Microsoft announced Windows 11 as a free upgrade to all the compatible Windows 10 PCs. After upgrading or reinstalling Windows 11. Free Windows 7 Ultimate Product Key (100% Working) [Free Keys 2021] The following activation keys will enable you to activate all. Process Lasso. Real-Time CPU Optimization and Automation. server-square. Download Free · Purchase Pro · v10.4.0 - November 11, 2021 (Now with CPU Sets!). {INSERTKEYS}{{{{{{\rm{V}}}}}}}_{{{{{{\rm{i}}}}}}-1}-{{{{{{\rm{V}}}}}}}_{{{{{{\rm{i}}}}}}}\right {{{{{{\rm{V}}}}}}}_{{{{{{\rm{i}}}}}}+1}-{{{{{{\rm{V}}}}}}}_{{{{{{\rm{i}}}}}}}\right

Are you looking for a Windows 10 activator? If the answer is yes, then you have come to the right place! In today’s guide, we’re going to share the best tool you can use to activate Microsoft Windows 10 for free.

Yes, this is the truth! The tools we are about to offer are 100% free and won’t cost you a penny. So keep reading this how-to article and leave nothing behind. If not, you might lose some information. Several tools on the Internet work well. But most of them are fake and do not activate for a lifetime.

I was looking for a windows 10 activator, but I spent about 4 hours searching for the right tool for myself. Do you know? Then I finally got the best tool. It works like a mantra! So I want to share all of the information on this tool here.

1. Windows 10 Product Key
2. Windows 10 ISO File
3. Microsoft Toolkit

### What is Windows 10 Activator?

Windows 10 Activator is a free tool that allows you to activate different versions of your operating system. In addition to that, this cool app will start activation with a quick one-click process.

It is mainly designed for this purpose and meets the OS administrator’s requirements. The administrator can configure or change the activation settings, so there is no possibility of corrupting files. Windows files are important.

Moreover, the application is compatible with most operating systems and can be run from an installed hard drive.

The booting process is pre-programmed, so there is no need to tweak or tweak the system configuration manually.

This lightweight and reliable software removes malware, viruses, and trojans and ensures a clean program at all times. Plus, you can safely use the app on any Windows operating system you want.

KMSPico is a new and popular tool that most people use on the Internet. This tool was developed by TeamDaz and had many contributions to these areas of activity.

If you are a previous user of these activators, you might have heard about them. This is because TeamDaz has created a lot of support tools for Windows 7 and Windows 8 / 8.1.

But after Windows 10 made so many changes, many people are concerned that they will not be able to get the full version without paying a fee. Thanks to this kit, we have this great KMSPico tool, and now we will use it to activate Windows 10.

This tool operates according to the guidelines recommended by Microsoft (abbreviated as KMS).

The reason behind this idea is that large business owners are not comfortable activating their products individually, so Bill Gates announced a KMS server technology that allows you to connect your computer to the server.

This server includes product keys for Windows and Office. Every time you install Windows and connect to the server, it is activated automatically.

The program’s functionality is equivalent to connecting to a server with many product keys. These keys are updated continuously for a period of time, so users who wish to activate for life must go online once within 180 days. Otherwise, the product key will not be updated, and the activation will be lost.

Now we think you know the value of this great tool. If so, do not waste your time and check out our download guide. Downloading the KMSPico from the official website is not difficult as we do not use links, pop-ups, or pop-under. Although, visitors can easily download Windows 10 Activator.

You can find details about the product on this page, and you will find buttons below this information. Click to open the Mega website.

You will see two options here:

• Import as (MEGA Icon)

Do you see the file name there? Now you have successfully downloaded KMSPico to your computer. Now let’s move on to another tutorial that shows you how to install it properly.

### How to install Window 10 Activator

Ok. After downloading the file, there are instructions. Follow the instructions to install the files on your computer properly, don’t miss a step, or else your personal files might be damaged. Read the entire procedure twice to avoid mistakes.
Note:
We recommend that you disable Windows Defender before following the installation instructions. Otherwise, the file will be deleted.

Here’s how to turn it off.

• Open the “Start” menu, search for Defender in the search box, then open WindowsDefenderSecurityCenter.
• Go to “Virus and threat protection” and open “virusandthreatprotection.”
•  Turn off Real-time protection here, then click Yes if necessary.

If you are using third-party antivirus software, disable it.

A new folder will be created with the name of the downloaded Zip file. Open the folder, right-click on KMSPico-setup, then click “Runasadministrator.”

It will ask for confirmation, so click “Yes,” and the program will open the installer window.

Follow all the steps shown here, and it will install to your system in seconds.

When the installation is complete, it will not appear on your screen. But you can see it by going to the Start menu and looking at the “Recently Added” menu.

This means that the activator is already installed on your computer. Now is the time to find out how you can use activation to activate Windows 10. Here are the complete instructions to follow.

This is the complete step-by-step guide on how to activate Windows 10 using KMSPico Activator, don’t miss a step, or else you might damage some personal files.

After installing KMSPico, restart your computer and then follow the steps above to disable Windows Defender.

Open the “Start” menu and right-click on KMSPico under “Recently added applications” (If you cannot find KMSPico in the “Recently added applications” section, you can search for KMSPico.)

When you see a dialogue asking for confirmation, click Run as administrator and then Yes.

A new window will appear with some options and a redbutton.

Don’t touch anything; just click the red button and wait. After a few seconds, you will hear the messages “Affirmative or YES” and “Programcompleted.”

When you hear the notification, you will see the Windows 10 logo with a green background; if you see this message, Windows is activated.

Now restart your computer, right-click on “My Computer” and open “Properties,” and pull up the Windows Activation section and you will see the message “Windows is activated.”

### Why use Windows 10 Activator?

Microsoft Windows is a very popular operating system in the world. Since it is a very simple and user-friendly interface, most people prefer Windows over other operating systems such as Mac OS or Linux.

However, one of the problems is that since this OS is not free like other OS, you have to pay to use it on your computer.

A free 30-day trial is available, and at the end of the trial period, you will be required to purchase a product key. If you don’t, the “Activate Windows” watermark will appear on the desktop.

Users are also prohibited from using certain features like “Change wallpaper,” “Get Microsoft updates,” “Annoying screen watermark,” etc.

This is why people search for Windows 10 Activator so they can get the full version for free because not everyone can buy a copy of the window.

Now, let me tell you this tool is 100% working fine and will be activated permanently. When you use one of the activators given, your window will always be activated, and it will always be there unless you reinstall Windows.

### Features

Here is a list of the main features of Windows 10 Activator. Since we are writing some of the main features, you may not find them all. You can see other features while using the tool that is not described here.

Try various tools, but can you get annoying popup ads from them? I also encountered this problem when testing a tool with too many annoying ads on the Internet. Now, this issue has been fixed as there isn’t a single ad. Hence we recommend it personally.

There is no virus

Some people on the Internet say they have viruses, some say they contain malware, and others say they stole personal information.

This is just a rumour that VirusTotal scanned, and the tool did not find any kind of malware.

Since we are using it since 2017, you can use it without worrying about anything. Your data is safe and absolutely free from virus/malware attacks.

Free

What you want is that you will never ask for payment. Yes, the tool is free and does not require any registration. It can be used with multiple computers without any problems.

### Is there a virus in Windows 10 Activator?

Of course, this tool is 100% safe and free from malware. Antivirus detects it as a virus. But in reality, all the activators are blacklisted from these programs. So, no matter which tool you use, it will always appear to contain a virus.

### How do I get a Windows 10 license key?

When using KMSPico or other tools, users are not required to provide the license key. However, if you prefer somehow to use a license key rather than a tool, KMSAuto also lets you create a key for Windows and Office.

Источник: https://cwiki.apache.org/confluence/pages/viewpage.action?pageId=181306341
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## Activation pathway of a G protein-coupled receptor uncovers conformational intermediates as targets for allosteric drug design

### Abstract

G protein-coupled receptors (GPCRs) are the most common proteins targeted by approved drugs. A complete mechanistic elucidation of large-scale conformational transitions underlying the activation mechanisms of GPCRs is of critical importance for therapeutic drug development. Here, we apply a combined computational and experimental framework integrating extensive molecular dynamics simulations, Markov state models, site-directed mutagenesis, and conformational biosensors to investigate the conformational landscape of the angiotensin II (AngII) type 1 receptor (AT1 receptor) — a prototypical class A GPCR—activation. Our findings suggest a synergistic transition mechanism for AT1 receptor activation. A key intermediate state is identified in the activation pathway, which possesses a cryptic binding site within the intracellular region of the receptor. Mutation of this cryptic site prevents activation of the downstream G protein signaling and β-arrestin-mediated pathways by the endogenous AngII octapeptide agonist, suggesting an allosteric regulatory mechanism. Together, these findings provide a deeper understanding of AT1 receptor activation at an atomic level and suggest avenues for the design of allosteric AT1 receptor modulators with a broad range of applications in GPCR biology, biophysics, and medicinal chemistry.

### Introduction

As the largest superfamily of cell surface proteins in the human genome, G protein-coupled receptors (GPCRs) represent the therapeutic targets of nearly one-third of all approved drugs1. These receptors share a conserved structural architecture of seven transmembrane (7TM) helices linked by three extra- (ECLs) and three intracellular loops (ICLs)2. The GPCR-mediated signal transduction is always triggered by an extracellular signal to the orthosteric site located in the extracellular region of the 7TMs bundle center, which then transduces the stimuli to the intracellular region, thereby leading to the engagement of the receptor with G proteins or β-arrestins3,4. In addition, some GPCRs can also transmit signals in the absence of an external stimulus or an agonist, through ‘basal’ (also known as ‘constitutive’) activity5,6,7,8. The orthosteric site is conserved across the members of a single GPCR subfamily and, thus, poses a significant challenge in the development of selective drugs that can bind to a unique receptor subtype. As an alternative strategy, targeting a binding site outside the conserved orthosteric site, also termed as an “allosteric site”, may provide avenues for the design of modulators with desirable selectivity profiles9,10,11,12,13,14, which is a long-standing bottleneck in GPCR drug discovery.

Recent technological breakthroughs in structural biology, such as cryo-electron microscopy (cryo-EM) or X-ray free-electron lasers, have led to the identification of increasing GPCR structures15,16,17, either in the inactive or active conformations, thereby providing mechanistic insights into the agonist-dependent receptor activation mechanisms that are useful for investigating structure-based drug design18,19. However, these high-resolution structures represent static snapshots obtained under specific experimental conditions; hence, they may miss important information pertinent to the conformational ensemble of GPCRs, as the receptors may have undergone a large-scale conformational transition during their (de)activation process20,21. Therefore, mechanistic and structural elucidation of the (de)activation pathway of the GPCRs is of paramount importance as distinct conformational states, such as intermediate, metastable, or transient states, present during the inactive-to-active transitions (or vice versa) of the receptors, are desired for a rational design of selective modulators.

The angiotensin II (AngII) type 1 receptor (AT1 receptor), a prototypical class A GPCR, offers an important model for mechanistic exploration as it is a prominent therapeutic target for hypertension and related cardiovascular diseases22. Moreover, its high-resolution structures in both inactive, antagonist-bound23 and active, agonist-bound, and nanobody-stabilized conformations have recently become available24,25,26. By comparing the structures of its inactive, antagonist-bound state (ZD7155; PDB ID: 4YAY)23 (Fig. 1A) and the active state, meaning the AT1 receptor complexed with a partial agonist S1I8 peptide, and a G protein mimetic nanobody to maintain the active conformation (PDB ID: 6DO1)24 (Fig. 1B), the most remarkable differences in the two structures are found in the TMs 5–7 and helix 8 (H8) (Fig. 1C). Notably, on the intracellular side, the active state structure exhibits an outward displacement of TM5 and TM6, inward movement of TM7, and substantial repositioning of H8 parallel to the membrane, relative to the inactive state structure (Fig. 1C). On the extracellular side, the major conformational changes include an inward shift of TM5 and TM7 in the active state structure compared with the changes in the inactive state structure. Although the static, active, and inactive states of AT1 receptor exhibit marked structural divergences, it has been challenging to completely capture the large-scale conformational transitions along the activation pathway of the AT1 receptor experimentally. Therefore, it remains unclear how a dynamic pathway connects the inactive-to-active conformational transitions of the AT1 receptor, thereby hindering a deeper understanding of the comprehensive landscape of the activation mechanisms of this receptor as well as for other GPCRs. Furthermore, despite the availability of the inactive and active structures of the AT1 receptor, there are no allosteric modulators of this receptor reported to date, suggesting a challenge for targeting potential allosteric binding sites in the two available snapshots. However, a cryptic allosteric site may exist in the transition pathway. Thus, it is advisable to capture key conformational substates along the activation pathway for the purpose of allosteric drug design27,28.

To uncover the activation pathway of GPCRs, biosensors, nuclear magnetic resonance (NMR), and computational methods have been widely applied29,30,31,32,33,34,35,36. Among these approaches, molecular dynamics (MD) simulations have become a well-established technique for probing the conformational landscapes at an atomic level and directly uncovering biomolecular mechanisms37,38,39. Integrating MD simulations with Markov state models (MSMs)40 has proven successful for understanding the molecular switches in β2 adrenergic receptor (β2AR)41, elucidating ligand-driven conformational changes in CC chemokine receptor (CCR) 242, and revealing a cryptic pocket in dopamine D3 receptor43. In the best structurally and biochemically characterized GPCRs, the rhodopsin receptor, the activation pathway, and the corresponding intermediate states have been elucidated by NMR44, Fourier transform infrared spectroscopy45, and MD simulations46.

Here, we use a computational framework including a transition pathway generation algorithm, extensive all-atom MD simulations (300 μs) of the AT1 receptor in the membrane-embedded environment, and MSM analysis for investigating the conformational landscapes of AT1 receptor activation. We found an intermediate state during activation and identified a cryptic allosteric pocket on it. Multiple mutagenesis experiments confirm both the intermediate state and the potential regulation ability for the pocket. Our study not only offers a deep atomic-level insight into AT1 receptor activation, but also provides an opportunity for the design of allosteric AT1 receptor modulators.

### Extensive unbiased MD simulations reveal the activation pathway of AT1 receptor

To understand the inactive-to-active conformational transition pathway of the AT1 receptor, we first generated a minimum energy path (MEP) by connecting the starting, antagonist-bound inactive (PDB ID: 4YAY), and the end, both nanobody- and agonist-bound fully active (PDB ID: 6DO1) states, by inserting a series of replicas between the two AT1 receptor structures using the string method with the nudged elastic band (NEB) (see “Methods”). Both the nanobody and agonist were removed from 6DO1 and the antagonist was excluded from 4YAY. Here, we defined the GPCR structures with only agonist-bound forms as active conformations and with both agonist- and G protein-, β-arrestin-, or nanobody-bound forms as fully active conformations. After a simulated annealing process, 15 initial structures distributed on the MEP were selected and subsequently embedded with a 1-palmitoyl-2-oleoyl-sn-glycero-3-phosphocholine (POPC) membrane and explicit water. Each structure had 2 μs × 10 independent runs with random initial velocities, leading to a cumulative simulation timescale of 300 μs. Such an extensive timescale has been proven efficient for exploring the GPCR activation process38. As shown in Fig. 1, the most remarkable intracellular transmembrane domain conformational variations of the AT1 receptor during activation are arguably the outward movements of TM5 and TM6, and the inward displacement of TM7. Thus, we defined two activation parameters (Fig. 1C) to project the simulated trajectories onto a two-dimensional (2D) space to comprehensively capture the conformational landscape of AT1 receptor activation. Notably, L5.55 (superscripts indicate the Ballesteros–Weinstein numbering for GPCR residues)47 undergoes a large and conserved rearrangement during GPCR activation and N7.46 localizes at the TM7 twist position, which reflects the inward movement of TM7 during activation. Thus, one parameter for defining activation was the distance between the Cα atoms of L5.55 and N7.46, which represented the conformational changes of TM5 and TM7. The other parameter was the angle among the Cα atoms of F6.34, S6.47, and V2.41, which reflected the outward movement of TM6, a crucial hallmark of class A GPCR activation that provides space to accommodate downstream signal proteins including G proteins or β-arrestins48. As shown in Fig. 1C, the corresponding distance and angle values were markedly distinct between the inactive and fully active AT1 receptor crystal structures, thereby highlighting the discriminatory power of the activation parameters.

Based on these activation parameters, we calculated the corresponding value of each snapshot during the simulations and plotted a free-energy landscape (Fig. 2A), depicting the inactive-to-active transition pathway of AT1 receptor activation. Since the initial inactive crystal structure of the AT1 receptor was located at a distance and angle of 21.3 Å and 36.7°, respectively, the largest free-energy basin with a distance of ~19–23 Å and an angle of ~35–42° represented the inactive region. Further, owing to the outward movement of TM6, the angle gradually increased, reflecting an open degree of TM6, whereas the movement of TM5 and TM7 caused a decrease in the interhelical distance. During the activation process, the AT1 receptor overcomes a relatively low-energy barrier and enters the intermediate state located at a free-energy basin with a distance of ~17–19 Å and an angle of ~46–51°, and it then crosses a high energy barrier at ~17 Å and 63° to arrive at the fully active state (17.4 Å, 69.4°), representing the coordinates of the fully active AT1 receptor crystal structure (PDB 6DO1). Overall, the convergence of the landscape was proven in both the timescale of a single trajectory and the number of rounds (Supplementary Note 1), confirming that the sampling has been sufficient to explore AT1 receptor activation.

Because of the recent breakthroughs in receptor crystallization, more than 440 class A GPCR structures of >70 receptors have been determined by X-ray or cryo-EM crystallography (Supplementary Note 2 and Supplementary Data 1). These structures represent the GPCR conformational ensemble covering all inactive, active, and intermediate states, including different inverse agonist- or antagonist-bound inactive conformations, agonist-bound active conformations, and their effector G protein-, β-arrestin-, or nanobody-stabilized fully active conformations. Considering the conformational heterogeneity of class A GPCR structures, we projected these experimentally solved structures onto the 2D conformational landscape of the AT1 receptor sampled by the simulations. The corresponding residues at positions 5.55, 7.46, 6.34, 6.47, and 2.41 of each structure were selected to calculate the distance and angle using their Cα atoms. Then, the fully active, inactive, and active states were mapped to provide insights into the dynamic conformational landscape of the receptors. The projection suggests that the AT1 receptor activation pathway samples a wide conformational landscape of GPCR ensemble with the major distribution of experimental structures within the reaction path (Fig. 2B). This result suggests that the computational model is in reasonable agreement with the experimental data, highlighting the ability of our simulations to reproduce the overall class A GPCR activation and suggesting that it is thus suitable for further investigations.

Interestingly, the inactive energy basin largely matched its crystal structures. This can perhaps be ascribed to the fact that the inactive structures have mostly similar TM5−TM7 conformation. The outliers—platelet-activating factor receptor (PDB 5ZKP) and P2Y12 receptor (PDB 4NTJ)—however, have an unusual anchor (TM2) movement and distinct TM6 twist, respectively (Supplementary Note 3)49,50. Since agonist binding shifts the receptor conformational ensemble toward the active state, resulting in increased conformational dynamics and the coexistence of multiple states along the activation pathway51,52, diverse structurally activated states can be captured under distinct conditions. For example, with the same ligand AM-841, the type 1 and 2 cannabinoid receptors (CB1R and CB2R; PDB 5XR8 and 6KPC, respectively) sample entirely different conformations. After binding another agonist, CP55940, and a negative allosteric modulator, ORG27569, CB1R (PDB ID: 6KQI) also situates at the inactive cloud (Supplementary Note 3). Indeed, half of the active structures are located at the intermediate cloud sampled by the simulation, indicating that the intermediate state extensively exists in class A GPCRs. Most fully active structures are proximal to the active cloud sampled by the simulation, except the rhodopsin receptor that has a larger distance (~22 Å) between TM5 and TM7 due to its unique activation mechanism45 (Supplementary Note 3). Of note, receptor-Gs complex has a larger angle than receptor-Gi complex, which is consistent with the different volumes of the two downstream proteins. The position of the G protein complex has a similar distribution with nanobody-stabilized structures, suggesting that the nanobodies used in crystallization have a minor influence on its conformations (Supplementary Note 4). Since our simulations are based on the apo structure, the active cloud is not highly correlated with fully active non-rhodopsin structures, indicative of the instability of active structures and the necessity for transducers to stabilize the fully active conformations. Collectively, the free-energy landscape illustrates the activation process of the AT1 receptor and uncovers a hidden intermediate state along the pathway linking the inactive and active receptors.

### Markov state model discovers a synergistic transition mechanism of AT1 receptor activation

To elucidate the mechanism of AT1 receptor activation, we built a kinetic network MSM using the activation parameters. From a statistical viewpoint, MSM provides summarized insights of the conformational ensemble of biomacromolecules at equilibrium. Therefore, the key intermediate states are identified more precisely and the thermodynamic properties, such as transition timescale, are also quantified40,41. Upon validation of the Markovian properties by implied timescale test (Supplementary Note 5), we constructed a 200-microstate MSM using the k-means algorithm and clustered them into three macrostates (Fig. 3A), using the Robust Perron Cluster Cluster Analysis (PCCA+) algorithm. Next, transition path theory (TPT) was applied to map out the transitions and their timescales among the active, inactive, and intermediate states (Fig. 3B). The prediction of MSMs in the three macrostates was also confirmed by the Chapman–Kolmogorov test (Supplementary Note 5).

The conformational space was divided into three regions, and the locations of yellow, green, and purple regions corresponded to inactive, active, and intermediate states, respectively (Fig. 3). Thus, the population ensemble determined by the 3-macrostate MSM was consistent with our state distribution. Notably, the proportion of the inactive state (43.1%) was the largest in the ensemble, in agreement with the fact that the apo AT1 receptor naturally stays in the inactive state. However, the active state also occupied 28.1% of the conformations during simulation, indicating that the apo AT1 receptor still has the ability to trigger downstream pathways and the AT1 receptor might have constitutive activity.

The transition from the inactive to the active states (401.63 μs) was significantly longer than the transition from the active to inactive states (266.84 μs), as shown in the transition timescale (Fig. 3B). This observation supports the concept that the apo AT1 receptor is normally inactive, and that the active conformer also exists in its ensemble. Based on the AT1 receptor conformational distribution, it is evident that both the active (86.46 μs) and inactive (93.95 μs) states are amenable to transfer to the intermediate state. However, it was difficult for the intermediate state to transfer to the active (257.87 μs) and inactive (125.66 μs) states. Hence, the kinetics of the activation pathway indicates that the rate-limiting step of AT1 receptor activation is the transition from the intermediate to the active states.

As our simulations were based on the apo AT1 receptor, the spontaneous activation reflects a constitutive activity of the wild-type (WT) AT1 receptor, which has been previously reported53,54. Using bioluminescence resonance energy transfer (BRET) assays in G proteins, we confirmed the constitutive activity of the AT1 receptor in Gq, Gi, and G12 compared with a control vasopressin 2 receptor (V2R). We first controlled the equal levels of the AT1 receptor and V2R on the cell surface (Supplementary Note 6). Next, the dissociation of Gα and Gβγ was represented by the decrease of ΔBRET. The ΔBRET decreased with increasing AT1 receptor levels in a dose-dependent manner, suggesting that downstream G proteins are dissociated from the apo AT1 receptor (Fig. 3C–E). In contrast, V2R levels had no influence on the ΔBRET value. Collectively, these data suggest a constitutive activity of the AT1 receptor.

To further elucidate the activation mechanism of the AT1 receptor, we evaluated key residue rearrangements. During GPCR activation, the emergence of residue rearrangements—or micro-switches—is fundamental to TM movement48. Common activation micro-switches include breaking of the hydrophobic lock composed of residue positions 3.43, 6.40, and 6.41, which loosens the connection between TM3 and TM6, and the rupture of a conserved ionic lock between residue positions 3.50 and 6.30, which releases TM6 outward48,55. In addition, the AT1 receptor harbors unique residue movements as well. In the inactive state, R3.50 and N6.30 form a relatively weak polar interaction instead of the conserved ionic lock and break upon activation. Further, F6.44 and F6.45 ratchet over F5.51 when the AT1 receptor is activated. In addition, Y7.53 and Y5.58 tend to approach each other during AT1 receptor activation, which reflects the movement of TM5 and TM7 (ref. 24).

We then clustered the representative conformations of the macrostates according to the root mean square deviation (RMSD) between structures (see “Methods”). The micro-switches are also shown in the inactive, intermediate, and active AT1 receptor states (Fig. 4A–C). Based on a representative simulation trajectory across the three states, we observed the dynamic positioning of our switches (Fig. 4D–F). Globally, TM6 moved outward, while TM5 and TM7 approached the inactive-to-active conformational transition. Meanwhile, the landscape coordinates of the inactive, intermediate, and active representative conformations were located at (23.3 Å, 40.3°), (19.0 Å, 50.6°), and (17.1 Å, 71.3°), respectively; matching the low-energy conformer and further strengthening the accuracy of our cluster methodology.

Other structural elements also appear functionally significant. Upon activation, ECL2 and extracellular TM6 moved toward the center of the endogenous AngII pocket and nearly closed the pocket in the active state, indicating that ECL2 and TM6 stabilize the pocket and maintain the binding of AngII during activation (Supplementary Note 7). The closure of the orthosteric pocket has also been observed in the active β2AR-nanobody complex structure56. Moreover, intracellular TM5 formed more α-helices in the active state, and the length of ICL2 decreased accordingly to help the binding of the downstream proteins, which is in line with the common GPCR activation process. In addition, H8 moved upward from the inactive to the active state to provide space for the downstream effectors. This upward movement produces a suitable accommodation for ligand binding among TM1, TM7, and H8, which may offer an opportunity for the formation of a cryptic pocket for drug design (Supplementary Note 7). We further explored the variation of secondary structural elements among macrostate trajectories using the Definition of Secondary Structure of Proteins algorithm (Supplementary Note 8) and community analysis for the signal transfer (Supplementary Note 9). During receptor activation, the ECL2 became stable, the intracellular TM5 was elongated, and connections to the transducer pocket increased. The upward movement of H8 was also observed.

In agreement with the observations revealed by comparing the crystal structures, the micro-switches generally and synergistically changed from the inactive state to the intermediate and active states. For example, V6.41 moved toward the membrane step-by-step and finally broke the hydrophobic lock in the active structure. This movement provided space for Y7.53, which finally approached Y5.58 (inactive: 18.9 Å, intermediate: 13.1 Å, active: 5.1 Å). In addition, the distance between R3.50 and N6.30 increased upon the outward movement of TM6 (inactive: 5.4 Å, intermediate: 15.3 Å, active: 22.8 Å), which in turn, rendered F6.44 and F6.45 to ratchet over F5.51. Meanwhile, the relative positions of the residues in the intermediate state were in between the active and inactive states, indicating that the activation process generally occurs. In a single spontaneous activation trajectory (Fig. 4D–F), the variations of the three micro-switches—breaking of the ionic lock (Fig. 4D), the opening of the hydrophobic lock (Fig. 4E), and rearrangement of phenylalanine ratchet (Fig. 4F)—synchronize, and thus reflect the synergistic activation mechanism of the AT1 receptor. The rationale for the measurement of residue pairs describing the hydrophobic lock is provided in Supplementary Note 10. Thus, the zoom-in views of the residue rearrangements were consistent with the global secondary structural movements.

To demonstrate the transition pathway, we employed gaussian accelerated MD (GaMD) simulations on the holo AT1 receptor systems using the intermediate structure as a model system. The inverse agonist olmesartan and the endogenous agonist AngII were docked to the orthosteric pocket of the intermediate state, respectively. Then, the two holo systems underwent 2 μs conventional MD simulations followed by 1 μs GaMD simulations for three independent runs. In the corresponding free-energy landscapes, AngII binding pushed the open of TM6 along with the inward movement of TM5 and TM7, resulting in the active state, while olmesartan binding led to a smaller TM6 angle and a larger TM5–TM7 distance, shifting the receptor conformation to the inactive state (Supplementary Note 11). Together, these results suggest the likely existence of an intermediate structure as captured from the transition pathway.

Since the AT1 receptor activates multiple G protein subtypes in its apo and holo states, we further explored whether this intermediate state could be commonly required by different G protein downstream signal pathways. By comparing the intermediate structure with the inactive and active macrostate structures, we identified several specific micro-switches for the intermediate state, such as polar contacts among K5.42, H6.51, and T6.55; a hydrogen bond between Y5.58 and I6.37; hydrophobic network among M6.38, W5.62, and F6.34; and hydrophobic contacts among V1.53, V1.56, and F8.50 (Fig. 5A and Supplementary Note 12). Based on these intermediate-specific interactions, we designed K5.42A, Y5.58A, W5.62A, F6.34A, and F8.50A mutations to specifically disrupt the intermediate state. The protein levels of the variants were similar to that of the WT receptor (Supplementary Note 13). Constitutive and AngII-induced activities for G proteins were then measured using BRET assays to evaluate the influence of these mutations (Fig. 5B–D).

Based on BRET assays, the disruption of micro-switch interactions in the intermediate state inhibited both the constitutive and AngII-induced activities of the AT1 receptor for G proteins, including Gq, Gi, and G12. The decrease of constitutive activity was more obvious than the AngII-induced activity. In the AngII-induced activity, K5.42A decreased the activity of Gq and Gi more than G12, suggesting that Gq and Gi activity may need a larger movement of extracellular TM5. Although W5.62A and F6.34A did not influence the Gq and Gi signal in Emax, the increase of the EC50 value reflected a weaker activation upon AngII binding (Supplementary Note 14). The remaining Y5.58A and F8.50A mutations similarly modulated the activity of the AT1 receptor in the three G proteins. In summary, these results indicated that the intermediate state is required for all G protein signaling pathways, highlighting its possibility to become a drug target.

### Distinct active conformations from global movement can induce biased signals

Apart from the dimension reduction accomplished by the features extracted from the biological process (such as the activation parameters shown in Fig. 1C), we also applied time-structure-based independent component analysis (tICA) to our system to analyze the global movement of the receptor during activation. tICA employs linear combinations to particular features, such as the phi/psi angle of the backbone, to maximize the decorrelation time among these features. Thus, tICA enables the capture of slow dynamic processes during simulations40,57. With the help of tICA, we projected the trajectories onto another 2D landscape, which was representative of the global phi/psi movement. Through the implied timescale test, dimensionality reduction using tICA has been proven to have Markovian properties. The tICA landscape was separated into eight macrostates according to the PCCA+ algorithm, and its prediction met the requirements of the Chapman–Kolmogorov test (Supplementary Note 15). To analyze the global movement during simulations, the representative conformation of each tICA macrostate was extracted and projected onto the distance-angle landscape based on their corresponding activation parameters (Fig. 6A). Finally, the TPT method was applied to calculate the transition dynamics between six major macrostates (Fig. 6B).

As shown in Fig. 6A, macrostates 1 and 5 were in the intermediate state, macrostates 2–4 and 8 were in the inactive state, and macrostates 6 and 7 were in the active state. Notably, the macrostates 3–8 occupied more than 96% of the snapshots during our simulations (Supplementary Note 15) and stayed close to the energy basins in our landscape (Fig. 2A), thereby suggesting that the coordinate parameters (distance and angle) elucidated the activation process just as the global movement did. Thus, the mechanism according to the activation parameters is credible. In addition, the representative conformations of macrostates 4 and 6 corresponded to the area with the most inactive and active crystal structures, respectively (Figs. 2B and 6A). Pathway 4–5–6 also takes the shortest transition time, thus it is the most preferential way to activate the AT1 receptor among all the analyzed pathways (Fig. 6B). Hence, macrostates 4 and 6 are crucial conformers along the activation pathway, that is, macrostate 4 is encountered before entering the activation process, and the typical activation process involves the macrostates 4–6.

Since the AT1 receptor has a constitutive activity for G proteins (Fig. 3C–E) and shows β-arrestin activity upon AngII binding or specific agonists25, we explored the connection between different active states and biased signaling. Because Gq is the major G protein activated by the AT1 receptor and β-arrestin 2 is commonly used in biased signaling38,58,59, we first constructed AT1 receptor−Gq and AT1 receptor−β-arrestin 2 complexes based on the active macrostates 6 and 7 to determine whether the two active conformations had a bias for these transducers (Supplementary Note 16). The models showed that macrostates 6 and 7 tend to initiate Gq and β-arrestin 2, respectively, suggesting that biased conformations naturally exist in the ensemble of the AT1 receptor.

Supported by this preliminary analysis, we further investigated biased signaling in the AT1 receptor based on tICA analysis. From the representative structures, we explored specific micro-switches of each macrostate (Fig. 7A). Y7.53 is close to hydrophobic residues at TM1 and TM2 in both macrostates 5 and 6, while it forms a hydrogen bond with R3.50 or Y5.58 in macrostates 1 and 7, respectively. Thus, by introducing a hydrophobic residue, the Y7.53I mutation could maintain the hydrophobic interactions, but disrupted the polar interactions, leading to the stability of macrostates 5 and 6, and the instability of macrostates 1 and 7. In addition, N3.35 and D2.50 form a tight hydrogen bond in macrostate 7, whereas it is weak in macrostate 6 and diminishes in macrostates 1 and 5. N3.35A mutation disrupted the hydrogen bonding interaction, preferred the conformation with a long distance between the residue at 3.35 and the polar residue at 2.50 (macrostates 1, 5, and 6). In turn, a moderate mutation D2.50N changes the charge of its sidechain, which may weaken the interaction between residue at 3.35 and residue at 2.50. Because macrostate 7 has a strong hydrogen bond, its conformation may maintain in response to the D2.50N mutation. However, macrostate 1, 5, and 6 may be disturbed owing to this mutation. Our mutation experiments for biased signaling were benchmarked against these micro-switches and the corresponding results (Fig. 7B–E). An operational model was also applied to determine the biased signaling with normalized Emax values (Fig. 7F).

In Fig. 7F, blue and light pink represent a weak signal and red reflects a strong signal, compared with the WT AT1 receptor. In particular, N3.35A mutation (benefits macrostates 1, 5, and 6, inhibits macrostate 7) obviously promoted Gq and Gi signals but inhibited G12 and β-arrestin 2 signals. In contrast, D2.50N showed relatively weaker Gq and Gi signals compared with G12 and β-arrestin 2 signals (Fig. 7F and Supplementary Note 17). Facilitating macrostates 5 and 6, Y7.53I also promoted Gq but inhibited β-arrestin 2. Thus, micro-switches facilitating macrostate 6 and restraining macrostate 7 (N3.35A, Y7.53I) led to the Gq activation and β-arrestin 2 inhibition, whereas the D2.50N mutation, boosting macrostate 7 and reducing macrostate 6, preferred β-arrestin 2 rather than Gq signal. This is in line with our hypothesis that macrostate 6 is biased to Gq pathway, whereas macrostate 7 is for β-arrestin 2 (Supplementary Note 16). As for Gi, it can be inferred that the intermediate macrostate 1 (inhibited by Y7.53I and D2.50N, facilitated by N3.35A) may play a critical role in the activation toward Gi since Y7.53I and D2.50N suppressed Gi signal but N3.35A increased it. Conversely, G12 activation may be related to the macrostate 5 (facilitated by Y7.53I, inhibited by N3.35A and D2.50N) because Y7.53I stimulated the G12 signal, while N3.35A and D2.50N repressed G12 activation. The effect of mutations can be confirmed by recent structural information. For instance, Y7.53I quenches polar interactions with R3.50 and Y5.58, which releases intracellular TM3 and TM5. The flexible TM3 can move downward and increase the interaction of ICL2 with transducers, while TM5 might extend toward G protein. Accordingly, the increased ICL2 interaction and extended TM5 are properties of GPCR-Gq/11 complex compared with GPCR-Gi/o complex60,61,62. Thus, Y7.53I promotes Gq but inhibits Gi signals in the AT1 receptor.

In summary, we observed hints for each biased signal pathway in our MD simulations for the apo AT1 receptor and confirmed them by site-directed mutation experiments. During biased signaling, different ligands stabilize distinct receptor conformers, reflecting the biased signaling of the AT1 receptor via a conformational selection mechanism. Taken together, tICA identified a transition pathway based on global movements, and the active and intermediate conformations provided indications of biased signaling.

### Identification of a cryptic allosteric site in the intermediate state

We also used a pocket prediction algorithm to identify potential sites on macrostates and guide the allosteric drug design. Using Fpocket, we identified and clustered nine pockets on the hidden intermediate state (Fig. 3A) and several pockets in other states. Fpocket defines alpha spheres by Voronoi tessellation and regards the alpha spheres that are concentrated on clefts or cavities and with the ability to bind small molecules as the pocket63. In Fig. 8A and Supplementary Note 18, the positions of the pockets are depicted and the overlapping pockets between the intermediate state and other states are also highlighted. Most of the detected pockets in the intermediate state corresponded to pockets in the inactive and active states, except for P6 and P9, which were the cryptic pockets only in the intermediate state.

Among the nine pockets, P1 was located at the center of the AT1 receptor and spanned from top to bottom. It revealed a large cavity inside the AT1 receptor that partly overlapped with the orthosteric site for AngII. Nevertheless, P1 was too large for the design of small-molecule modulators. In addition, the P2–P6 pockets laid on the membrane side, whereas the intracellular P7–P9 pockets were in the proximity of the transducer site (Fig. 8A). The volume of P2–P9 pockets could fit possible allosteric modulator sites, and P2, P4, P8, and P9 matched previous allosteric drug sites as well (Fig. 8B, C). In particular, P2 was mostly identical to the LY2119620 pocket in the M2 muscarinic acetylcholine receptor (PDB 4MQT)64. P8 also partially overlapped with the allosteric agonist AP8 pocket in free fatty acid receptor 1 (PDB 5TZY)65. In addition, BPTU interacted with P2Y1 receptor (PDB 4XNV)66 in the vicinity of P4, confirming the accuracy of our site prediction. P9 overlapped with three allosteric sites observed in GPCR-modulator structures as an intermediate-only potential pocket. The sites were as follows: the CCR2-RA-[R] site in CCR2 (PDB 5T1A), the vercirnon site in CCR9 (PDB 5LWE), and the Cmp-15PA site in β2AR (PDB 5X7D), and were not confined to the intermediate state67,68,69. Thus, P9 is a common allosteric site in the intermediate state of the AT1 receptor. However, since P2, P4, P8, and P9 emerged as the allosteric sites in other class A GPCR-modulator structures, the sites shown in Fig. 8A are highly likely allosteric sites.

Since P9 was excluded because of its universality, P6 remained the only cryptic allosteric site in the intermediate state. As P6 has never been reported in previous GPCRs and allosteric modulator structures, our findings indicate that it may be an allosteric site for drug design. We also conducted molecular screening on P6 using a GPCR allosteric compound library and found a binding pose highly overlapping with P6 from the best-scored ligand (Supplementary Note 19), further suggesting the targetable potential of P6.

### Mutagenesis and FlAsH-BRET assay confirm the predicted cryptic binding site

Next, we confirmed that P6 was an allosteric site firstly using clustered alanine-scanning mutagenesis. Three groups of mutations (Fig. 9A, B) were independently introduced to directly investigate the effect of allosteric perturbations of P6 on the transducer activity of the AT1 receptor. The cluster 1 mutations included F1.48A, L1.52A, and I1.57A, located on the intracellular side of TM1. The cluster 2 mutations were N7.49A, P7.50A, and F7.55A on TM7. The cluster 3 mutations (K8.49A, F8.50A, K8.51A, Y8.53A, and F8.54A) are on the top of H8. Since the AT1 receptor acts as a model system for biased signaling, we further investigated whether these cluster mutations had effects on AngII-induced Gq activation as well as on β-arrestin 2 recruitment using BRET assays upon similar expression (Supplementary Note 20). Cluster 2 and 3 mutations completely depleted the two activation pathways of the AT1 receptor, whereas cluster 1 mutations decreased both two Emax by 22–34% (Fig. 9C, D, Supplementary Note 20). Thus, the direct perturbation at P6 influenced the G protein and β-arrestin pocket.

To further determine the potential of P6 on our intermediate state to regulate the binding of downstream transducers to the G protein pocket, we applied the AlloSigMA algorithm, which evaluates allosteric effects using the Structure-Based Statistical Mechanical Model of Allostery70,71. AlloSigMA analysis indicated that ligand binding on P6 influenced the G protein pocket of the receptor and that stable/bulky or unstable/tiny mutations on several P6 residues changed the dynamics of the G protein pocket (Supplementary Note 21). Thus, we designed bulky mutations (G1.49L, F7.55W, and F8.54W, Fig. 10A) and tiny mutations (F1.48A, N7.49A, Y7.53A, and F8.50A, Fig. 10B) in the AT1 receptor and used BRET assays to test the regulation ability of P6 on both G protein and β-arrestin pathways. Under similar protein levels (Supplementary Note 22), point mutations disturbed both Gq and β-arrestin 2 activities (Figs. 10C, D, and Supplementary Note 22). In all mutations, although the F8.54W mutant retained the same Emax for Gq as the WT AT1 receptor, the higher EC50 value suggested a weaker effect of AngII. Thus, point mutations guided by AlloSigMA suggested the existence of P6.

To investigate the effects of mutations at the potential allosteric site on the AngII-induced AT1 receptor conformational change, we performed intramolecular FlAsH-BRET assays using our recently developed AT1 receptor conformational sensor, which was generated by incorporation of Rluc at the C-terminus and FlAsH motif (CCPGCC) into the ICL3 (referred to as AT1R conformation sensor)72,73. As expected, AngII stimulation on the AT1R conformation sensor decreased the BRET signal in a dose-dependent manner, suggesting an increase of the distance between the C-terminus and the FlAsH motif inserted at the ICL3 of the AT1 receptor (Fig. 10E, F). Notably, albeit with similar total and cell surface expression levels compared with the WT sensor, mutations at all seven sites led to the impairment of AngII-induced AT1 receptor conformational change, as revealed by the increased EC50, whereas G1.49L, F1.48A, N7.49A, Y7.53A, and F8.50A mutants also displayed decreased maximal response (Supplementary Note 22). These mutational effects on the downstream Gq signaling are consistent with the Gq BRET assay, suggesting that the AT1R conformation sensor can primarily reveal the conformational changes required for the AT1 receptor Gq signal. These data suggested that the potential allosteric site participated in AT1 receptor signal regulation through conformational modulation. Hence, considering the shared existence of P6 in the class A GPCR family, it is possible to develop general allosteric modulators targeting P6 and regulating GPCR activation.

### Discussion

GPCRs have gained increasing attention as therapeutic targets for the treatment of sensory, circulation, and central nervous system disorders. Despite intensive research and the accumulation of considerable amounts of structural and functional data, it is still challenging to completely understand the mechanism of GPCR activation, elucidate the complex signaling machineries that can adopt a slew of distinct conformations to modulate multiple downstream signaling pathways (known as biased signaling), and design small-molecule allosteric modulators that can distinguish between closely related receptor subtypes1,22. As a quintessential model system for GPCR research, the AT1 receptor not only shows biased signaling by regulating its endogenous ligand AngII, but also shows great promise as a therapeutic target for treating hypertension. Recently, the antagonist-bound (inactive), AngII-bound, and biased agonist-bound (active) AT1 receptor structures have been determined23,24,25; however, the dynamic process of receptor activation and its biased signaling through a conformational selection mechanism are still not fully understood10,74.

In the present study, we employed extensive MD simulations (300 μs) of the AT1 receptor and used the NEB method to define a pathway from the inactive to the active state. A free-energy landscape was identified and confirmed by MSM by referring to the common GPCR activation properties. MSM extracted representative conformations of the inactive, intermediate, and active states; consequently, a synergistic activation mechanism was proposed by comparing these conformations and confirming the existence of the intermediate state. Then, tICA was applied to analyze the global movement, and it identified distinct active states, which provide hints for uncovering the biased signaling pathway mediated either by G proteins or β-arrestins. NMR experiments in future may further confirm the macrostates and the biased signaling for the AT1 receptor. Between TM7 and H8, a cryptic allosteric site was discovered in the intermediate state and confirmed by site-directed mutagenesis and biosensor. Cumulatively, our simulations and experiments make a significant contribution to the currently available knowledge as it provides insights into the AT1 receptor activation mechanism and reports an allosteric pocket.

In general, the activation mechanism of the AT1 receptor is similar to that of other class A GPCRs. During activation, TM6 moves outward to provide space for the transducers, and TM5 and TM7 approach to form the pocket. Notably, H8 experiences an upward movement that creates a cryptic allosteric site during transition21,48. However, upon activation, the AT1 receptor has several other unique properties due to its particular residue composition. For instance, R3.50 and N6.30 show weak polar interactions rather than a strong ionic lock as observed in other GPCRs, which facilitate the conformational rearrangements of TM6 to engage with the downstream effectors and increase the proportion of the active conformations in the apo ensemble24. This may lead to a relatively high constitutive activity. Interestingly, the approach of Y7.53 and Y5.58 promotes the formation of the active state, which is unique to the AT1 receptor. Thus, the AT1 receptor not only harbors a common activation mechanism, but also maintains its uniqueness through the movement of specific residues.

Notably, intermediate states and active states exist naturally in the apo AT1 receptor. This population distribution has been proven in other apo GPCRs as means of basal activity75,76,77,78. Thus, agonists select and stabilize the active conformation of GPCRs and promote the binding of transducers. Once the transducers bind to the GPCRs, the active conformation harbors lower energy, hereafter the fully stabilizes the active state52,77,79, and proves the conformational selection mechanism of GPCR activation. The active AT1 receptor showed two distinct conformations, as determined by tICA, and thus supports the hypothesis that both the Gq-bound and β-arrestin 2-bound active conformations exist in the ensemble of the AT1 receptor. The two conformers are stabilized by Gq and β-arrestin 2 biased agonists, respectively38, suggesting a population shift mechanism in the biased signaling80,81. In contrast, the balanced agonists equally stabilize the two conformers, and thus lead to no preference for the downstream proteins75. Therefore, the emergence of biased signaling occurs via a conformational selection mechanism. In other words, the regulation of GPCR activity is attributed to a generalized allosteric modulation82,83,84; hence, it is natural to design modulators targeting allosteric sites other than endogenous orthosteric sites85,86.

In the intermediate state, we discovered P6 as a site for allosteric regulation. Hidden between TM7 and H8, P6 was only observed during the upward movement of H8. Community analysis indicated that the impeding signal transmission between H8 and other parts impeded the formation of an active conformation (Supplementary Note 9). Mutagenesis experiments further confirmed that the residues around P6 transmit signals that modulate both G proteins and β-arrestin, and suggest that the allosteric perturbation from P6 has the potential to modulate the activity of the AT1 receptor. Several GPCRs currently lack features of endogenous ligands to initiate specific biased signaling. Thus, directly targeting intracellular TMs (such as P6) by allosteric modulators may provide a strategy to selectively regulate GPCRs75,87. In particular, as an unexploited allosteric site, P6 can be targeted in future allosteric drug designs for GPCRs. In order to fully realize P6 as an allosteric target for GPCRs, further structure-based drug design methodology and experimental investigation will be deployed to discover lead compounds. These studies are not only expected to provide insights into the activation mechanism of the AT1 receptor, but also offer versatile applications for GPCR biology, biophysics, and medicinal chemistry.

### MD simulation system setup

The inactive AT1 receptor structure in complex with an antagonist, ZD7155, and a co-crystallized apocytochrome b562RIL (PDB 4YAY), as well as the fully active structure complexed with a partial agonist, S1I8, and G protein mimetic nanobody (PDB 6DO1), were downloaded from the Protein Data Bank (PDB). Both the nanobody and agonist, as well as the antagonist and b562RIL were removed from 6DO1 and 4YAY, respectively. In both structures, the truncated N-terminal, ECL2, and ICL3 were recovered by a loop building program, according to the sequence of the WT AT1 receptor. The hydrogens were added, while the termini were capped with acetyl and methylamide groups. Referring to the common sequence in the solved structures, the 9–317 residues in 4YAY and 6DO1 were extracted as the input structures for the following processes.

### NEB sampling

To characterize the activation process of the AT1 receptor, the NEB method, a compelling approach to determine the transition pathways among conformations, was performed. During the NEB process, a series of replicas (or images) with the same atom composition but different atom coordinates are generated between the initial and final states. The replicas are linked to their nearest neighbors via elastic bands, in order to confirm that they are evenly distributed along the pathway and prevent them from directly sliding down to the energy basin88. Consequently, the replicas constitute a certain pathway that connects the start and end conformations.

Then, a simulated annealing process was applied to the replicas to find an MEP between the fixed initial and final states. In this process, the elastic bands were denoted as 3 × Natoms dimensional vectors [R0, R1, …., RN], corresponding to the pathway from the initial state R0 and end state RN. To avoid interference of spring forces caused by the restraints and potential forces caused by the force fields, a tangent vector τi at each replica position was introduced. With respect to the energy of replica i (Vi) and its neighbors, τi is defined in Eq. (1).

$${{{{{{\boldsymbol{\tau }}}}}}}_{{{{{{\bf{i}}}}}}}=\left\{\!\!\!\begin{array}{c}{{{{{{\bf{R}}}}}}}_{{{{{{\bf{i}}}}}}{{{{{\boldsymbol{+}}}}}}{{{{{\bf{1}}}}}}}-{{{{{{\bf{R}}}}}}}_{{{{{{\bf{i}}}}}}},{V}_{i+1} \, > \, {V}_{i} \, > \, {V}_{i-1}\\ {{{{{{\bf{R}}}}}}}_{{{{{{\bf{i}}}}}}}-{{{{{{\bf{R}}}}}}}_{{{{{{\bf{i}}}}}}{{{{{\boldsymbol{-}}}}}}{{{{{\bf{1}}}}}}},{V}_{i+1} \, < \, {V}_{i} \, < \, {V}_{i-1}\\ \left({{{{{{\bf{R}}}}}}}_{{{{{{\bf{i}}}}}}{{{{{\boldsymbol{+}}}}}}{{{{{\bf{1}}}}}}}\!-\!{{{{{{\bf{R}}}}}}}_{{{{{{\bf{i}}}}}}}\right)\Delta {V}_{i}^{{\max }}+\left({{{{{{\bf{R}}}}}}}_{{{{{{\bf{i}}}}}}}\!-\!{{{{{{\bf{R}}}}}}}_{{{{{{\bf{i}}}}}}{{{{{\boldsymbol{-}}}}}}{{{{{\bf{1}}}}}}}\right)\Delta {V}_{i}^{{\min }},{V}_{i} \, > \, {V}_{i+1} \, > \, {V}_{i-1}\,{{{{{\rm{or}}}}}}\,{V}_{i+1} \, > \,{V}_{i-1} \, > \, {V}_{i}\\ \left({{{{{{\bf{R}}}}}}}_{{{{{{\bf{i}}}}}}{{{{{\boldsymbol{+}}}}}}{{{{{\bf{1}}}}}}}\!-\!{{{{{{\bf{R}}}}}}}_{{{{{{\bf{i}}}}}}}\right)\Delta {V}_{i}^{{\min }}+\left({{{{{{\bf{R}}}}}}}_{{{{{{\bf{i}}}}}}}\!-\!{{{{{{\bf{R}}}}}}}_{{{{{{\bf{i}}}}}}{{{{{\boldsymbol{-}}}}}}{{{{{\bf{1}}}}}}}\right)\Delta {V}_{i}^{{\max }},{V}_{i} \, > \, {V}_{i-1} \, > \, {V}_{i+1}\,{{{{{\rm{or}}}}}}\,{V}_{i-1} \, > \, {V}_{i+1} \, > \, {V}_{i}\end{array}\right.$$

(1)

where \(\Delta {V}_{i}^{{\max }}={{{{{\rm{max }}}}}}\left(\left{/INSERTKEYS}

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