit of the average cumulative deformation of 254.1 mm were taken as the critical control warning values of the catastrophic deformation of monitoring points D1 and D2. It provides a basis to quantitatively study the deformation index in similar engineering risk assessments.


However, IF1 is not ubiquitously expressed in mammals, showing tissue-restricted expression in humans and mice and large expression differences between the two species in some tissues. Herein, we summarized key regulatory functions of IF1 for tissue homeostasis, with special emphasis on the deleterious effects that its genetic ablation in neurons has in learning. The development and characterization of tissue-specific mouse models with regulated expression of IF1 will be crucial to disentangle the contribution of the ATP synthase/IF1 axis in pathophysiology.The ATP synthase is an essential multifunctional enzyme complex of mitochondria that produces most of cellular ATP, shapes the structure of the inner membrane into cristae and regulates the signals that control cell fate or demise. The ATPase Inhibitory Factor 1 (IF1) functions  Mitochondria are highly dynamic organelles that play crucial metabolic functions in cellular physiology, the control of intracellular signaling and cell fate . The mitThe ATPase inhibitory factor 1 (IF1) is the physiological regulator of the ATP synthase, inhibiting both the synthetic and hydrolytic activities when it is bound to the enzyme . Besideso domain, which contains the rotor and the proton channel, and the catalytic matrix-protruding F1 domain, which is responsible for the synthesis of ATP . The assembly of both domains is assisted by different assembly factors , indicating that the interactions between both monomers within a dimer are also dynamic   . Therefo1 domain  . And indignaling  and cryoignaling   is increased , caused by a splice mutation found in Leigh syndrome patients also reduces the content of ATP synthase dimers (Notably, the genetic ablation of IF1 in forebrain neurons impairs learning in mice, while its overexpression promotes long-term memory, indicating a key role for IF1 in the regulation of neuronal function and cognition . IF1-knoe dimers . Mitochoe dimers , thus sue dimers . Hence, e dimers .β-F1-ATPase, the catalytic subunit of the ATP synthase, is reduced in a mouse model of Parkinson disease (1 and Fo domains of the enzyme that destabilizes the complex (1 domain (The expression of other subunits of the ATP synthase is also altered in different neurological diseases. For instance, OSCP is downregulated in Alzheimer disease patients and mouse models . The exp disease . In both complex , favorin1 domain . Moreove1 domain - promote1 domain . This wac-ring (1 domain of the ATP synthase and an increase in free c-rings, which augment mitochondrial proton leak (It has been described that the ATP synthase harbors the uncoupling channel that contributes to the PTP in the membrane-embedded c-ring . An excec-ring  and fragc-ring . It is cton leak . The excton leak . Intereston leak . We haveton leak . Converston leak . Therefoton leak .Finally, we have recently shown that IF1 also regulates neuronal function through the control of mtROS production in neurons  Figure . mtROS pNeurons and glial cells work together in the brain forming an intricate association that is necessary for proper neuronal and higher-order brain functions . In thisin vivo as a physiological inhibitor of the ATP synthase. The IF1-mediated inhibition of the ATP synthase is a physiologically relevant mechanism to adapt ATP production by OXPHOS with the cellular energy demand, control ΔΨm and mitochondrial retrograde signaling mediated at least by mtROS. Moreover, it regulates the oligomerization of the ATP synthase and the organization of mitochondrial cristae. However, very little is known about the precise regulation of the ATP synthase by IF1. In this regard, it will be interesting to address whether the IF1-inhibited fraction of ATP synthase promotes localized mitochondrial hyperpolarization and mtROS production by the electron transport chain in functionally independent cristae, or even within different regions of the same cristae, rather than by overall increase of ΔΨm in the whole mitochondrion.The ATP synthase is a crucial hub in mitochondria that integrates OXPHOS, the organization of mitochondrial cristae, the permeabilization of the IMM, and the regulation of intracellular signaling. Using both pharmacological and genetic approaches, we have shown that IF1 functions in vitro or in vivo yields different outcomes depending on the cellular type. In neurons, which show ample expression of the inhibitor in both humans and mice, IF1 is a crucial protein regulating synaptic transmission and memory as well as for protection from excitotoxic insults (Moreover, the biological relevance of the ATP synthase/IF1 axis is far from being fully understood. The tissue-restricted expression of IF1 may explain why its overexpression either  insults . Hence,  insults . However


S1211). Both monomeric and dimeric forms of the highly flexible apo enzyme were reconstructed from a single dataset. The N‐ and C‐terminal domains of monomeric sACES1211 were resolved at 3.7 and 4.1 Å, respectively, while the interacting N‐terminal domains responsible for dimer formation were resolved at 3.8 Å. Mechanisms are proposed for intradomain hinging, cooperativity, and homodimerization. Furthermore, the observation that both domains were in the open conformation has implications for the design of sACE modulators.Hypertension (high blood pressure) is a major risk factor for cardiovascular disease, which is the leading cause of death worldwide. The somatic isoform of angiotensin I‐converting enzyme (sACE) plays a critical role in blood pressure regulation, and ACE inhibitors are thus widely used to treat hypertension and cardiovascular disease. Our current understanding of sACE structure, dynamics, function, and inhibition has been limited because truncated, minimally glycosylated forms of sACE are typically used for X‐ray crystallography and molecular dynamics simulations. Here, we report the first cryo‐EM structures of full‐length, glycosylated, soluble sACE (sACE The highly dynamic nature of the somatic isoform of the blood pressure regulator, sACE, as revealed by CryoEM provides new mechanistic insights. EC 3.4.15.1) is well‐known for its crucial role in blood pressure regulation and fluid homeostasis via the renin–angiotensin–aldosterone system (RAAS), where it catalyzes the hydrolysis of angiotensin I (Ang I) to the vasopressor angiotensin II . The centrality of ACE in the RAAS led to the early development of ACE inhibitors for the treatment of hypertension  is a 1,277‐amino acid type‐I integral membrane protein with a short C‐terminal cytoplasmic tail  but display distinct differences in substrate and inhibitor binding specificities. A wide range of substrates can be cleaved by sACE through endo‐ or exopeptidase activity  determine the apo structure of full‐length soluble sACE; (ii) gain insight into intradomain hinging, interdomain cooperativity, and allostery; and (iii) determine the mechanism of sACE homodimerization. While protein and glycan flexibility can hinder structure solution via X‐ray crystallography, this is not the case for cryo‐electron microscopy , which describes the continuous structural flexibility of proteins imaged by cryo‐EM, and confirmed computationally using NMA. Bending, breathing, and swinging motions of the domains were observed for the first time. Cavity detection with analysis of their motion correlation allowed the identification of promising allosteric sites. We conclusively show that homodimerization under basal conditions occurs via N‐domain protein–protein and glycan‐glycan interactions. These interactions are proximal to an allosteric site, dramatically alter C‐domain dynamics, and induce rotamer changes in key N‐domain catalytic residues to potentially reduce sACE activity. This study represents an important milestone in understanding sACE and heralds a new era of research into its structure–function relationship, which will facilitate improved ACE inhibitor design.S1211) was recombinantly expressed in CHO‐K1 cells and purified to homogeneity. This construct encodes L1‐S1211 of sACE (138 kDa protein‐only molecular weight) and thus includes the two ectodomains, an interdomain linker, and a part of the juxtamembrane stalk, but lacks the transmembrane (TM) domain and intracellular tail  template‐based autopicking of a subset of micrographs using 2D class averages obtained during screening, (ii) 3D reconstruction, (iii) a second round of template‐based autopicking from all 7,689 micrographs, and (iv) iterative 2D and 3D classifications in RELION , and monomer classes with clear clefts in both domains . Although the protein seemed mostly monomeric in solution, the dimer was detected in both 2D and 3D, and showed clear density for three of the domains (classes f and q). Correlation artifacts likely affected sACES1211 separation and caused particles from distinct conformations to pool together, for example, a single monomer 3D class (class m) with no extra density separated into ~40% dimer (class q) and ~60% monomer (class r) upon further heterogeneous refinement.Following a single round of 2D classification, the stack of ~1.2 million Topaz‐picked particles was used directly for 3D reconstruction in cryoSPARC  with overall root‐mean‐square deviation (RMSD) values of 1.4 and 1.8–1.9 Å, respectively. The N‐domain cleft is slightly more open than that of the C‐domain, and both active sites are large with respective volumes of 6,020 and 6,634 Å3 according to the Computed Atlas of Surface Topography of Proteins (CASTp) server . All C‐domain structures to date showed the active site cleft occluded by a lid‐like region composed of the first three N‐terminal helices, with a small N‐terminal pore of ~9 Å in diameter. Until recently, all N‐domain structures were also in this conformation despite the ability of sACE to cleave large substrates. The latest apo N‐domain structure revealed a more open conformation  calculations, substrate cleavage by sACE requires zinc coordination by conserved N‐/C‐domain residues H361/959, H365/963, and E389/987, with activation of a zinc‐coordinated water molecule for nucleophilic attack by E362/960 . The interdomain linker and C‐domain lid N‐terminus thus acted like a spring. These results clearly illustrate the dynamic nature of sACES1211 and emphasize that its full ensemble of states cannot be described by MD simulations/crystal structures of the truncated domains.Second, we used NMA of the models to assess whether the motion observed by 3DVA represents the protein's intrinsic continuous flexibility, or if it was biased by noise/suboptimal particle alignment during reconstruction. Horizontal swinging and deep bending of the interdomain linker elicited alternative rotation of the two domains, and movement of N‐domain subdomain II towards C‐domain subdomain I or II Movie , respectet al, loop‐1): N‐domain 238–261/C‐domain 836–859; Cloop‐2: N‐domain 472–498/C‐domain 1,070–1,096; and Cloop‐3: N‐domain 589–617/C‐domain 1,187–1,201  was identified at the C‐terminus of the N‐domain formed by the 461–471 helix, the adjacent Cloop‐2 residues 474–5 and 478–9, and Cloop‐3 residues 597–600, while Site 2 (Z‐score 0.82) was identified between the N‐domain Cloop‐3 and the C‐domain N‐terminus  but weakly correlated to the C‐domain (Z‐score 0.63) orthosteric site. Site 6, which only affected the N‐domain orthosteric site (Z‐score 1.15), was identified near the N‐domain antiparallel beta‐sheet loop and lid tip on subdomain I, while Site 7, which affected both domains , was identified near N‐domain hinges 2 and 4. Site 8 is in a similar region on the C‐domain surface but does not affect the N‐domain orthosteric site. Since Sites 6–8 are regions that undergo substantial shifts upon active site closure  affects Site 4 (C‐domain cleft) with a left Fig , distal sure Fig , their pet al, et al, et al, S1211 homodimer structure was determined in the present study. Although the C‐domains were unresolved due to high flexibility, the N‐domains resolved to a high resolution and allowed for the identification of the dimerization interface. To investigate how biological or “native‐like” the modeled interface is, the dimer's protein interface score  of −7.29 kcal/mol upon formation of the complex .sACE homodimerization has previously been investigated using reverse micelles, a split‐ubiquitin assay, fluorescence energy resonance transfer and small‐angle X‐ray scattering  of 0.27 suggests that the interface is specific. Furthermore, the TEMPy Segment‐based Manders' Overlap Coefficient (SMOC) score . The active sites are in an open conformation for both states. Since the interface is essentially identical to that of ligand‐bound crystal structures, ligand‐binding and active site closure also do not alter N‐domain dimerization  and Cloop‐3 across the dimer interface, and consequent changes in their dynamics. Overall, the altered N‐domain active site suggests that dimerization induces a shift from an active to a resting or inactive state, since H361 and K489 are essential for activity  of this ~170 kDa glycoprotein was studied as it occurs naturally in the blood, seminal fluid, and cerebrospinal fluid  the protein had a low signal‐to‐noise ratio due to its small size; (ii) the protein domains are globular and resembled noise at close‐to‐focus, even in denoised micrographs; (iii) two neighboring monomer top views could look like a monomer side view and only be picked once, thus potentially leading to a loss of particles. Once particle picking was optimized, 2D reconstruction also posed a challenge, since: (i) the size of the orthogonal monomer projections showed large disparity; (ii) the signal‐to‐noise ratio was low ; (iii) the dimer showed high flexibility and thus poor alignment; (iv) the discrete monomer and dimer states were pooled together into the same class averages, and (v) a sizeable proportion of the mass of the particle consists of flexible glycans.Unlike the crystal structures, where most glycans are removed and the protein is truncated to the N‐ or C‐domain, the cryo‐EM structures produced in the present study were of the full‐length soluble enzyme in a fully glycosylated and apo state. The protein was catalytically active, and native PAGE revealed the presence of a small fraction of dimeric sACEab initio reconstruction and heterogeneous refinement) of the entire Topaz‐picked particle stack in cryoSPARC  wraps around the N‐domain surface between Cloop‐1 and ‐2, which extends into the active site and harbors the critical substrate‐anchoring Q259, K489, and Y498 residues. Cloop‐3 likely regulates interdomain cooperativity with ligand binding and active site closure of one domain affecting the other. Our study revealed that Cloop‐3 is highly flexible and undergoes conformational changes upon active site breathing, and that the active site clefts allosterically regulate each other since their motions are highly correlated. Ligand binding to Q259 (Cloop‐1), K489, and Y498 (Cloop‐2) in the N‐domain active site likely alters the dynamics of the nearby Cloop‐3, to either elicit a change in bending between the domains or in the C‐domain lid breathing motion. Likewise, ligand binding to Cloop‐1 and Cloop‐2 in the C‐domain active site may affect the N‐domain by causing C‐domain active site closure and lid motion adjacent to the N‐domain Cloop‐3. Experimentally, there is a strong link between these three loops and sACE activity. Mutation of the C‐domain equivalents of K489 and Y498 (K1087A and Y1096F) or a key salt bridge between Cloop‐1 and ‐3  was previously shown to drastically decrease sACE activity and inhibitor binding  analysis, and protein–protein docking of N‐domain crystal structures  flexibility was significantly increased upon dimerization and resulted in large C‐domain swinging motions. This increased flexibility could explain why basal dimerization in cell culture was less efficient for sACE than for truncated N‐ or C‐domain  with K489 (Cloop‐3), and ultimately cause the K489 rotamer flip. Dimerization appears to activate the proximal allosteric switch (Site 1) and shift sACES1211 from an active to an inactive or resting state, although this should be investigated further. This could be the ideal pocket to target for stabilization or induction of dimerization. The single free N‐domain cysteine C474 is located in this pocket and is separated by ~17 Å between interacting N‐domains. The design of a molecular tether based on 1,8‐ANS to covalently link these pockets could allow studies into the effect of dimerization on sACE activity and intracellular signaling. If dimerization indeed decreases sACE activity and elicits beneficial signaling, a molecule that shifts the equilibrium from monomer to dimer could be a viable novel therapeutic agent.We observed close associations between Cet al, et al, et al, The results from this study also have diagnostic applications. Although sACE glycosylation occurs to varying degrees in different tissues  cells . Harvested culture medium was loaded onto the column at a flow rate of 3 ml/min after which the column was washed using equilibration buffer at a flow rate of 1.5 ml/min. The protein was subsequently eluted using 50 mM Boric acid (pH 9.5) at a flow rate of 3 ml/min. Elution was monitored by a Bio‐Rad Econo UV Monitor  and Ross Chart Recorder. Fractions were collected and assayed for activity . The protein was dialyzed for 6 h at 4°C in 5 mM HEPES (pH 7.5) followed by second dialysis in 50 mM HEPES (pH 7.5) for 24 h at 4°C. The dialyzed protein was concentrated to 3 mg/ml using an Amicon® Ultra 30 kDa molecular weight cut‐off centrifugal filter (Millipore), filtered using a 0.45 μm Nanosep MF GHP centrifugal filter , and stored as single‐use aliquots in liquid nitrogen.Protein purification from the culture medium was performed using lisinopril–Sepharose affinity chromatography, as previously described . The native state of purified sACES1211 was analyzed using native PAGE. Stacking and resolving gels were prepared to 4 and 6% acrylamide, respectively, and the running buffer consisted of 25 mM Tris–HCl (pH 8.3) with 125 mM NaCl. The sACES1211 sample, bovine serum albumin (BSA) standard, and gel filtration standards (Bio‐Rad Laboratories Inc.) were prepared in sample buffer containing 125 mM NaCl and, for sACES1211, 10 μM ZnCl2. Gels were run on ice at 25 and 100 mA for the stacking and resolving stages, respectively, and stained with AcquaStain (Bulldog Bio Inc). Native sACES1211 was further analyzed by size‐exclusion chromatography on an Agilent 1260 Infinity HPLC system with a BioSEP™ SEC S3000 column (Phenomenex). The column was equilibrated using 5 mM Hepes (pH 7.5) with 125 mM NaCl and operated at a flow rate of 0.5 ml/min. Absorbance was monitored at 280 nm and compared between sACES1211, gel filtration standards (Bio‐Rad Laboratories Inc.), and truncated glycosylated N‐ or C‐domain. The truncated proteins were expressed and purified separately as detailed in  and a fluorescence spectrophotometer (Cary Eclipse Varian Inc.). Briefly, protein samples were diluted in 100 mM K2HPO4/KH2PO4 (pH 8.3), 300 mM NaCl, 10 μM ZnSO4, and incubated with 1 mM Z‐FHL for 15 min at 37°C. The reaction was stopped by the addition of 0.28 M NaOH, followed by derivatization of the HL product with o‐phthaldialdehyde (Sigma‐Aldrich® Co.), incubation for 10 min at room temperature, the addition of 3 M HCl, and reading of fluorescence intensity at excitation and emission wavelengths of 360 and 385 nm, respectively. One milliunit (mU) of sACES1211 activity equals 1nmol of HL product per min.The catalytic activity of purified sACEer et al  using thS1211 at a final concentration of 1.5 mg/ml in 50 mM Hepes (pH 7.5), 125 mM NaCl and 10 μM ZnCl2 was incubated for 30 min on ice. These essential cofactors were added to ensure that the protein is imaged in an active state. Quantifoil® 200‐mesh R2/2 copper grids were glow discharged for 30 s after which 2.5 μl of sACES1211 was applied and incubated on the grid for 30 s in a Vitrobot Mark IV (Thermo Fisher Scientific) chamber at 100% humidity and 4°C. The grid was blotted for 3.5 s and vitrified by plunging into slushy ethane. Grids were stored under liquid nitrogen. Electron micrograph movies were recorded using a Titan Krios (Thermo Fisher Scientific) equipped with a K3 direct electron detector (Gatan) and high‐brightness field emission gun (X‐FEG) operated at 300 kV in super‐resolution mode at a magnification of 81,000× corresponding to a calibrated pixel size of 0.53 Å per pixel. A total of 11,628 micrograph movies consisting of 40 frames each were automatically acquired from 35 grid squares of a single grid using EPU (Thermo Fisher Scientific) with aberration‐free image shift (AFIS), defocus range of −1.8 to −3.0 μm, dose per frame of 1.075 e−/Å2, and exposure time of 3 s for a total dose of 43 e−/Å2.Purified sACEet al, et al, T = 2, K = 200, 25 iterations) were performed to remove junk particles. Again, since most class averages appeared monomeric, dimeric class averages were discarded. A coarse search  was required as the initial coordinates were not centred. Additionally, the expectation/alignment step (E‐step) was limited to 10 Å resolution. A stack of 1,254,521 monomer‐like particles were selected and underwent a third round of 2D classification . The resulting 2D class averages still showed evidence of junk  and dimer‐like particles (three or four globular domains) in addition to the monomer top (single globular domain) and side (two connected globular domains) view particles. Thus, dimer and monomer particles were pooled into the same 2D class averages and not adequately separated. The elongated nature of monomeric sACE and difference in top and side view diameters  likely also resulted in suboptimal alignment of the particles when using a uniform box size, as previously seen for alcohol dehydrogenase  with mask diameters of 120 and 200 Å for the top and side views, respectively. Top views underwent one round of 2D classification, while side views underwent four rounds of 2D classification during which dimer classes still separated out. The resulting stacks of 275,142 top view particles and 397,765 side view particles were combined. An initial 3D reconstruction was generated, centred, and 3D classification performed with coarse sampling  for 11 iterations and finer sampling  for 13 iterations. A wide circular mask diameter of 311 pixels was set to allow detection of any remaining dimer classes. Two 3D classes  were selected, pooled, and the particles re‐extracted with recentering at 1.06 Å/pixel in a 360‐pixel box. The particles underwent two further rounds of 2D classification  to yield 536,833 monomer‐like particles. A 240 Å mask diameter was used for all monomer views, i.e., top and side views were not separated. One round of 3D classification was performed  for 22 iterations. One class  was selected and refined  to yield a 10 Å resolution reconstruction. A final round of 3D classification was performed as before, except with K = 3. One 3D class  was selected and refined as before but with local angular searches from 1.8°, a soft mask around the protein, and solvent‐flattened FSCs. This resulted in a ~7 Å resolution reconstruction. To test whether the resolution was limited by the box size, these refined particles were re‐extracted in a 560‐pixel box and refined as before. No improvement was observed and the resolution was thus not limited by box size.The first round of image processing was performed in RELION version 3.1 . The set of 7,689 curated micrographs was denoised using the pretrained Topaz‐denoise  to remove junk particles. A total of 1,167,502 particles were selected for further processing. Reconstruction was first attempted using iterative 2D and 3D classification in RELION  and heterogeneous refinement. This was followed by duplicate removal from each class and re‐extraction at 2.12 Å/pixel in a 256‐pixel box. This box size allowed for the removal of particles with nearby neighbors or micrograph edges/contaminants. Class 0  and class 2  seemed to only consist of a single globular domain each. Ab initio reconstruction (two classes and class similarity of 0) and heterogeneous refinement, however, separated each of these seemingly single‐domain classes into a 2‐domain class  showed two linked globular domains and was separated into 4 classes by a second round of ab initio reconstruction (class similarity of 0) and heterogeneous refinement. Class 1 subclasses 0  and 2  showed two linked globular domains while subclasses 1  and 3  showed three and one globular domains, respectively. Class 1 subclass 1 was subjected to a third round of ab initio reconstruction (class similarity of 0) and heterogeneous refinement, which allowed for the separation of 28,895 junk particles . A third round was also performed for Class 1 subclasses 0, 2, and 3. For the latter, classes g  and h  were obtained and thus discarded. Subclasses 0 and 2 yielded two‐domain classes i‐l  and subjected to ab initio reconstruction (class similarity of 0.1) and heterogeneous refinement into four classes. Classes n and o were discarded as n  contained junk while o  showed two globular domains with a slightly longer interdomain linker, which could not be subclassified or further refined likely due to remaining heterogeneity  showed two linked globular domains and clear glycan stumps on the surface. Class m  also showed two linked globular domains. The latter underwent another round of ab initio reconstruction (class similarity 0.1) and heterogeneous refinement into two classes. Surprisingly, this yielded a two‐domain class r  and a four‐domain class q , each with clear glycan stumps on the surface  and heterogeneous refinement to generate four classes of which two were discarded. Particles from the two selected classes (each ~7–8 Å resolution) were pooled and re‐extracted at 1.06 Å/pixel in a 256‐pixel box, and underwent one round of 2D classification  with a mask  around each respective domain. One class was selected for each domain and imported to cryoSPARC  of the two stable domains. A single class of 107,816 particles was imported to cryoSPARC  with the auto‐tightened FSC mask from the final cryoSPARC nonuniform/local refinements applied with a default cone angle of 20.0, FSC cut‐off of 0.143, sphericity threshold of 0.5, and high‐pass filter of 150.0 Å.Global B‐factor sharpening was performed in cryoSPARC  was therefore used as the initial model for the dimer. All glycans and ligands (except the active site zinc) were removed, residues mutated in the crystal structure restored to the wild‐type sequence of sACE , and the protein rigidly docked into the 3.63 Å single‐domain dimer map using Chimera  applied at the zinc coordination site, followed by bulk flexible fitting of the entire chain. After release of the adaptive restraints, localized flexible fitting was performed with inspection of each residue at least once. The density‐modified, globally sharpened, and unsharpened maps were used. Adjustments were made where necessary, and missing residues and glycans were built using ISOLDE  as an initial model. Residues were renumbered to sACE numbering and bulk/localized fitting into the 4.08 Å C‐domain map performed as for the N‐domain. A chloride ion was modeled at the Chloride (II) site in the C‐domain, while zinc was removed from the final C‐domain structure as there was no density to support it. The interdomain linker was built using the AlphaFold structure prediction for sACE  server  was performed using EADock at the SwissDock server  are available from the following databases:ihttps://www.ebi.ac.uk/empiar)ahttps://www.ebi.ac.uk/empiar/EMPIAR‐10980)EMPIAR‐10980 ahttps://www.ebi.ac.uk/pdbe/entry/pdb/7q3y)7Q3Y (full‐length soluble monomer) (bhttps://www.ebi.ac.uk/pdbe/entry/pdb/7q49)7Q49 (monomer N‐domain) (chttps://www.ebi.ac.uk/pdbe/entry/pdb/7q4c)7Q4C (monomer C‐domain) (dhttps://www.ebi.ac.uk/pdbe/entry/pdb/7q4d)7Q4D (2‐domain dimer) (ehttps://www.ebi.ac.uk/pdbe/entry/pdb/7q4e)7Q4E (single‐domain dimer) (Atomic coordinates: PDB (iiihttps://www.ebi.ac.uk/emdb/) ahttps://www.ebi.ac.uk/emdb/EMD‐13797)EMD‐13797 (monomer consensus refinement) (bhttps://www.ebi.ac.uk/emdb/EMD‐13799)EMD‐13799  (chttps://www.ebi.ac.uk/emdb/EMD‐13801)EMD‐13801  (dhttps://www.ebi.ac.uk/emdb/EMD‐13802)EMD‐13802 (dimer consensus 4‐domain refinement) (ehttps://www.ebi.ac.uk/emdb/EMD‐13803)EMD‐13803  (fhttps://www.ebi.ac.uk/emdb/EMD‐13804)EMD‐13804  (Cryo‐EM maps: EMDB (Lizelle Lubbe: Conceptualization; data curation; formal analysis; validation; investigation; visualization; methodology; writing – original draft; writing – review and editing. Bryan Trevor Sewell: Resources; supervision; funding acquisition; writing – review and editing. Jeremy D Woodward: Resources; supervision; funding acquisition; methodology; writing – review and editing. Edward D Sturrock: Conceptualization; resources; supervision; funding acquisition; writing – review and editing.CRediT author contributions listed above, the contributions for LL and JDW in detail are:In addition to the LL purified the protein, performed biochemical analyses, prepared cryo‐EM grids, collected cryo‐EM data, performed all image processing, model building, refinement, validation, data deposition, wrote the first draft of the manuscript, and edited the manuscript. JDW provided guidance during cryo‐EM grid preparation and initial image processing in RELION, assisted with cryo‐EM data collection, and edited the manuscript.The authors declare that they have no conflict of interest.AppendixClick here for additional data file.Expanded View Figures PDFClick here for additional data file.Table EV1Click here for additional data file.Movie EV1Click here for additional data file.Movie EV2Click here for additional data file.Movie EV3Click here for additional data file.Movie EV4Click here for additional data file.Movie EV5Click here for additional data file.Source Data for Figure 2Click here for additional data file.


Posterior screw-rod constructs can be used to stabilize spinal segments; however, the stiffness is not absolute, and some motion can persist. While the effect of crosslink-augmentation has been evaluated in multiple studies, the fundamental explanation of their effectiveness has not been investigated. The aim of this study was to quantify the parameters “screw rotation” and “parallelogram deformation” in posterior instrumentations with and without crosslinks to analyze and explain their fundamental effect. Biomechanical testing of 15 posteriorly instrumented human spinal segments (Th10/11—L4/L5) was conducted in axial rotation, lateral bending, and flexion–extension with ± 7.5 Nm. Screw rotation and parallelogram deformation were compared for both configurations. Parallelogram deformation occurred predominantly during axial rotation (2.6°) and was reduced by 60%  by the addition of a crosslink. Simultaneously, screw rotation (0.56°) was reduced by 48%  in this loading condition. During lateral bending, 0.38° of parallelogram deformation and 1.44° of screw rotation was measured and no significant reduction was achieved by crosslink-augmentation . During flexion–extension, parallelogram deformation was 0.4° and screw rotation was 0.39° and crosslink-augmentation had no significant effect on these values . In axial rotation, crosslink-augmentation can reduce parallelogram deformation and with that, screw rotation. In lateral bending and flexion–extension parallelogram deformation is minimal and crosslink-augmentation has no significant effect. Since the relatively large screw rotation in lateral bending is not caused by parallelogram deformation, crosslink-augmentation is no adequate countermeasure. The fundamental understanding of the biomechanical effect of crosslink-augmentation helps better understand its potential and limitations in increasing construct stiffness. The acting bending moments can be separated into three major rotational motion planes: flexion–extension, lateral bending and axial rotation. The hypothezides construct deformation due to the generated internal stresses are illustrated in Fig. During physiological loading, notable deformation of posteriorly instrumented spinal segments can occur Fig. A1, which4 and their effect was evaluated in a multitude of studies, as summarized in a recent systematic review3: in axial rotation, a relatively consistent positive effect on construct stiffness was observed, while the effect on lateral bending was more variable and in flexion–extension, only minimal effect was recorded. While the biomechanical effect was largely similair for the whole spinal column, clinical benefit has only been shown for C1/2 instrumentations6. For the posterior instrumentation in adolescent idiopathic scoliosis, no clinical benefit was shown8 and no clinical data is available for other clinical situations at the lumbar or thoracic spine. To help direct clinical studies and the clinical practice towards the optimal use of crosslinks, a more fundamental understanding of the biomechanical effect on the construct charachteristics is required.Crosslinks, connecting the two sides of the screw-rod construct, were proposed as a measure to increase rotational construct stiffnessWe hypothesize that while relevant screw rotation and parallelogram-deformation can occur in posterior screw-rod instrumentations, crosslinks are only able to reduce parallelogram-deformation and are ineffective in reducing screw rotation. To quantify the biomechanical effect of crosslink-augmentation on these parameters, posteriorly instrumented single-level human segments were analyzed during axial rotation, lateral bending, and flexion–extension.The experimental protocol was approved by the local ethics institution  and conforms to the relevant guidelines and regulations. All medical information about the donors was fully anonymized and the original written informed consent for donation, in accordance with applicable law and regulation are on file at the offices of Science Care . Biomechanical experiments were performed on 15 human spinal segments (Th10/11—L4/L5) originating from four fresh frozen cadavers  or the cortical bone trajectory (8 segments)9 with specific insertion guides10 . Screw diameter and length were maximized during preoperative planning11 and ranged from 5 to 6 mm in diameter and 40 mm to 55 mm in length . Vertical rods  were inserted on either side  to prevent dehydration. A camera system with a telecentric objective  was used to image the screw-rod construct in the configuration with and without crosslink-augmentation for each loading direction  with a specific spine testing setup14 Fig. A was useion Fig. B.Figure To image the deformation during lateral bending (in which the instrumentation faced downwards), a mirror oriented at 45° relative to the floor was used Fig. B. The teFor every segment, each of the four screw heads and the two vertebral bodies were furnished with a specifically designed crosshair-label that allowed for the assessment of the rotational and translational position in the obtained pictures Fig. C. Image Statistical evaluation was performed with MATLAB. According to the Shapiro–Wilk parametric hypothesis tests of composite normality, not all results were normally distributed. Therefore, the Wilcoxon signed-rank test was used (α = 0.05).All data generated or analysed during this study are included in this published article and its . AnalysiThe median value of pedicle screw rotation in the configuration without crosslink was 0.56° in axial rotation, 1.44° in lateral bending and 0.39° in flexion–extension. The addition of a crosslink reduced this values to 0.29° in axial rotation  and to 1.25° in lateral bending . No reduction was measured in flexion–extension  . In this loading condition, the addition of a crosslink is no effective measure to reduce screw rotation, as it is not primarily caused by parallelogram deformation. We postulate that the specific behavior of the construct is largely caused by the boundary conditions of the biomechanical test setup: in our case, the setup allows for coupled motion in the translational plane orthogonal to the loading axis, which prevents the build-up of large lateral shear forces and which results in pure bending moments around the loading axis. Without the generation of relevant lateral shear forces, only limited parallelogram deformation is generated, which explains the small parallelogram deformation in our results. However, in a setup that does not allow coupled motion in the translational plane orthogonal to the loading axis , relevant lateral shear forces can occur, which can, in turn, result in larger parallelogram deformation. In such a situation, crosslink-augmentation could result in an increase in construct stiffness by reducing this parallelogram deformation. This consideration can help explain the large variability of the effect of crosslink-augmentation during lateral bending in the literature3. While we evaluate neither pure nor constrained bending moments to perfectly represent in-vivo loading conditions and while both boundary conditions can be useful for specific research questions, knowledge about the characteristics of the test setup is nevertheless crucial for the correct interpretation of biomechanical results.In contrast, large screw rotation and only minimal parallelogram deformation were observed during lateral bending loading. The large screw rotation can be an important factor for the relatively large motion of an instrumented segment in this loading situation Fig. A1. In th3.In flexion–extension, only minimal screw rotation and very small parallelogram deformation were recorded, which can be explained by the small torque around the screw axes and the minimal lateral shear forces in this loading condition. In consequence, crosslink-augmentation is not able to affect construct stiffness in this loading condition, which is in line with the findings of the literature1 and does not lead to the conclusion that any of the two trajectories is more prone to benefit from crosslink augmentation in the clinical routine. Furthermore this finding allows for the combined analysis of all spinal segments, which helps achieve higher statistical power. While in the configuration without crosslink, parallelogram deformation was significantly larger in the CBT-instrumented segments, the absolute values were comparibly small for both trajectories. Therefore we evaluate the measured difference to be irrelevant for the clinical routine.The comparison of the two pedicle screw trajectories did not reveal significant differences in the effectiveness of crosslinks in regard to parallelogram deformation and screw rotation. This finding is in line with the similar effectiveness of crosslink-augmentation on segmental range of motionThe here presented study is associated with certain limitations: screw rotation and parallelogram deformation were evaluated in the 2D-plane of the telecentric objective. While this plane was oriented in the coronal plane of the segments, small variability between loading conditions and between specimens cannot be excluded. Also, some inaccuracy during image processing must be assumed, which increases the variability of the results. Furthermore, the experiments were conducted on cadaveric specimens at room temperature loaded with well-defined but largely simplified loading conditions compared to the in-vivo situation. These aspects limit the direct transferability of the findings to the clinical situation. Nevertheless, we evaluate the qualitative conclusions of this study to be robust against these potential sources of error. Finally, a rather simple configuration with only single-level instrumentations without additional decompression procedures or interbody cage insertion was chosen to keep additional parameters minimal. This experimental setting limits definitve conclusions on more complex configurations.In axial rotation, crosslink-augmentation can reduce parallelogram deformation and with that, screw rotation. This explains the effectiveness of crosslinks in this loading condition. In unconstrained lateral bending and flexion–extension, parallelogram deformation is minimal and therefore, crosslink-augmentation has no significant effect. Since the relatively large screw rotation in lateral bending is not caused by parallelogram deformation, crosslink-augmentation is no adequate countermeasure for this problem. The fundamental understanding and the quantitative information about the biomechanical effect of crosslink-augmentation on construct stiffness and construct deformation helps to evaluate its potential benefit in the clinical practice and to better understand the limitations of its effectiveness.Supplementary Information.Supplementary Legends.


We introduce a mathematical model, based on networks, for the elasticity and plasticity of materials. We define the tension tensor for a periodic graph in a Euclidean space, and we show that the tension tensor expresses elasticity under deformation. Plasticity is induced by local moves on a graph. The graph is described in terms of the weights of edges, and we discuss how these weights affect the plasticity. Further, a standard net is characterized by the tension tensor. We show that the Cauchy stress tensor is also expressed by the tension tensor. Furthermore, if the graph structure is preserved, the elasticity at the macro-scale is also determined by the tension tensor; otherwise, a departure from elasticity, known as plasticity, occurs. To describe the manner in which the graph structure changes, we consider two types of local moves: contraction and splitting. We define a condition for a local move and introduce two models of deformation concerning plasticity. This enables us to draw the stress–strain curve.In this paper, we suggest that the energy of a net induces a model of hyperelastic materials. Here, hyperelasticity is the property from which stress under deformation is derived using an energy density function. To describe the deformation of a net, we introduce the notion of a Our model is motivated by the structure of thermoplastic elastomers (TPEs). A TPE is a polymeric material with rubber elasticity and is remoldable at high temperatures. A typical TPE consists of ABA triblock copolymers, in which monomers of types A and B are arranged in a sequence such as In our model, the hard and soft domains correspond to the vertices and edges, respectively. More precisely, a hard domain is a vertex, and a polymer chain in the soft domain is an edge. The endpoints of an edge are the (possibly single) hard domains that contain monomer A of the copolymer. The obtained graph may have loops and multiple edges.The network structure of polymers induces rubber elasticity. The random motion of polymer chains in the soft domain gives rise to entropic forces. A hard domain functions as a cross-link. In our approximation, we ignore the maximal length of the chain and the interaction between chains. If a chain moves randomly, the tension on the chain is proportional to the distance between the endpoints. This setting is consistent with the definition of the energy of a net. Suppose that the polymers can move freely while preserving the network structure. Additionally, we obtain a harmonic net in equilibrium. Since harmonicity is preserved by affine deformation, the affine assumption in the classical theory of rubber elasticity holds. Additional details of rubber elasticity are provided in the book by Treloar .The hard domains of a TPE are less robust than the cross-links of vulcanized rubber because each hard domain is aggregated by intermolecular forces. The network structure of a TPE may change under deformation, as observed in simulation ,9 and byIn §2, we give preliminary definitions of nets. The graphs we use are weighted, and their weights can be regarded as the number of edges. Different types of polymers may contribute different weights.In §3, we introduce the tension tensor. This is visualized by an ellipsoid.In §4, we consider the elasticity of nets under deformation. Based on a physical argument, the Cauchy stress tensor is derived from the tension tensor. We also consider the stress under uniaxial extension. Young’s modulus of a standard net is determined by the energy per unit volume.In §5, we define local moves. The contraction of two vertices is a natural operation. A splitting of a vertex is an inverse operation of contraction. The sum of weights is preserved in our model. The conditions for the occurrence of these moves are provided by the realization of a graph. The local tension tensor is used to determine whether a vertex splits or not. We suspend the physical validity of the conditions.In §6, we introduce two models we call fast and slow deformations. Although these models reflect the dependence on the speed of deformation, we consider only the two extreme cases. Subsequently, we obtain the stress–strain curve, which is merely piecewise continuous. We suppose that the local moves under deformation finish in finitely many times. For example, after a vertex splits, the inverse contraction should not occur immediately. We show only a sufficient condition to avoid such repetitions.Section 7 is devoted to mathematical results. In there, we consider the extent to which the weight of an edge affects the harmonic realization. For the sake of theoretical consideration, we allow weights to be non-negative real numbers and continuously deformed. When the weight of an edge in a harmonic net becomes large, the limit of nets is obtained by the contraction of this edge. In theorem 7.2, we show a mathematical result on the relation between the edge length and the difference of the tension tensor. Moreover, in theorem 7.6 we obtain a lower bound for the edge length.energy loss ratio to measure the plasticity of nets. This provides an estimate of the permanent strain for uniaxial tension. We observe simple examples, which suggest the following: (i)a material has lower plasticity if the proportion of loops is large; (ii)a material with lower plasticity is obtained by blending two materials. Continuous deformation of weights highlights these tendencies.In §8, we define the number called the In §9, we give examples of deformation. We use a periodic graph obtained from the hexagonal lattice. The nets obtained by deformation depend on the stretching direction.. 2Based on the formulation in ,5, we prA graph periodic graph, and period lattice for periodic realization of  (i) (ii) Condition (i) means that period homomorphism and the period lattice for We consider an infinite connected graph net in The pair A periodic realization We define the energy of a net and consider energy-minimizing realizations. Note that our definition of the energy is slightly different from that in , §7.4, wenergy (per period) of a net The harmonic if the energy A periodic realization standard if the energy A periodic realization Remark that when Clearly, a standard realization is harmonic. A harmonic realization is characterized by a local condition. We characterize a standard realization in §3.A periodic realizationis harmonic if and only iffor anyFrom this theorem, it directly follows that a linear transformation of a harmonic representation is also harmonic.Suppose thatis a periodic realization andis a linear transformation Then the compositionis a harmonic realization if and only ifis harmonic.One might think that in definition 2.2, the rest lengths of the springs should be positive, not zero. However, we have assumed the rest lengths to be zero for two reasons. The first reason is due to statistical mechanics. A chain in a TPE is not taut like a helical spring, but it fills space randomly. Therefore, the tension on the chain is proportional to the distance between the endpoints. The second reason is a mathematical one. We will transform harmonic nets by continuous linear transformations in the following sections. However, in order for the result of any linear transformation to be harmonic again, the natural length of the spring must be zero (see corollary 2.6).In this paper, we use the term ‘net’ as a periodically realized network in of Wells , who iniof Wells , a net i. 3In our mathematical model, a net represents the structure of TPE chains. Consider the tension caused by the structure. Indeed, a stretched TPE must have tension in the direction in which the structure is stretched. For example, the net in tension tensor. Essentially, the tension tensor denotes the energy of a net with information of the direction along which the net is stretched, or its ‘directed energy’. We will observe that the tension tensor can be visualized as an ellipse (or ellipsoid), such as the ellipses in figures To answer this question, we introduce a matrix named a  (i)when we stretch a net and the graph structure does not experience any change, the ellipse of the tension tensor also stretches in the same direction; and (ii)when the graph structure changes, the ellipse of the tension tensor becomes round. The former immediately follows from the definition. The latter can be verified by theorem 7.2.Through computer simulation for the deformation of two-dimensional nets, we make the following observations: (a)tension tensors as follows: For a vertex For a net It holds thatThe following characterization of a standard realization follows from Let To visualize the tension tensor, we define an ellipsoid by. 4In this section, we consider the stress experienced by a net by using the tension tensor. Fix a periodic graph on. . (a)By compositing rotational isometry, we may assume that the tension tensor per period is a diagonal matrix Cauchy stress tensor for a net as We apply a physical argument to define stress for nets. Let us consider a macro-scale object of which the shape is an orthotope  umption . In othdeviatoric stress tensor.Furthermore, we suppose that the deformation preserves the volume. In other words, If the external forces A material is hyperelastic (or Green elastic) if the stress under deformation is determined by a strain energy density function. In our setting, consider the affine deformation of a standard net  (b)true stress under this uniaxial extension is defined byengineering stress  is measured using the cross-sectional area before deformation, and it is defined by Consider a harmonic net LetThenSince permanent strain is the number The It holds thatYoung’s modulus for a standard net Consider the case in which . 5We introduce three local moves for nets: contraction and splitting. A local move for a graph is an operation to obtain a new graph by replacing some vertices and edges. When we say that we replace an edge  (a)contraction of Let 0 and v1 . Then, tFor a periodic graph To introduce deformation in §6, we define a condition for contraction using a realization Note that the weight  (b)A splitting of a vertex is an inverse operation of contraction. This is not determined only by the vertex. When a vertex For a periodic graph  (i)when a vertex  (ii)the way in which the edges originating from  (iii)the way in which the weights are assigned. Fix constants We define a condition for splitting using a realization  (i)the eigenspace associated with  (ii)there are no edges Because it is difficult to decide how the weights are assigned, we use an ad hoc setting: suppose that To obtain the unique division into two classes immediate realizationFor a realization Suppose thatis a graph obtained fromas a result of splitting under the above condition. Letbe a harmonic realization ofwith the same period asThen, Clearly, In the above two subsections, the graph In this paper, it does not matter because we only consider the shape of the realized graph and its energy. However, if one wants to discuss such as the displacements of nodes before and after a local move, this ambiguity should be removed. One idea is to assume that the centre of mass of the periodic cell is fixed.. 6We introduce two models: fast and slow deformation. The difference of these two models reflects the strain rate sensitivity, that is, the dependency of stress on the speed of deformation. A harmonic realization of a periodic graph is regarded as an equilibrium state. Let  (a)Let fast deformation.We continue this procedure by supposing that these procedures finish with finitely many local moves. In the end, we obtain a harmonic net  (b)Slow deformation is a limit of sequences of fast deformation. For a continuous family of linear transformations, take approximations by discrete families of small ones. They induce sequences of fast deformation. We obtain slow deformation by the limit as the approximations get arbitrarily fine.Equivalently and more precisely, slow deformation is defined as follows. Suppose that After the exhaustion of local moves for slow deformation.We continue this procedure by supposing that these procedures finish with finitely many local moves. Ultimately, we obtain a harmonic net  (i)two local moves do not occur simultaneously, and (ii)two vertices equivalent by the period do not contract. If the net is highly symmetric, the genericity is difficult to hold. For genericity, we arbitrarily choose a single local move at a time, and we ignore any contraction that violates the period.A condition of genericity is given as follows:stress–strain curve for a uniaxial extension. LetWe define the  (c)We suppose that the above procedures in deformation finish with finitely many local moves. In general, splittings and contractions may cause an infinite sequence of local moves. We give only a partial result for this problem.Let Suppose that the weights are non-negative integers. Then,Let Let LetandSuppose thatandThen, We find the maximum . 7In this section, we consider the extent to which the harmonic realizations and their energies depend on the weights, which vary in non-negative real numbers. We describe a contraction as the limit by increasing the weight of an edge. Note that we do not suppose that the sum of weights is preserved, which differs from the assumption in §5.Let Fix a period homomorphism The realizationsconverge toasIn other words, converge tofor eachandconverge toIn particular, Consequently,Consider a (not necessarily harmonic) periodic realization Since Suppose that There areandsuch thatandwhereanddo not depend onanddoes not depend onConsequently,We showed that Suppose that We regard the tension tensor equality , we haveLet the vectorand the numberbe as in theorem 7.2. Suppose thatThen,SinceAs in the proof of theorem 7.2, we have Letbe a symmetric matrix. Suppose thatis invertible. ThenUsing the adjugate matrix, we haveLetbe a symmetric matrix. Suppose thatfor anyandfor anyThen, is positive semi-definite.The proof is by induction on Letbe a realization ofsuch thatfor anyandis harmonic aroundThen, Since Since . 8The plasticity of a material is its ability to undergo permanent deformation. For a fixed periodic graph, no external force is applied to a net if and only if its realization is standard, as shown in §4. Hence, if no local moves occur in the deformation of a net, then it returns to its initial state by unloading, and it is perfectly elastic. In general, however, local moves cause plasticity. To measure plasticity, we introduce the energy loss ratio of a net under deformation. This is defined by comparing the energy with that of a net in which local moves do not occur.energy loss ratio of Let If no local moves occur, then Consider the uniaxial extension bySuppose thatis standard, no contractions occur in the deformation, andThen, the permanent strainsatisfies thatApproximationsWrite In the remainder of this section, we consider the simplest case of splitting; let For After we show general behaviour, we give some examples of the energy loss ratio It holds thatandThe condition of harmonic realization We remark that By way of example, we suppose that the weights are given by a function of the lengths of edges.Letbe a non-negative function onsuch thatPutforSuppose that the sumis convergent. Letwhich is regarded as the probability that a loop changes into a non-loop edge. Suppose that there issuch thatThen, the energy loss ratiosatisfieswhere we suppose that the three integrals are finite. Moreover, ifandthenfor any fixedBy the second assertion, we may understand that the material has lower plasticity if the proportion of loops is large.Since Suppose that For example, we use the normal distribution, given by the function Next, we consider the weight functions given by the linear sums Suppose thatThenThe equality holds if and only ifand the vectorsandare parallel. For fixedandthe minimum ofis attained whenWe may understand that a material with lower plasticity is obtained by blending two materials. Note that if It is easy to check the presentation of We remark that the linear sums of weights with different energy loss ratios do not yield smaller ratios in general. For instance, the linear sums of Under the assumption that For example, we consider the cube lattice. Suppose that We give another example by using the linear sums of normal distributions. Suppose that . 9In this section, we give examples of deformation for two-dimensional nets. LetFor the splitting of a vertex Let  (a) (i)for  (ii)representatives  (iii)The non-loop edges Let  follows :(i) forThen, the volume per period is  (i)Consider the slow deformation by Furthermore, when Similarly, more splittings of vertices may occur. Subsequently, the length of a path of edges increases. For  (ii)Consider the slow deformation by . 10 (1)The definition of the tension tensor fits with our purely mathematical interest, and we can apply it to topological and discrete geometric properties of graphs and nets. To define an energy, we have used  (2)In §5, we have arbitrarily given the threshold values  (3)The validity of our model for deformation requires the assumption that infinite repetition of local moves does not occur. However, mutually inverse splittings and contractions may continue alternatingly in some cases. Theorem 6.1 gives a sufficient condition that such alternating repetition does not occur. Is another kind of infinite repetition possible? If it is possible, what condition is sufficient to avoid such repetition? (4)In §9, we have given simple examples for two-dimensional nets. What about three-dimensional nets? (5)Do local moves occur in unloading? If not, then it reproduces the Mullins effect : the str (6)We have worked in continuous weights of edges, which is useful for mathematical arguments. Of course, the weights corresponding to actual polymers are integers. However, it would not be meaningful only to restrict ourselves to integer weights. Our presented settings preserve periodicity. As a result, stress–strain curves are not continuous. To obtain a continuum limit of such discrete matters, stochastic formulation of local moves will be effective. The stochastic formulation in integer weights may induce continuous stress–strain curves and a nonlinear constitutive equation that reflects elastoplasticity. We expect that this is also appropriate to describe fracture of materials.We discuss some open issues and propose questions for further research.