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674dc5277be152b1d0af2339
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To test the activity of (+)-JQ1-ASO conjugates, we initially chose a splice-switching ASO (SSO). The 18mer SSO we used contained a fully phosphorothioated (PS) backbone and all sugars with a 2′-OMe modification (Figure ). We used the gold-standard SSO sequence developed for the HeLa pLuc/705 cell line , which expresses a luciferase-encoding gene interrupted by a mutated ß-globin intron. The mutation generates a 5′-splice site which activates a cryptic 3′-splice site, resulting in incorrect mRNA splicing and non-functional luciferase production (Figure ). The SSO binds to the mutant 5′-splice site and promotes the exclusion of the aberrant intron, restoring the pre-mRNA splicing to produce functional luciferase.
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To test our (+)-JQ1-SSO conjugates in the HeLa pLuc/705 cell line, we compared their activity to the well-studied SSO, without the 5′ (+)-JQ1 modification. Cells were treated with varying concentration (25-200 nM) of either SSO (unconjugated and (+)-JQ1 modified) for 24-hours and quantified by luminometry, serving as a measure of splice correction and SSO efficacy. Our (+)-JQ1-SSO conjugate showed 2.0, 1.8, 1.9, and 1.7 fold higher splice-switching activity compared to the 20 unconjugated SSO, at 25, 50, 100, and 200 nM respectively (Figure ). Covalent addition of (+)-JQ1 did not significantly increase the toxicity of the SSO at any concentration, relative to its unconjugated form -assessed through total protein production quantified by BCA (SI Figure ) and Cell-Titer Glo (SI Figure ).
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To confirm that the improved splice switching was induced by BET protein-mediated nuclear import rather than nonspecific binding to cellular proteins, we carried out a competition assay with excess small molecule, (+)-JQ1. At low concentrations of (+)-JQ1 (5 nM) the enhanced activity of the (+)-JQ1-SSO conjugate at 25 and 50 nM was effectively inhibited (Figure ). Consistent with the idea that the (+)-JQ1-ASO conjugate acts via an interaction with BET proteins, even higher concentrations of 200 nM (+)-JQ1 were required to inhibit the enhanced activity of the (+)-JQ1-SSO at 100 nM and 200 nM (Figure , SI Figure ). Thus, we demonstrated that splice-switching ASO activity can be doubled via a specific interaction with the BET proteins for enhanced nuclear import. Once we achieved improvement in splice-switching activity, we wanted to test whether this approach could be extended to RNase H-mediated gene knockdown (Figure ). Our test system was a 20mer ASO that targets the gold-standard knockdown target metastasis-associated lung adenocarcinoma transcript 1 (MALAT1), a nuclear-enriched long non-coding RNA. MALAT1 plays key roles in gene regulation and metastasis in cancer. Implementing the current state-of-the-art in ASO design, this MALAT1-ASO had a gapmer design, containing a fully PS backbone with terminal wings of five 2′methoxy-ethyl (MOE) sugar modifications (Figure ). In the gapmer design, the central region of DNA oligonucleotides are recognized by RNase H, while the flanks of 2′-modified sugars are RNase H-inactive, but enhance nuclease stability and target binding. As with the SSO, we prepared the (+)-JQ1 conjugate utilising copper-catalysed click chemistry, using the same (+)-JQ1-alkyne and a 5′-azide-modified MALAT1 gapmer. The 5′-azide MALAT1 gapmer was also prepared using azidoacetic acid-NHS ester functionalisation of a 5′-terminally amine-modified gapmer ASO. Following copper-catalysed click chemistry, the (+)-JQ1-MALAT1 gapmer was produced in >90% reaction yields and >95% purity, after HPLC 20 purification (SI Figure ).
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To test the activity of the (+)-JQ1-MALAT1 gapmer conjugate, we measured MALAT1 transcript levels using reverse transcription-quantitative polymerase chain reaction (RT-qPCR) at 24 hours, comparing the knockdown to the unconjugated MALAT1 gapmer, normalised to the house keeping gene GAPDH. As with the SSO conjugate, the (+)-JQ1-MALAT1 gapmer conjugate outperformed the unconjugated MALAT1 gapmer at all tested concentrations. We measured 20.1%, 25 30.2%, 56.8%, and 54% less transcript with the (+)-JQ1 modified gapmer treatment, compared to the unmodified gapmer, at 5, 50, 100, and 200 nM respectively (Figure ). Again, similar to the SSO, this enhanced activity was found to depend on specific (+)-JQ1-BET bromodomain protein interactions, as it was lost in the presence of excess quantities of small molecule (+)-JQ1. At 200 nM (+)-JQ1, the enhanced activity of the (+)-JQ1-MALAT1 gapmer at 50-200 nM was completely inhibited (Figure , SI Figure ). Furthermore, no increase in toxicity was observed for the (+)-JQ1-MALAT1 gapmer conjugate compared to the unconjugated MALAT1 gapmer at all concentrations (SI Figure ). This demonstrated that (+)-JQ1 conjugation can effectively increase both the splice-switching and gene-knockdown activity of ASOs. While MALAT1 ASOs are being evaluated for their therapeutic effect, Oblimersen (G3139), an 18-mer ASO containing a fully PS backbone designed to target apoptosis factor BCL-2, reached phase III clinical trials. BCL-2 is a crucial inhibitor of apoptosis that is overexpressed in various cancers. Despite promising phase I-II results as a sensitiser for chemotherapy, G3139 failed to show efficacy in multiple phase III trails. As this drug was well tolerated by patients , the limiting factor is likely target engagement and efficacy. Therefore, given that G3139 functions as an RNase H-active ASO (Figure ), like the MALAT1 gapmer, we aimed to measure whether a (+)-JQ1-G3139 conjugate could result in a more potent drug molecule (Figure ).
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| 8 |
We synthesised the (+)-JQ1-G3139 conjugate using the same copper-catalysed click methodology as the previous ASOs. The 5′-azide-G3139 was synthesised from the commercially-obtained 5′-amine PS ASO using azideoacetic acid-NHS ester functionalisation, and the (+)-JQ1 conjugate was synthesised through a copper-catalysed click reaction with the (+)-JQ1-20 alkyne (SI Figure ). We transfected ASOs into HEK293T cells and measured the BCL-2 transcript and protein levels, using RT-qPCR and western blotting, respectively, after 24 hours. Our (+)-JQ1-G3139 conjugate dramatically outperformed the unconjugated G3139 ASO at all concentrations tested. We measured 43.9%, 51.3%, 50.9%, and 64.5% less transcript using the conjugated (+)-JQ1-G3139 compared to the unconjugated G3139, at 50, 100, 200, and 500 nM respectively (Figure ). We also observed a concomitant marked reduction in comparative protein levels, especially at lower ASO concentrations (Figure ). We then conducted a competition assay using excess small molecule (+)-JQ1, similar to the approach used with the other conjugates. At 5 nM (+)-JQ1, the enhanced activity of (+)-JQ1-G3139 at 100 nM showed a significant reduction, and at 200 nM (+)-JQ1, the enhanced activity of (+)-JQ1-G3139 was fully inhibited. As observed with the previous (+)-JQ1 conjugates, no increase in cellular toxicity was observed for (+)-JQ1-G3139, compared to the unconjugated ASO (SI Figure ). two most prominent mechanisms of action: splice switching and RNase H-mediated gene knockdown. These findings build upon the previous literature surrounding the development of (+)-JQ1 as a nuclear importer. Our data underscores the importance of precisely targeting therapeutic agents to their site of action. By inducing ASO enrichment in the nucleus, we have improved their functional efficacy and therapeutic potential. In this way, we have shown that nuclear localization is crucial for maximizing the therapeutic potential of ASOs for splice switching and RNase H-mediated knockdown.
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| 9 |
Our approach is simple, versatile, and broadly applicable, as evidenced by the successful enhancement of ASO activity across different backbone chemistries, different targets, and different mechanisms of action. Thus, our technology is robust and potentially applicable to a wide range of therapeutic scenarios. Notably, we have taken Oblimersen, an 'almost-therapeutic' ASO, and used our approach to dramatically improve its therapeutic efficacy. This improvement suggests that our work has the potential to transform other suboptimal ASO drugs into more effective therapeutics.
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The modular nature of this technology opens exciting possibilities for future applications. By demonstrating that small molecules can be potent effectors of ASO cellular compartmentalization and thus, functionality, we pave the way for the development of advanced therapeutic strategies through small molecule conjugation. The integration of small molecules to manipulate cellular environments and target sites offers a promising approach to enhance therapeutic outcomes. Our work aligns with the burgeoning field of bi-functional molecules, which combines distinct functionalities into a single entity. This 30 intersection with bi-functional technologies allows our approach to be integrated into the broader context of therapeutic innovation, by leveraging the synergies between small-molecule ligands and nucleic acid-based therapeutics.
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In summary, our study provides a compelling demonstration of how small molecules can be harnessed to enhance ASO activity in multiple mechanisms of action, through improved nuclear localization. The modular nature of this technology, combined with its compatibility with emerging bi-functional modalities, positions it as a promising platform for future therapeutic advancements.
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618872138ac7a20d766c1337
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The application of lithium-ion batteries (LIBs) for energy storage has attracted considerable interest due to their wide use in portable electronics and promising application in high-power electric vehicles. Nowadays, many studies focus on inorganic materials and their carbon-involved composites. But their fabrication consumes a huge amount of energy and releases a large amount of CO2, which undermines their environmental sustainability. Moreover, the ultimately limited availability of lithium on earth has forced researchers to evaluate alternative electroactive materials for batteries. Compared to inorganic materials, organic compounds are attractive alternative electrode candidates for nextgeneration LIBs because of their distinct advantages such as lightweight, abundance, non-toxicity, sustainability, flexibility, and simple structure optimization. Therefore, replacing inorganic electrodes with organic materials in rechargeable batteries is ideal to alleviate the environmental and sustainability challenges.
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| 1 |
Over the past several decades, remarkable progress has been achieved with electroactive organic compounds for LIBs. A number of diverse structural motifs, such as organic free-radicals, conjugated carbonyls, imine compounds, phenothiazine, phenoxazine, benzothiadiazole, dibenzothiophenesulfone, and azobenzenes have been explored as electrodes for batteries. So far however, most of these organic materials are still not competitive enough with high-performing inorganic electrode materials. Their practical application is severely hampered by their high solubility in organic electrolytes and low electronic conductivity, which leads to poor cycling stability and rate performance. Moreover, the charge capacity is generally limited to either one or two electrons per molecule or unit, which limits the overall capacity and energy density of the battery.
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Pyridinium salts demonstrate chemically reversible electrochemical behavior upon reduction. One representative example is the viologen family (quaternized 4,4'-bipyridiniums) that has been widely utilized in electrochromic devices and energy storage, due to their desirable redox properties. Another notable example includes the NAD + /NADH couple that functions as an electron-transfer catalyst in the respiratory chain. The key active motif is the carbonylpyridinium unit, capable of accepting two electrons per unit. By virtue of its two reversible redox events, high theoretical capacity, and its unique structural feature mentioned above, the use of benzoyl-N-methylpyridinium (BMP) derivatives has been proven to be very promising for redox-flow batteries by Sanford and coworkers. However, the exploration of this appealing bio-derived redox-active structure for application in LIBs is considerably underdeveloped, largely due to its high solubility in commonly employed electrolytes.
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The traditional design involves the integration of redox-active moieties in the main chain by conjugation with aromatic linkers. On the other hand, such design has inevitable drawbacks for providing stable redox potentials, due to large charge repulsion from the generated delocalized polarons and bipolarons in the backbone, therefore leading to a low battery performance. Very recently, we reported a rationally designed conjugated polymer with pendant benzoylpyridinium (BMP) redoxactive units that simultaneously reduce the solubility of the materials and mitigate the charge repulsion to enhance the overall battery performance. The carbonyl linker could effectively minimize the overlap of the HOMO and LUMO orbitals in the materials, allowing the redox centers to operate in a relatively interference-free manner.
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Herein, we now report a new rational molecular design strategy and comprehensive study on the performance of next-generation carbonylpyridinium-based redox active materials (Scheme 1). As part of this investigation, we developed the two di-cationic carbonylpyridinium-derivatives 1 and 2 that can stably store up to four electrons per molecule. By increasing the size of aromatic systems, the solubility of the materials is greatly reduced as the result of enhanced π-π stacking potential as well as their dicationic nature. Our studies have revealed that the donor-acceptor (D-A) design in 2 with an electron-donating bithiophene bridge helps to enhance intramolecular charge transfer that improves the electronic conductivity, and the final battery performance. Moreover, to further extend the conjugated system within a polymer backbone, we also developed a new dibrominated building block 3, for the synthesis of a series of conjugated polymers CP1 and CP2. The main chain of these conjugated polymers is expected to reduce the twisting between the thiophene units because of the reduced steric repulsion between bithiophene and the alkynyl linkers, leading to a fairly co-planar extended backbone allowing for an efficient electron-conduction pathway. At the same time, the redox activity will largely be localized on the spatially separated, "pendant" carbonylpyridinium groups. As cathode materials for LIBs, the prepared polymers present remarkably improved cyclability and rate performance, because of their high electrochemical activity and effective suppression of dissolution. Impressively, CP2 delivers not only the highest capacity but also the best cycling stability in this series, reaching up to 540 mAh g -1 after 240 cycles at 0.5 A g -1 , and as such even competing with the best inorganic cathode materials. This work highlights the considerable potential of bio-derived carbonylpyridinium redox-active units for sustainable energy storage applications and provides strategy for how they can be improved in LIBs. Scheme 1. Evolution of carbonylpyridinium based two-electron catholytes from small molecules to conjugated polymers.
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Methylation of S1 and S2 by reaction with excess MeI in DMF at 90 ºC gave the desired carbonylpyridinium products (1 and 2) as pale-yellow solids with quantitate yields. Conveniently, due to the high reaction efficiency, no tedious column chromatography is needed to isolate the intermediates and final products in pure form. All the intermediates and final products were fully characterized by 1 H and C NMR spectroscopy, as well as high-resolution mass spectrometry (HRMS) (see SI). The structures of 1 and S2 were further confirmed by single-crystal X-ray crystallography (Figure and). Common features are a highly planar central biphenylene or bithienylene bridge, while the carbonylpyridine (or pyridinium) units are twisted out of the central plane. The crystal data and structural refinement parameters are summarized in Table . To further extend the basic conjugated framework, a dibrominated compound 3 was designed that allows for subsequent cross-coupling reactions. The two-step synthesis of 3 is similar to that of 2.
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The chemical structures of the polymers were verified by Fourier transform infrared (FTIR) spectroscopy and solid-state C nuclear magnetic resonance ( 13 C NMR) spectroscopy. From the FTIR spectra (Figure ), CP1 and CP2 exhibit strong peaks at υ = 1640 and 1650 cm -1 , corresponding to the C=O stretching mode. Both polymers also display distinct a C≡C stretching vibration signal at 2180-2190 cm -1 , confirming the successful polymerization by connecting two monomers through a C≡C-bond. In addition, the 728 cm -1 band of 3 assigned to C-Br stretching, is absent in CP1 and CP2, manifesting the completeness of the polymerization reaction. As observed from the Scanning electron microscopy (SEM) images show that CP1 and CP2 have a relatively uniform flower-like morphology (Figure ), constructed from 2D nanosheets. The formation of this morphology can be explained by the synergistic polymerization and self-assembly process of the formed amphiphile polymers through hydrophobic-hydrophobic interaction and - stacking. Such unique morphology is beneficial for exposing more redox-active sites and also minimized diffusion length for the transport of ions/electrons, thus potentially delivering a higher capacity and enhanced rate performance. In contrast, undesirable fine-grained bulk morphologies were observed for 1 and 2, probably resulting from tight molecular stacking, which is confirmed by their high crystalline nature.
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Powder X-ray diffraction (PXRD) patterns of 1 and 2 display several sharp reflection peaks at 2θ values of 5-30°. However, CP1-CP2 are amorphous, as revealed by broad peak in the range of 20° to 25° in the PXRD pattern (Figure ). The Brunauer-Emmett-Teller (BET) surface areas for CP1 and CP2 were determined to be 24.1 and 41.3 m 2 /g, respectively (Figure ). Such small surface area was tentatively attributed to the strong stacking as a result of the largely linear and extended conjugated system. Thermogravimetric analysis (TGA, Figure ) of the as-prepared polymers under N2 unveils modest-to-high thermal stability. The decomposition temperatures (defined as the temperatures of 5% mass loss) were determined to be ca. 150 °C. while those of 1 and 2 are 0.25 and 0.05 mg mL -1 , respectively. These results validate our design concept that the stronger intermolecular interaction (π-π stacking) can reduce the overall solubility, even for relatively small molecules. Polymerization further suppresses the solubility and two polymers (CP1 and CP2) gratifyingly exhibit undetectable solubility. On the other hand, as shown in Figure , the conductivity measurement confirms that the electrical conductivity of 2 (8.9 10 -8 S cm -1 ) is slightly higher than that of 1 (6.2 10 -8 S cm -1 ), consistent with the D-A design that leads to relatively stronger intramolecular charge transfer.
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Ag/Ag + ), and the profile is almost identical with that of BMP, except a slight positive shift (Figure ). This suggests that the two redox-active carbonyl pyridinium units in 1 remain completely independent, which is in line with related systems. Compared with 1, each pair of reduction waves in 2 is split into two, leading to the appearance of four reversible reduction waves, with the reduction potentials (vs. Ag/Ag + ) determined to be E1/2(1) = -0.97 V, E1/2(2) = -1.07 V, E1/2(3) = -1.60 V and E1/2(4) = -1.72 V. The reason could again be attributed to the intramolecular charge transfer due to the introduction of stronger electron-donating bithiophene linker. The peak-to-peak separation (ΔEp-p) for each wave is ca. 60 mV, indicating their good reversibility. In Figure , the differential pulse voltammograms (DPV) further confirm the four-electron reduction processes of 1 and 2, and the patterns are consistent with the CV. The CV of 3 is almost identical with that of 2 (Figure ). In addition, a negligible loss of signal is observed in both cases after 10 continuous CV cycles at a scan rate of 100 mV/s, suggesting their excellent electrochemical stability. Overall, each carbonyl pyridinium unit can serve as two-electron storage unit. As such, compounds 1 and 2 can store up to four electrons per molecule. To make it more intuitive, the four-electron transfer processes are depicted in Figure . The first two-electron reduction event leads to the formation of neutral radical species, and the second two-electron reduction produces the formation of anionic species, in line with the electrochemical data for BMP reported in the literature. In dilute DMF solution, the π to π* absorption peak maximum of 2 is located at max= 402 nm, which is 80 nm red-shifted compared to 1 at max = 322 nm (Figure ). Moreover, 2 exhibits an additional low-energy absorption shoulder at 525 nm, which is attributed to intramolecular charge transfer from electron-donating bithiophene to carbonylpyridinium units. From the onset of the absorption (1: onset= 375 nm; 2: onset= 600 nm), a narrower energy band gap (Eg = 2.07 eV) is determined for 2 compared to 1 (Eg = 3.31 eV). Based on the equation ELUMO = -[𝐸 𝑜𝑛𝑠𝑒𝑡 𝑟𝑒𝑑 -E(Fc/Fc + ) + 4.8] eV, the lowest unoccupied molecular orbital (LUMO) energies were calculated to be -3.8 eV for both 1 and 2. While the same LUMO energy level means that both compounds have identical electron affinity, the narrow band gap for 2 indicates its high electron conductivity, which is confirmed by the conductivity measurement.
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To obtain a better understanding of the photophysical and electronic properties, density functional theory (DFT) calculations for small molecules and polymers were performed at the B3LYP/6-31G(d) level of theory using the polarizable continuum model (PCM) solvation model (solvent: CH3CN). The polymers were modeled as short chains containing three structural units (CP1-T and CP2-T) to keep the calculations manageable. The frontier orbitals diagrams, optimized structures and the coordinates are included in the SI (Figures S10-S14, Table -S6). For small molecules 1 and 2, the highest occupied molecular orbitals (HOMOs) are localized over the biphenyl or bithiophene linker. The calculated energy gap between HOMO and LUMO for 1 is larger than 2, consistent with the experimental trend. The contour plots of the five lowest unoccupied orbitals (LUMO to LUMO+4) reveal that LUMOs are largely distributed on the carbonylpyridinium units. Representative frontier orbitals HOMO, and LUMO to LUMO+4 as well as the calculated orbital energy levels are shown in Figure . The energy gaps between LUMO and LUMO+1, LUMO+3 and LUMO+4 are very small (less than 0.35 eV, indicative of their degenerative nature). Interestingly, the LUMO+2 orbitals display a large contribution from the central linker, indicating that the extended linker may also play a role in accepting electrons. It is well-known that the LUMO energy level of n-type electroactive materials is a useful parameter for estimating its relative redox potential (a lower LUMO energy level corresponds to a higher reduction potential). Assuming that the electron configurations are rigid, electrons will successively fill the unoccupied orbitals from the lowest to the highest-energy orbital during the reduction process.
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The optimized structures of CP1-T and CP2-T reveal high planarity of the extended conjugated backbone (main chain. Figures ), where majorly contribute to the HOMOs. Analogous to small molecules 1 and 2, the LUMOs of the two polymers are also mostly located on the carbonylpyridinium units. Both polymers exhibit a dozen of low-lying orbitals from LUMO to LUMO+11 that could potentially be involved in relevant reductive processes (the lower/upper values are -3.8/-2.1 eV for CP1-T, -3.9/-2.7 for CP2-T, Figure ). Moreover, the introduction of an electron-withdrawing benzothiadiazole unit significantly lowers the LUMO in CP1-T.
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For the investigation of 1, 2, CP1, and CP2 as active materials in LIBs, we fabricated composite electrodes containing 60 wt% of active material, 30 wt% carbon black (ECP-600JD) as the conductive additive, and 10 wt% polyvinylidene fluoride (PVDF) binder. A lithium metal disk was used as the counter electrode and LiPF6 (1.0 M) in a mixture of ethylene carbonate (EC), dimethyl carbonate (DMC) and ethyl methyl carbonate (EMC) (1:1:1, v/v) with 1% vinylene carbonate (VC) was used as the electrolyte. The cells were first evaluated using cyclic voltammetry (CV) over the voltage range of 0.05-3.0 V vs Li/Li + at 0.2 mV s -1 . As shown in Figure -b, the two polymers show two quasireversible anodic peaks at 1.35 and 2.20 V (vs Li/ Li + ), that are comparable to the performance of small molecules and related polymers in our previous report. In addition, the peak-to-peak area under the electrochemical event at 2.20 V were enhanced for CP2 as result of the introduction of benzothiadiazole unit as an additional redox center. This could consequently improve the capacity of the battery. In comparison, the two small molecules 1 and 2 exhibit less-obvious peaks (Figure ), probably due to their undesired bulk morphologies, which hampers the effective full utilization of the redox-active units.
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As an important parameter in LIBs that is greatly influenced by the solubility, the long-term cyclability was first investigated. Figure presents the cycling performance with a high coulombic efficiency of 1, 2, CP1 and CP2 at a current density of 0.5 A g -1 . During the first 30 cycles, all electrodes showed a steady decay in capacity, and the capacity values in a certain cycle follow the trend of 1 ˂ CP1 ˂ 2 ˂ CP2. The capacity loss in the first few dozen cycles is mainly ascribed to the electrolyte decomposition and the formation of solid electrolyte interphase (SEI) film, which is often observed for organic battery electrodes. The electrodes based on compounds 1, 2, CP1 and CP2 deliver reasonable capacities of 196, 293, 291 and 390 mAh g -1 after 30 cycles. (Figure ). In subsequent cycles, the capacity of 2 stabilized at 260 mAh g -1 , while 1 further underwent gradual capacity decay to 100 mAh g -1 after 240 cycles.
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The relatively poor cycling stability of 2 can easy be ascribed to its partial dissolution of in the electrolyte. However, its cycling stability is still much better than that of BMP, which exhibits a rapid capacity fading with only 50 mAh g -1 after 480 cycles (Figure ). In comparison, the CP1and CP2-based electrodes display a gradually increased capacity in the following cycles, measured to be 360 mAh g -1 (CP1) and 540 mAh g -1 (CP2) after 240 cycles, respectively. This phenomenon can be attributed to electrode conditioning that provides more efficient conductive pathways as the electrode repeatedly swells and expels ions during charging and discharging cycles. Such behavior has also been observed in other lithium-organic hybrid batteries before.
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Notably, the capacity of CP1 outperforms 2 after 45 cycles, and a new trend 1 ˂ 2 ˂ CP1 ˂ CP2 occurs during cycles 45-240. These results suggest that conjugated polymers deliver the best cycling stability, resulting from high electrochemical activity and effective suppression of dissolution. We attribute the highest capacity observed for CP2 to the introduction of benzothiadiazole unit as an additional redox center, which has already been demonstrated in our previous work. Moreover, their Coulombic efficiencies are close to 100% in all cases, verifying the high reversible redox stability of the carbonylpyridinium units. Overall, the varying dissolution behavior of the four materials agrees well with their different cycling performance, thus leading us to believe that unfavorable dissolution (and not chemical decomposition of the materials) is the dominating factor in capacity fading.
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At a low current density of 0.2 A g -1 , electrodes based on 1, 2, CP1, and CP2 exhibited stable discharge capacities of 350, 520, 360, and 520 mAh g -1 , respectively. The discharge capacities monotonically decrease with increasing current densities but more drastically in the case of two small molecules 1 and 2. The detailed data are summarized in Table . This observation could be attributed to kinetic factors, such as lithium-ion diffusion and electron-transfer rates that limit the amount of charge extracted from the electrode at higher currents. Impressively, the two conjugated polymers again display superior rate performances than the small molecules. Relative to their specific capacity at 0.2 A g -1 , their capacity retentions at 1 and 5 A g -1 are 46 % and 22 % for 1, 50 % and 22 % for 2, 64 % and 40 % for CP1, 70 % and 44 % for CP2, respectively. Although CP1 displays lower capacity than 2 at low current densities (˂ 1 A g -1 ), it catches up with increasing current, which is consistent with the observed cycling performance. The excellent rate performance of the conjugated polymers is not only the result from their absolute insolubility, but also their enhanced electronic conductivity. In addition, their amorphous morphology in the microscale range probably allows the full utilization of the redoxactive sites and also minimizes diffusion length for the transport of ions/electrons, thus delivering a higher actual capacity and enhanced rate performance. Importantly, when the current density is shifted back from 5 A g -1 to 0.2 A g -1 , nearly quantitative recovery of the initial capacities is observed in all the cases, demonstrating the superior stability. Overall, the stepwise improvement of the rate performance and cycling stability on going from 1 via 2 to CP1 and finally CP2, is consistent with their inherent electrical conductivity, solubility, and morphology. Impressively, CP2 delivers not only the highest capacity but also the best cycling stability (Figure ), suggesting high electrochemical activity and effective suppression of dissolution.
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our knowledge, the rate capability and cycling stability of CP2 is one of the best reported for organic Li-ion batteries. Compared to many other well-known cathode materials for Li-ion batteries, such as inorganic LiCoO2 (140 mAh g -1 ), TEMPO (radical) functionalized polymethcarylate (PTMA, 111 mAh g -1 ), 8 polyanthraquinone (263 mAh g -1 ) , poly(chalcogenoviologen)s (463 mAh g -1 at 0.5 A g - 1 ), and other previously reported bio-derived polymeric cathode materials (< 150 mAh g -1 ), our polymer provides among the highest specific capacity, making this current system highly desirable as an organic cathode material for lithium-ion battery applications. . Furthermore, to better understand the Li + diffusion process, the diffusion coefficients (DLi + ) of the four electrodes were calculated from the Warburg region (see the SI for details). The DLi + values of 1, 2, CP1, and CP2 were calculated to be 6.64×10 -14 cm 2 S -1 , 8.54×10 -14 cm 2 S -1 , 1.42×10 -13 , and 3.12×10 , respectively. The highest DLi + value of CP2 indicates faster lithium-ion diffusivity, consistent with the battery test data.
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the reduction and oxidation reactions. These XPS data are consistent with the FTIR data, demonstrating the relatively good stability during the charging and discharging cycles. In addition, SEM analysis of the electrodes after 90 cycles revealed no obvious morphology changes compared to the fresh cells for both small molecules and polymers, further confirming their high electrochemical stability (Figure ).
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With the aim to improve the energy and high-power density of Li-organic hybrid batteries, the development of organic molecules with multiple redox centers, limited solubility and high conductivity is highly desirable. In the current work, we report a rational design strategy to suppress the dissolution and enhance the conductivity of such active materials from small molecules 1 and 2 to conjugated polymers CP1 and CP2 in stepwise fashion toward realizing high performance LIBs. Compounds 1 and 2 can reversibly store up to four electrons and show greatly decreased solubility compared to BMP as a result of strong intermolecular interactions. The donor-acceptor (D-A) character of 2 helps enhancing the intramolecular charge transfer, thereby improving the electronic conductivity and ultimately, battery performance. Both cycling stability and rate performance are further enhanced through the polymerization, as a result of high electrochemical activity and effective suppression of dissolution. Impressively, polymer CP2 delivers not only the highest capacity, but also the best cycling stability, reaching up to 540 mAh g -1 after 240 cycles at 0.5 A g -1 . We believe that this work provides a promising carbonylpyridinium-based building block featuring multiple redox centers, on the way to competitive high performance Li-organic batteries.
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65e5b6839138d23161a2d109
| 0 |
Liver cancer was one of the top five causes of cancer death in 185 countries in 2020, with an estimated 906,000 new cases and 830,000 deaths globally. The number of liver cancer deaths is precited to reach approximately 1.3 million people by 2040 (56.4% more than in 2020) , whilst the number of new cases of liver cancer per year is estimated to increase by 55% between 2020 and 2040, with a possible 1.4 million people diagnosed in 2040. Liver cancer is caused by viral infections including hepatitis B virus (HBV), hepatitis C virus (HCV), and hepatitis D virus (HDV), and is becoming one of the most challenging and urgent problems in Vietnam today. An estimated 25,000 new cases of liver cancer were reported in 2020, making Vietnam the fifth highest country in the world for incidence [2] . Since 2018, liver cancer has risen above lung cancer to become the leading cause of cancer death in Vietnam [2] .
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The herb Centella asiatica (L.) Urb. (Apiaceae) is extensively grown throughout Vietnam. This medicinal plant possesses a wide range of biological properties, including anti-oxidant, anticancer, memory stimulation, and wound healing . Madecassic acid is one of the three main pentacyclic triterpenoid acids produced by the plant, together with asiatic acid and terminolic acid . The compound has been reported to have valuable pharmacological activities, such as wound healing , antioxidant , anti-inflammatory , and antidiabetic effects . Recently, madecassic acid has been showed to induce apoptosis in the mouse colon cancer cell line CT26, and much attention has been paid to increasing the anticancer activity of this molecule through chemical modification of its structure: new madecassic acid derivatives have been reported to have significantly enhanced cytotoxicity towards cancer cell lines .
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strong antioxidant potential and anti-lipid peroxidation action . Recent studies have found that silybins possess cytotoxic activity in a mouse model of prostate cancer , in the human nonsmall-cell lung carcinoma H1299, H322 and H460 cell lines , and the SW480 human colorectal cancer cell line . In addition, they have proven cytotoxic potential in bladder, skin, prostate, colon and lung cancers by a mechanism of regulating the cancer cell cycle, apoptosis and autophagy, as well as inhibiting tumor-inducing factors . Thus, structural modifications of silybin could provide further interesting information on silybin's applications.
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Hybrid synthesis of naturally bioactive compounds into a single molecule has been regarded as an effective strategy to produce novel and better active substances for the treatment of cancer . It is believed that bio-conjugated molecules could have superior efficacy in comparison with a single drug due to the minimization of the unwanted side-effects as well as the synergism of two or more active moieties in one molecule .
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Inspired by the bio-conjugated compounds, herein we report the synthesis of madecassic acidsilybin conjugates with an ester bond between a hydroxy group of silybin and the carboxyl group of madecassic acid, or over an amino acid as a spacer, and evaluation of their antiproliferative in vitro potential against the HepG2 human liver cancer cell line. The most potent active compound 8 was further evaluated for cytotoxic activity on the Hep3B, Huh7, and Huh7R liver cancer cell lines together with cell cycle analysis.
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Silybin ( ) is a flavonoligan containing five functional hydroxyl groups, thus the selective esterification between the carboxylic acid group of madecassic acid (MA, 2) and silybin moiety may be somewhat problematic (Figure ). Initially, we aimed at conjugating madecassic acid with hydroxy at C-23 of silybin through an ester. Accordingly, silybin was converted into 3,5,7,20-Otetraacetyl silybin as described by Armando , followed by reaction with 2,3,23triacetylmadecassic acid (3)/or its amide derivatives, which were linked to a glycine amino acid unit.
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Due to the failure of using 3,5,7,20-O-tetraacetylsilybin as precursor for the selective esterification, free silybin, without any protection was used as starting material. As reported by Li and Decker , under Mitsunobu conditions, a regioselective 23-OH esterification of silybin moiety was formed preferentially. However, in our case the esterification did not occur when using Mitsunobu conditions (DEAD/PPh3 or DIEA/PPh3).
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As illustrated in Scheme 1, treatment of acid 3 with either thionyl chloride or oxalyl chloride to produce an intermediate MA-chloride acid followed by exposure to silybin in the presence of triethylamine (TEA) led to the formation of esters 4 and 5, in which compound 5 showed a dehydration of hydroxyl group at C-3 on silybin by the appearance of a signal of singlet proton H-3 at ẟH 6.56 and carbon atom C-3 at ẟC 104.38 . The selective attachment to the 7-OH group of silybin was confirmed based on NMR analysis. This result was in agreement with the report by Křen and Decker . On the other hand, under Steglich conditions using DCC/DMAP, esterification of acid 3 with silybin 1 provided ester 6 with a selective attachment to the 3-OH group of silybin. The NMR analysis of ester 6 confirmed an elimination of protons H-2 and H-3 on the silybin moiety, which took part in an oxidation under mild basic conditions to create 2,3- dehydrosilybin derivatives. The conclusion was based on the disappearance of proton signals at ẟH 5.0 , together with carbon signals at δC 84.7 (C-2) and at δC 73.7 , accompanied by the appearance of two new carbon signals upfield at ẟC 156.7 (C-2) and 132.5 (Scheme 1).
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Scheme 2. Synthesis of dehydrated madecassic acid conjugated with silybin. Reagents and conditions: a) Pyridine, SOCl2, pyridine, 2 h, rt, 47%. B) i) oxalyl chloride, DCM, rt, 14 h, ii) amino acid (glycine; or β-alanine), TEA, DCM, rt, 18 h; iii) Silybin, dicyclohexylcarbodiimide (DCC), DMAP, THF, 0 o C, rt, 18 h; c) Ac2O, pyridine, rt, 12 h. Conjugation of madecassic acid to silybin via an amino acid as a spacer (glycine, β-alanine, or 11-aminoundecanoic acid, 11-AUDA), was carried out in a two-step reaction sequence. Treatment of triacetyl madecassic acid with oxalyl chloride to furnish the intermediate acyl chloride acid was followed by reaction with one of the three amino acids, and then subsequent coupling with silybin 1 under Steglich conditions, using DCC/DMAP in dry tetrahydrofuran (THF) at room temperature. The corresponding conjugates 8-10 were obtained in overall yields of 20-26%. Analysis of NMR spectroscopy of these conjugates 8-10 confirmed the regioselective esterification to the 3-OH group of silybin moiety. A comparison of the 1 H NMR spectra of silybin ( ) with its conjugated ester exhibited distinctly that the chemical shift of proton H-3 in silybin moiety was downfield shifted from a range of 4.65-4.64 up to 5.82-5.84 ppm. Only a few silybin ester derivatives in positions C-3 have been reported so far. Antoszczak et al. reported the synthesis of conjugates of silybin with the antibiotics salinomucin and monensin. The authors obtained the conjugates through an ester linkage at the 23-OH group with the yields of 43% and 35%, respectively and no conjugates at the 3-OH group of silybin . However, our results are in good agreement with the results by Křen who reported a selective attachment on the 3-OH group of silybin moiety under Steglich esterification conditions to form the 3-O-galloylsilybin. Acetylation of compounds 6, 8-10 with acetic anhydride in pyridine at room temperature provided the corresponding acetylated products . Their structures were evidenced by the analysis of the NMR and MS spectroscopic data.
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65e5b6839138d23161a2d109
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In order to evaluate the role of the 6-OH group in madecassic acid on the cytotoxicity to HepG2 cells, a series of the conjugates 15-18 were synthesized (Scheme 2). Triacetyl madecassic acid was first treated with thionyl chloride in the presence of pyridine to give the dehydrated compound 14 in 47% yield after a silica gel column chromatography. In a similar two-step sequence reaction, conversion of the acid 14 into the intermediate chloride acid followed by coupling with amino acids (glycine; or β-alanine), and subsequently esterified with silybin (1) under Steglich conditions provided esters 15-16 . Treatment of these esters with acetic anhydride provided products 17 and 18 in 68% and 58% yields, respectively, after silica gel column chromatography (Scheme 2).
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All compounds synthesized from silybin and madecassic acid were initially screened for their antiproliferative activity on the HepG2 cell line using the MTT assay before testing on other hepatocarcinoma cell lines. As shown in Table , the GI50 values for the 18 compounds on HepG2 cells was diverse. A number of compounds showed a significantly higher antiproliferative activity on HepG2 versus the original compounds silybin (1) and madecassic acid (2), whilst others exhibited weaker activity. To illustrate, compounds 3, 8, 11 and 14 showed lower GI50 values and therefore improved antiproliferative activity in comparison to that of the two original compounds silybin and madecassic acid (2) and notably neither 3 nor 14 contain a silybin unit. In contrast, compounds 4, 5, 7, 10, 12, 16 and 18 showed reduced antiproliferative activity (higher GI50 values), versus the two starting compounds and 4 and 5 (conjugated to silybin's 7-position) had GI50 values greater than 500 µM. The other group including compounds 6, 9, 13, 15, 17 presented improved antiproliferative activity versus silybin, but not madecassic acid.
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Table . Cytotoxic activity of madecassic acid (2), silybin and their conjugated madecassic acidsilybin derivatives on HepG2 cancer cell lines. a Alcohol on the silybin unit through which conjugation is achieved. b +Ac = acetylation, -H2O = dehydration on 2,3 positions on silybin (S) or 5,6 positions on madecassic acid (M). Ellipticine was used as a positive control and all values were generated using in a 96 hour MTT assay (n ≥ 3 assay repeats).
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Of all the compounds evaluated on HepG2 cells, compound 8 exhibited the highest cytotoxic activity and was chosen for the further evaluation on a variety of hepatocyte cancer cell lines: Hep3B, Huh7, and Huh7R, with the results compared with those of madecassic acid (2), using the MTT assay (Table ). The antiproliferative activity of compound 8 was much greater than that of madecassic acid, with the GI50 decreased by multiples of 15.1 to 18.5 respectively for Hep3B, Huh7, and Huh7R
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versus the parent compound. Compound 8 suppressed the antiproliferative activity of the Hep3B line the most effectively versus Huh7 and Huh7R cell lines. The effectiveness of compound 8 on the these three cell lines was noted to be in the same order of ranking by GI50 value as for madecassic acid (2)
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Based on the strong anti-proliferative activity of compound 8 on the four hepatocarcinoma cell lines, its ability to induced apoptosis was investigated. This was evaluated through measurement of Annexin V-FITC and PI staining of HepG2 cells, 24 hours after incubation with compound followed by flow cytometry for detection of these markers. The results (Table , Figure ) showed that HepG2 cells treated with madecassic acid (2) at 1x GI50 exhibit a higher proportion of cells in early and late apoptosis when compared with the untreated control (9.00 and 4.29% respectively) and that increasing the MA concentration to 3x GI50 increased the proportion of cells in both early and late apoptosis further . This result is in accordance with the reported induction of apoptosis by MA when colon cancer cells are treated with this compound. Compound 8 at 1 x GI50 resulted in induction of early and late apoptosis of 10.6% and 4.92% respectively along with a strong necrotic signal of 13.32%, whilst at 3 x GI50 early and late apoptosis values were lower than the untreated control, whilst necrosis was still increased (13.81%) Because cysteinyl aspartate specific proteinase-3 (caspase-3) plays an important role in the apoptotic pathway, the capacity of madecassic acid (2) and compound 8 to induce caspase 3 activity was measured at 1, 6, and 24 hours post treatment. The results in Table demonstrate that compound 2 caused a significant increase in caspase 3 activity in the HepG2 cells at 6 h post treatment, and at 3
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According to Pucci and college , there is an important linkage between the apoptosis and the cell cycle because mitosis and apoptosis express close morphological characteristics and so the effects of madecassic acid (2) and compound 8 on the cell cycle profile of HepG2 cells was investigated using PI staining, 24 hours post compound treatment. With 1 x GI50 of madecassic acid (2), there was a reduction in the proportion of cells in G0/G1 and G2/M with a concomitant increase in cells in S phase (Table and). The effects at 3 x GI50 of compound 2 were different, with the cell number at both S phase and G2/M reduced and a high number of cells in G0/G1, when compared the vehicle control (0.5% DMSO). In contrast to madecassic acid (2), compound 8 induced cell cycle arrest in S phase with a concomitant decrease in the number of cells in G2/M whilst the number of cells in G0/G1 phase was unchanged compared with the vehicle control at both 1 and 3x GI50 of compound 8.
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Sixteen novel conjugates of madecassic acid and silybin have been synthesized through esterification or using amino acids as linkers. The formation of esters at positions 3 or 7 could be controlled depending on the reaction conditions. The new conjugates were evaluated for their cytotoxicity on the HepG2 liver cancer cell lines HepG2. Nine compounds showed higher cytotoxic activity than that of silybin and five compounds exhibited greater cytotoxic activity than madecassic acid itself. Among dosage, but also giving biological properties which are enhanced or distinct from the two molecules individually.
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were recorded on a Bruker AVANCE 500 MHz with tetramethylsilane (TMS) as the internal standard for 1H and solvent signals for 13 were measured on a 1100 Agilent LC/MS ion trap. Reactions were monitored by thin-layer chromatography using silica gel G60 F254 (Merck). Silica gel 300-400 mesh (Merk) was used for column Chromatography.
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A solution of 2α,3β, in dry DCM (5 mL) was treated with oxalyl chloride (0.07 mL, 0.8 mmol). After stirring for 14 h at room temperature, the solvent was removed under reduced pressure to dryness and the resulting residue was redissolved in dry DCM (5 mL) followed by addition of amino acid (glycine; or glycine methyl ester or; or β-alanine; or 11-aminoundecanoic acid; or methyl 11-aminoundecanoic acid methyl ester;
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62bd7c24d66f683b75b7a07a
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The photochemical [2+2]-cycloaddition between alkenes proved to be a powerful strategy to construct cyclobutanes. In this context, we wondered if diene 1 (easily obtained from the commercially available starting materials, please see next paragraph) would undergo an intramolecular cyclization into the needed 2-oxabicyclo [2.1.1]hexane core. Direct irradiation of diene 1 in acetonitrile under different wavelengths gave only traces of products (entries 1-4, Table ). Irradiation with a Hanovia broad wavelength mercury lamp gave the needed product along with many side products (entry 5). Next, we tried the addition of available organic ketones for the triplet sensitization of the styrene moiety. Indeed, smooth formation of the needed products 1a was already observed. The best result was obtained with benzophenone (entry 7), whereas acetophenone and substituted benzophenones also worked, but provided the product with lower yields (entries 6, 8, 9). Among all tested solvents (entries 10-13), the best outcome was obtained in acetonitrile. Without irradiation, the reaction did not take took place neither at room temperature nor under heating (entries 14, 15).
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62bd7c24d66f683b75b7a07a
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Even though under optimized conditions, cyclization of diene 1 led to rather a clean formation of a diastereomeric mixture ofproducts 1a (d.r.=4:1), the pure major isomer 1a was isolated by column chromatography in only 56% yield. The separation of isomers by column chromatography was problematic and led to a significant loss of the yield, which needed to be solved Scaled-up synthesis. The whole optimized synthetic protocol is shown in Scheme 1. For us, it was important to elaborate on a method that employed only available and cheap starting materials. The synthesis started from propargyl alcohol (2).
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62bd7c24d66f683b75b7a07a
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Copper-catalyzed reaction with phenyl magnesium bromide gave alcohol 3 in 71% yield, following the reported procedure. Michael-addition of the latter with methyl propiolate (4) in the presence of DABCO afforded the needed diene 1. We mentioned that compound 1 partially decomposed during column chromatography and even under storage at room temperature. Therefore on the scale, we generated crude diene 1 in situ (please, see SI) and used it directly in the photochemical step. A mixture of isomers 1a was obtained. After extensive experimentation, we found a solution on how to avoid column chromatography, and not lose the yield. The crude reaction mixture after irradiation (isomers 1a and benzophenone) was saponified with sodium hydroxide. Standard workup (removal of benzophenone) followed by crystallization from hexane-MeOtBu mixture (removal of the minor isomer) allowed isolation of pure major isomer 1b in 71% yield after three steps from alcohol 3. Product 1b was obtained on a tengram scale with no column purifications involved. Scope. Next, we studied the scope of the developed method. The photocyclization tolerated well various substituents on the aromatic core (Scheme 2). Among them were the alkyl groups (5a-8a), fluorine (9a-11a) and chlorine atoms (12a, 13a), methoxy groups (14a-16a) and trifluoromethyl groups (17a-19a). The reaction was also compatible with various substituted pyridines (20a-24a). In all cases, we isolated analytical quantities of intermediate esters 5a-24a by column chromatography to characterize them. On the gram scale, however, we directly used crude reaction mixtures with 5a-24a after photocyclization in the subsequent saponification step. In half of all cases, we could obtain the final carboxylic acids by simple crystallization of crude reaction mixtures from various solvents (Scheme 2). In another half of the cases, column chromatography was still needed. The structure of carboxylic acids 5b and 9b was confirmed by X-ray crystallographic analysis (Figure ). Chemical stability. We also checked the chemical stability of three representative carboxylic acids 1b, 19b and 22b (Scheme 2), because we suspected that some of them could decompose via a retro-Michael type reaction. Treatment of them with aq. 1M hydrochloric acid, or aq. 1M aq. sodium hydroxide at room temperature for one day did not lead to any decomposition. All products were crystalline solids, and we stored all of them in closed vials at room temperature on the shelf. 1 H-NMR, LC-MS inspection after three months did not reveal any decomposition either.
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Crystallographic analysis. Next, we compared the geometric parameters of 2-oxabicyclo[2.1.1]hexanes with the orthosubstituted phenyl ring and their previously suggested saturated bioisosteres, bicyclo [1.1.1]pentanes and bicyclo [2.1.1]heptanes. For that, we employed the exit vector plots tool. In this method, substituents at the disubstituted scaffold were simulated by two exit vectors n1 and n2 (Figure ). The relative spatial arrangement of vectors is described by four geometric parameters: the distance between C-variation atoms r, the plane angles 1 (between vectors n1 and C-atom) and 2 (between n2 and C-atom), and the dihedral angle defined by vectors n1, CC and n2. An additional important parameter -distance d between two carbon substituents (Figure ) -was also measured.
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We calculated the values of d, r, 1, 2, and of 2oxabicyclo [2.1.1]hexanes from the X-ray data of compounds 5b, 9b. The related parameters for bicyclo [1.1.1]pentane 25 12b and bicyclo [2.1.1]heptanes 26, 27 were calculated from their X-ray data published in the literature. The corresponding parameters for ortho-substituted phenyl rings were calculated from the reported crystal data of two antihypertensive drugs -Valsartan and Telmisartan (Figure ). Analysis of this data revealed that geometric properties of 2-oxabicyclo[2.1.1]hexanes in general were indeed similar to those of the ortho-substituted phenyl ring. In particular, distance r in 2-oxabicyclo [2.1.1]hexanes was ca. 0.2 Å longer than that in the ortho-phenyl ring: 1.56-1.57 Å vs 1.38-1.44 Å (ortho-phenyl). The distance d between substituents in 2-oxabicyclo [2.1.1]hexanes was also ca. 0.5 A longer than that in the ortho-phenyl ring: 3.6 Å vs 3.0-3.1 Å (ortho-phenyl).
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Synthesis of the saturated analogue of Fluxapyroxad was undertaken from carboxylic acid 11b (Scheme 4). The standard Curtius reaction followed by acylation of the intermediate amine with the substituted pyrazole carboxylic acid gave the needed compound 29. Using an analogous tactic, compound 31, -a saturated analogue of Boscalid, -was also obtained from carboxylic acid 17b (Scheme 4). The saturated analogue of Phtalylsulfathiazole was obtained by converting carboxylic acid 16b first into the methyl ester followed by the oxidation of the phenyl ring. Amide coupling of the formed acid with the substituted aniline followed by saponification of the methyl ester gave the final compound 33 (Scheme 4). Amide coupling of carboxylic acid 19b with the correspondingly N-substituted 4-aminopiperidine gave compound 35a saturated analogue of Lopitamide (Scheme 4).
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Replacement of the phenyl ring with bicyclo [2.1.1]heptane led to either increase of clogP (Fluxapyroxad, Boscalid), or did not affect it significantly (Phthalylsulphathiazole, Lopitamide) (Scheme 4). However, in all four bioactive compounds incorporation of 2-oxabicyclo[2.1.1]hexane instead of the orthosubstituted phenyl ring led to a decrease of clogP index by ca. one unit.
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Bioactivity. Finally, we wanted to answer a key question, -if 2-oxabicyclo[2.1.1]hexanes could indeed mimic the orthosubstituted phenyl ring in real-world bioactive compounds? Therefore, we measured the antifungal activity of the marketed fungicides Fluxapyroxad (BASF), Boscalid (BASF) and their saturated analogues 28-31. In strict contrast to medicinal chemistry, the use of racemic mixtures in agrochemistry is common; 23 therefore for the primary validation of the proof-ofconcept, we directly studied the biological activity of the available racemic compounds 28-31 (Scheme 4).
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Fluxapyroxad, and its saturated analogues 28, 29 showed a similar level of activity at inhibition of Fusarium oxysporum growth (Figure ). In the inhibition of growth of Fusarium verticillioides, the oxygen-containing saturated analogue 29 was almost the same potent as Fluxapyroxad at high concentrations but showed a reduced potency at low concentrations (Figure ).
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Boscalid and both saturated analogues 30, 31 also effectively inhibited the growth of Fusarium oxysporum (Figure ). Moreover, at low concentrations, the oxygen-containing saturated analogue 31 exhibited a potency identical to that of Boscalid. In the inhibition of growth of Fusarium verticillioides, compound 31 was still active but exhibited a lower potency compared to Boscalid (Figure ). Summary. The ortho-substituted phenyl ring (as well as metaand para-isomers) is a basic structural element in chemistry, and children learn about it already at school in a general chemistry class. In this work, we developed water-soluble saturated bioisosteres of ortho-substituted phenyl ring: 2oxabicyclo[2.1.1]hexanes (Figure ). These scaffolds were synthesized from available starting materials on a multigram scale. Crystallographic analysis revealed that these structures and the ortho-substituted phenyl ring indeed have similar geometric properties. Moreover, replacement of the orthosubstituted phenyl ring in bioactive compounds with 2oxabicyclo[2.1.1]hexanes, in most cases, improved water solubility (up to more than ten times), reduced lipophilicity, improved metabolic stability and most importantly -retained bioactivity.
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Biological networks are often extremely complicated structures, often made of ordered arrangements of biopolymers. These structures can suffer damage or Persistence Diagrams to Visualise Damage to Biological Networks degrade over time, with serious consequences for the functionality of the network and the organism containing that network. It is important to understand the precise nature of damage over the lifespan of a network; a better understanding of how defects or tears propagate over time could be used to inform methods to either halt or repair the damage. One such biopolymer network of scientific interest that loses function over time is that of collagen in the ocular lens capsule. The loss of flexibility owing to damage-induced changes in the lens capsule has consequences for the visual system losing its ability to accommodate different focal depths ), or the lens capsule can suffer immediate damage when ruptured during cataract surgery ) or by exposure to focussed high energy sources.
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Some biological systems consist of layers of two dimensional networksdepending on the network, these layers could themselves be highly ordered (made up of single repeating geometric units, said by to be hexagons) or highly disordered as shown in microscope images from and . To avoid the computational complexity of considering the position of every atom, it can be a powerful approach is to simplify the biological network with a graph theory approach, treating molecules as edges and their interaction sites as nodes. The use of graph theory opens up an avenue to analyse the mid-range structure of 2D networks, an approach that has seen great success when used by Le to analyse inorganic 2D networks such as glasses or by Ormrod to study amorphous graphene.
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The rings in a 2D network can be conceived of as being the next level up in a hierarchy of network features: a node at an interaction site is a zerodimensional feature, an edge between two nodes is a one-dimensional feature, and the rings formed edges are two-dimensional features as shown in Figure . A ring representation of a 2D network allows for analysis of medium range order, including how those rings are correlated to one another (for example, are rings with many sides adjacent to rings with few sides or vice versa as measured by ) or characterising networks by the distribution of the rings seen in them (referred to by authors such as or Le as 'ring statistics'). Ring statistics are also used in studies of 3D networks, although the definition of a ring is less well defined and the ring structure is considerably harder to visualise. The most intuitive set of 3D features to treat with a persistence diagram are pores in zeolites as discussed by , although those are outside the scope of this work.
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Framing biological networks like this is reminiscent of the persistent homology frameworks used to characterise proteins, 2D atomic glasses and zeolite structures (see refs. ; Ormrod ; ). Persistent homology approaches work by analysing objects called homology groups, generally constructed by placing a sphere of size r on each data point in a point cloud and expanding the size of r. When two spheres overlap, an edge is drawn between them and the simplices (polygons, polyhedra, or hyper-polyhedra) that are formed are analysed. The homology groups generated that way are often unwieldy ), and are most often simplified into a few useful outputs (among others): Betti numbers β n which measure the number of n-dimensional features; persistence diagrams which show how features appear and disappear as functions of r; and birth-death diagrams which show clusters of features appearing or disappearing. We can map the persistent homology framework onto the established network theory simply, as the first three Betti numbers correspond to the number of zero-, one-and two-dimensional features in the networks. In most persistent homology approaches, a persistence diagram charts at which value of a length scale (called a filtration parameter) various features first start to exist ('are born', in the language of persistent homology) or stop existing ('die') ). Persistence diagrams have been criticised (see, for example, refs. and Ormrod ) as they may be hard to interpret quantitatively, and they often reflect properties of a network that are difficult to visualise.
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Furthermore, analysing biological networks as a function of a length scale at a single point in time can potentially lose critical information. For a biological network, a meaningful length scale is already defined as approximately the length of one biomolecule, and the most meaningful information is in how those networks vary across time. To resolve this, we can look at persistence diagrams generated at different timescales and a fixed length scale. Related work by discusses using persistence diagrams that are functions of both length scales and time scales to measure similarity between zeolites. In this work, we take the idea of a persistence diagram (and the associated metrics for birth-death diagrams and ring lifetimes) and show their qualitative power for analysing ring statistics over time.
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While criticism has been levelled at persistence diagrams for how much they rely on qualitative analysis (see refs. ; Ormrod ; ), if that is what's desired they can be extremely useful. The analysis of persistence diagrams is simplified if instead we track whether particular features in a network persistent across time instead of length scale. The medium-range order as represented by a ring structure in a network is especially interesting to us, as it is often hardest to track. In this analysis, we define a ring R as being constructed of a unique set of moleculesfor example, a square might be made up of molecules {1, 2, 3, 4} and a second triangle could be made up of molecules {4, 5, 6, 7}. We then can calculate an existence function f (t; R) for each ring R over time t during a trajectory of interest. This function f (t; R) has a major difference from the usual filtration functions used for persistent homology in that a single ring can be born, die, and live again. For example, the square {4, 5, 6, 7} could exist at t = 1, die at t = 2, be reborn at t = 3 and die a final time at t = 4 as seen in the right bar of Figure . This leads to the persistence diagram in Figure striped in both directions, which will be useful for analysis of repair phenomena. The lifespan of a given ring can be defined as L(R) = ∞ 0 f (t; R)dt, and birthdeath diagrams can be plotted using the first birth and final death of a given ring.
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To generate trajectories of interest, we use a simplified biopolymer model to mimic the self assembly process of collagen IV, first used by . The biopolymers are made up of a linear chain of beads with beadbead bonds modelled using a Morse potential to allow for bond breaking. The beads at either end of the polymer chain are marked as 'head beads' and experience Lennard-Jones attractions towards other head beads. The beads within the polymer are marked as 'body beads' and a truncated Lennard-Jones potential is used to mimic repulsion between molecule bodies and provide an excluded volume. This model has been successfully by used to mimic self-assembly phenomena of biological networks.
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For the ordered hexagonal networks we first started with an ordered hexagonal network with each edge being made up of one polymer. To break the symmetry of the network, a single polymer was removed to create a tensided ring. Over a time-scale of 100 µs this hexagonal network was stretched sinusoidally with a time period of 5.00 µs per cycle and the cycles increasing in amplitude from 10.0 % to 50.0 %. This method has been used by to mimic damage due to stretching in the ocular lens capsule.
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To generate disordered networks we used an established self-assembly process that starts with a collection of the model polymers ). The polymers are initially placed on a random square grid, and allowed to form a proto-network, which was then equilibrated under a barostat at a constant stress ('2D pressure') p and temperature 100 K to fix the simulation cell size, and thus area per molecule. The use of different stresses allows networks with different areas, and hence topologies, to be constructed. This proto-network was heated to 1000 K using a Langevin thermostat to create a polymer liquid. This polymer liquid was cooled back down to 100 K over 100 µs to re-form a network, and ring statistics were tracked during the network forming process. Once the networks were cooled, we repeated the stretching simulation for used for the hexagonal networks to induce rupturing and once again tracked the ring statistics.
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61d714b0e7b75196248f5ca6
| 9 |
Ring statistics in a 2D graph can be tracked easily, as the excluded volume of the beads in the polymer model means that the polymers naturally form a planar graph with a defined embedding in a periodic plane under all but the most extreme conditions. The rings are then found as being nodes in the dual graph of the polymer graph; the dual graph can be found either by computing an anticlockwise ordering of edges around each node (de Berg ( )), or by computing the Delaunay triangulation of the points and re-joining the triangulation into rings ). In this work we use the latter method as we often wish to compute the Delaunay triangulation anyway for analysing networks ), and it can be simpler to treat algorithmically for a periodic simulation cell.
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61d714b0e7b75196248f5ca6
| 10 |
With the simulation protocol established, we simulated both ordered and disordered networks to analyse their formation and damage. We will demonstrate the use of persistence diagrams to analyse ring structure first with individual snapshots to link the visualisation method with the simulated networks, and then move on to showing how persistence diagrams over time can be used to analyse the formation and rupturing behaviour of networks.
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61d714b0e7b75196248f5ca6
| 11 |
Snapshots of a single stretching simulation of an ordered hexagonal network are shown in Figure over the first 50.0 µs of simulation time. The snapshots were taken when the networks passed through an unstretched state during the simulation, and the maximum amount of stretching increased between each snapshot. For the first 20.0 µs the network topology remains fundamentally the same, with only small distortions to the positions of nodes in Figure ). After 30.0 µs however we see the ring structure start to break down, and rings with many sides begin to appear in Figure . As the stretching simulation progresses, we see in Figure (f) that by 50.0 µs almost all hexagons have been destroyed and mostly extremely large rings remain.
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61d714b0e7b75196248f5ca6
| 12 |
The information accessible can be useful in identifying broad mechanisms by which key processes are taking place, but may be insufficient for obtaining sufficient statistical information. A systematic approach can be enabled by using a persistence diagram with the time filtration function f (x, R) as shown in Figure . This persistence diagram correlates well with the qualitative analysis performed earlier -the hexagon lifetimes are solid from 0 µs to 20.0 µs, and begin to break from 25.0 µs to 50.0 µs, with a few 'stragglers' lasting for a long time. Next, mid size rings form, as shown by redder lines, some of which live for a long time. For example, the 16-sided shape in the top left of the snapshots at 30.0 µs to 50.0 µs is represented as a long red line. A few decagons form as the stretching gets more intense, but they have short lifetimes. For stretching times from 40.0 µs onwards, we observe that very large rings with extremely short lifetimes are observed; this matches what we see in the snapshots
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61d714b0e7b75196248f5ca6
| 13 |
The analysis process can be repeated for more complex disordered networks, showing the power of barcode diagrams inspired from persistent homology approaches. The networks in this section were first self-assembled then ruptured as discussed in Section 2. Pairs of birth-death diagrams and barcodes are shown in Figure for networks assembled under different initial conditions (here, the area available per molecule when the network formed, chosen systematically by applying a '2D pressure' to fix the simulation cell size).
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61d714b0e7b75196248f5ca6
| 14 |
The birth-death diagrams can then be used to identify a qualitative difference in the way the networks form at different available areas per molecule. In Figure (a) and Figure , only small rings with short lifespans are formed over the time-span 0 µs to 40.0 µs. Snapshots of the networks at 40.0 µs are shown in Figure ), just before the critical formation point. We can see that in the high area network there are many loose molecules which will soon assemble into rings.
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61d714b0e7b75196248f5ca6
| 15 |
Many rings with long lifespans then form over the time-span 40.0 µs to 60.0 µs with few rings formed in the final 60.0 µs to 100 µs as the system is A magnified view of the ring life lines is shown in Figure and emphasises a key difference this approach has from the traditional simplicial complex approach used in persistent homology. As discussed in Section 1, a dynamical system varying over time there is no guarantee that a given ring R that is born time t will still exist at a later time t ′ . Furthermore, dying at time t ′ does not preclude a ring from being born again even later at time t ′′ . The loss of the mathematical continuity of simplices prevents us from using some further persistent homology analysis techniques (Stolz-Pretzer ( )), such as using the barcodes to measure similarity of trajectories as performed by or to identify spatially important information such as binding sites in the manner of . However, it is useful to identify showing detail on the life-lines of some rings are dead for extended periods of time but are reborn and continue to exist afterwards. For the higher density network shown in detail in Figure (b) we see a key difference from the low density network. For a high density network, the rings are close to one another and the polymer edges are packed in tightly. This means that few rings with long continuous lifespans are observed, but many microscopic damage / repair events happen continuously. This can be quantified with the data in Table , showing there are more unique hexagons observed for as the area per molecule decreases and the hexagons that are observed are reborn more often.
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61d714b0e7b75196248f5ca6
| 16 |
Turning our attention to the top right quartile of Figure ) and the top half of Figure ) we can see how this framework is useful to analyse the difference in rupturing phenomena. We can see in Figure The rupturing at approximately 150 µs is seen for networks at all densities, although differences can be seen in how many new rings are formed during the rupturing process. For the low density network, only very few entirely new rings are formed during the rupturing process but for the high density network many rings are formed (seen as more points in the top right corner of the birth-death diagram) owing to the increased density of polymers allowing for easier repair and reforming of rings.
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61d714b0e7b75196248f5ca6
| 17 |
Persistence diagrams are useful in that they allow us to identify patterns in data that exist in small regions of interest (usually spatial, here temporal) or large regions of interest; either of these regimes may be most useful for further analysis depending on the topic at hand (Stolz-Pretzer ( )). As samples of trajectories are taken with finite gaps of time between them and disk space is not infinite, the temporal sampling rate of a trajectory is a controllable variable. Similarly, the sampling rate for our persistence diagram analysis can be some integer fraction of the sampling rate from the trajectory. Analysing networks at different sampling rates can reveal important patterns in the data, for example low-frequency sampling rates can smooth out ephemeral patterns in the data, and high-frequency sampling rates can draw attention to short-lived phenomena. Alternatively, low-frequency sampling rates can save the computational expense of tracking many unique and short-lived rings. Importantly, the similarity between Figure ) reinforces the idea that high resolution temporal data is not required for persistence diagrams to be a useful analysis tool. Even at the coarsest sampling frequency, key behaviours can still be seen in the persistence diagram and the bars can show which regions of time are interesting enough to study in more detail.
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61d714b0e7b75196248f5ca6
| 18 |
In Figure we can see the effect of different sampling frequencies. While the overall pattern remains the same, they differ in a few ways that are useful for overall analysis of a trajectory. For example, the same critical points of formation beginning at about 40.0 µs and rupturing at 150 µs exist in all diagrams. While the lines of critical behaviour become clearer with more samples, the fact that they exist at all timesteps reinforces their importance. At a very small timestep, short-lived rings (or absences of rings) can be missed out. A short timestep will count a superset of the rings identified using a large timestep, but include many more rings with lifespans less than the window size ∆t. This also means that repair phenomena can be missed if the timestep is too short compared to the timescale of repair. This is especially obvious if we highlight the number of times a given ring is repaired against the timescale, as is shown in Figure . These show that if the timescale is increased, repair phenomena can be missed entirely; this manifests as the graphs trailing off to show on average one birth per ring and that they are alive continuously over their lifespan for long sampling timescales. However, at lower sampling timescales the repair phenomena become clear as each ring is repeatedly reborn and is only alive for a fraction of its total lifespan. This does not plateau at short timespans, indicating that there are possibly infinitely many microscopic damage and repair phenomena before a ring is destroyed for good as a consequence of the thermal motion of the connecting sites.
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61d714b0e7b75196248f5ca6
| 19 |
Finally, we can use the birth-death diagrams to analyse families of simulations and see how critical behaviour emerges from them. The self assembly then rupture simulations were repeated 20 times with different random seeds and the birth death diagrams (as first seen in Figure )) for each trajectory were combined. The resulting combined birth-death diagrams are shown in Figure , and the critical behaviours are now more obvious. We can see in all three subfigures lines just after 100 µs in x and y axes, representing minor damage to rings when the rupturing process begins.
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61d714b0e7b75196248f5ca6
| 20 |
In conclusion, we have demonstrated how the tools of persistent homology, especially the persistence diagram and the concept of tracking births and deaths of individual features, can be useful to analyse temporally varying data. The use of temporal persistence diagrams analysing unique rings being born and dying (possibly repeatedly) over the course of a simulation gives us the ability to clearly identify which rings are long-lived and which are ephemeral. It also allows us to observe how medium-range structure (as represented by rings) changes over time, and which topological features are favoured when. The key difference from traditional persistence diagrams, in which time is not a factor, is the fact that given rings can be born, die, and be born again. This is critical for analysis of repair phenomena in biological networks which have, until now, been difficult to visualise and effectively quantify. Critically, the density of the temporal data does not appear to mask significant network changes from being identified. The method employed aids as a lens to analyse the networks, and this provides further insights into how a network changes over time, allowing us to assess the timescale of network behaviours. We anticipate that this analysis can be extended from 2D biological networks to other network formation and damage analysis tasks, such as glass ageing or zeolite formation.
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6286cbd1d555504c5fa060fd
| 0 |
Accurately and efficiently simulating the quantum dynamics of an open quantum system remains a challenging task in modern quantum physics and chemical physics, due to the unfavorable exponential scaling of the quantum problem. To resolve this challenge, mixed quantum-classical (MQC) approaches have been developed by treating the crucial part of the system as the quantum subsystem, and the other parts as classical degrees of freedom (DOFs). This type of approach is particularly successful for describing non-adiabatic molecular dynamics that involve electronic-nuclear interactions, given the fact that the nuclear DOFs are in general anharmonic and their interactions with the electronic DOFs are non-Markovian, thus rendering many approximate master equations not directly applicable. One major category of these mixed quantum-classical methods is surface hopping, most notably Tully's fewest-switches surface hoping approach and several later additions to this method. Another major category is the mean-field Ehrenfest (MFE) approach, or more generally, semiclassical mapping approaches based on the Meyer-Miller mapping formalism. All of these approaches explicitly propagate the classical DOFs and captures their influence on the quantum DOFs through the parametric dependence of the quantum equations of motion (EOM) on the classical trajectories.
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6286cbd1d555504c5fa060fd
| 1 |
At the same time, the quantum subsystem, in principle, can also interact with other environmental DOFs. Example of these interacting environmental DOFs include the interaction of the electromagnetic field with molecules that causes spontaneous emission and farfield electromagnetic modes that couple to a quantized radiation mode inside an optical cavity that causes phoa) Electronic mail: ekoessle@ur.rochester.edu b) Electronic mail: pengfei.huo@rochester.edu ton leakage. It is often desirable to implicitly capture the environmental influence on the dynamics and avoid simulating these environmental DOFs explicitly. Another example is the presence of high frequency vibrations in molecules that are essentially Markovian for the subsystem dynamics. These high frequency vibrations, however, cause difficulties for MQC simulations due to the inadequate description of them using classical trajectories. A number of approaches seek to phenomenologically incorporate these effects of population decay from a higher energy state to a lower energy state through an imaginary Hamiltonian approach, incorporating ad-hoc first-order decay processes, or a posteriori probabilistic collapse into the ground state. These approaches do not capture the full nature of the decay dynamics as they generally do not describe the dynamics of the ground state at the wavefunction level, or do not account for proper decoherence processes during the population decay dynamics.
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6286cbd1d555504c5fa060fd
| 2 |
When a Markovian approximation of the systemenvironment interaction is valid, the Lindblad master equation offers the most general and quantum mechanically valid description of an open system's quantum dynamics, as it guarantees the positivity and preserves the trace of the reduced density matrix. Thus, it is ideal to use this computationally efficient and analytically simple Lindblad master equation to describe the influence of those Markovian environmental DOFs on the quantum subsystem, while simultaneously using the MQC approximation to incorporate the influence of the anharmonic, non-Markovian nuclear DOFs on the quantum subsystem. Indeed, the density matrix hybrid method, which is based on this idea, has been developed recently to study non-adiabatic dynamics by combining MQC methods and master equation approaches. Still, solving the Lindblad master equation with many states in the quantum system is computationally challenging, due to the need to propagate the density matrix in the Liouville space.
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6286cbd1d555504c5fa060fd
| 3 |
To reduce this unfavorable scaling related to density matrix propagation in the Liouville space, one can equivalently describes dynamics as a stochastic wavefunction in the open system's Hilbert space that exactly reproduces the reduced density matrix of the open system, thus reducing the cost of solving the same dynamics by utilizing an ensemble of trajectories in the Hilbert space. This is referred to as the unravelling of the master equation. It is well known that the Lindblad mater equation can be equivalently expressed as the stochastic Schrödinger equation (SSE) with jump processes that captures the projective action of the environment on the system, which randomly collapses the system wavefunction into a pure system state. Through the trajectory average of a large number of realizations of the jump trajectories, the SSE generates identical reduced density matrix dynamics as the Lindblad master equation. However, due to the large fluctuations in population of the system, the SSE encounters numerical instability when combined with mixed quantum-classical approaches. The Ehrenfest+R method was recently developed to simulate the electronic quantum subsystem coupled to the classical electromagnetic field in order to accurately describe spontaneous emission processes. It effectively captures Lindblad dynamics with a deterministic change of the magnitude of the quantum coefficients, and stochastic changes of the phases. The Ehrenfest+R method, however, cannot exactly reproduce the Lindblad master equation due to a specific choice of the off-diagonal decay rate.
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6286cbd1d555504c5fa060fd
| 4 |
In this paper, we develop a general framework for simulating Lindblad-type dynamics with a wavefunction description. As opposed to the SSE approaches that stochastically change both the magnitudes and the phases of the quantum expansion coefficients, our new method only stochastically changes the phases of the expansion coefficients, and exactly reproduces the jump operator dynamics of the Lindblad master equation. In particular, we give the derivation of a method to propagate the coefficients of the MFE method that fully agrees with Lindblad dynamics. The coefficient-propagation method derived using this framework is inspired by the Ehrenfest+R method; however, our approach exactly recovers Lindblad dynamics whereas the Ehren-fest+R approach does not. Additionally, we derive a simple fix to the approximations present in the Ehren-fest+R method and give rigorous justifications for particular algorithmic choices. Our wavefunction description of Lindblad dynamics can be seamlessly integrated into mixed quantum-classical methods that simulate coupled electronic-nuclear dynamics, such as the mean-field Ehrenfest approach, and we refer to this particular approach as the L-MFE method. We have tested the accuracy, efficiency, and robustness of the L-MFE method through numerical simulations. We envision that our work will provide a general framework for future theoretical work into incorporating Lindblad dynamics into more accurate semiclassical and mixed quantum-classical approaches.
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6286cbd1d555504c5fa060fd
| 5 |
where ĤS is the system Hamiltonian, ÎS is the identity in the system Hilbert space H S , ĤE is the environment Hamiltonian, ÎE is the identity in the environment Hilbert space H E , and ĤI is the interaction Hamiltonian between the system and the environment. Note that the partitioning of the system and environment is not unique. Any part of the total system that we do not want to explicitly simulate or is difficult to simulate can be treated as the environment. The time evolution of the density matrix ρT describing a quantum state of the entire system plus the environment is governed by the following quantum Liouville equation
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6286cbd1d555504c5fa060fd
| 6 |
where L T [•] is the Liouvillian superoperator that acts on operators and [•, •] is the commutator between two operators. When the quantum system of interest interacts with a large environment, the dynamics of the composite system and environment is generally too complicated and computationally expensive to keep track of in full detail. Instead, the dynamics of the system alone can be tracked using the reduced density matrix ρS (associated with ĤS in Eq. 1) defined as follows
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6286cbd1d555504c5fa060fd
| 7 |
For a closed quantum system (isolated system) with no entanglement or interaction with the environment ( ĤI = 0 in Eq. 1), the total density matrix can be factorized as ρT = ρS ⊗ ρE and the quantum Liouville equation for the reduced density matrix of the system is expressed as
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6286cbd1d555504c5fa060fd
| 8 |
When there are explicit system-environment interactions ( ĤI ̸ = 0), however, the dynamics of the reduced density matrix of the system ρS (t) can no longer be described by the unitary evolution of Eq. 4. An exact calculation of ρS (t) could be performed using Eq. 3 if ρT (t) is fully known, but this is generally infeasible to obtain due to the large number of environmental DOFs. Approximations of the environment and system-environment interactions can be made to calculate ρS (t) under certain conditions.
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6286cbd1d555504c5fa060fd
| 9 |
In situations where there are many system DOFs, further approximations must be made to calculate ρS (t). A common approach when there are several nuclear DOFs present in a molecular system is to separate the system into a classical subsystem and a quantum subsystem. The corresponding system Hamiltonian ĤS in this case is
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6286cbd1d555504c5fa060fd
| 10 |
where the quantum subsystem belonging to the Hilbert space H Q , with Hamiltonian ĤQ and identity ÎQ , and the classical subsystem (typically nuclear DOF) belonging to the Hilbert space H C , with Hamiltonian ĤC and identity ÎC , interact through ĤQC . In reactive molecular systems, the Hamiltonian ĤC is usually anharmonic and cannot easily be directly considered as ĤE . When possible, the system dynamics of Eq. 5 can be evaluated fully quantum mechanically; however, when the classical subsystem is too large, the semiclassical approximation can be made. Under the semiclassical approximation, the classical subsystem evolves using classical mechanics while the quantum subsystem evolves using quantum mechanics.
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6286cbd1d555504c5fa060fd
| 11 |
where the trace is performed over the classical nuclear DOFs R (or whatever the classical DOFs happen to be). When the mixed quantum-classical approximation is used, 1 the nuclear coordinates evolve classically and the quantum subsystem evolves using the time-dependent Schrödinger equation governed by ĤQ + ĤQC (R). In this case, since only states in the quantum subspace are evolved using quantum mechanics, ρ(t) can be computationally evaluated without performing a trace on the much larger system Hilbert space which greatly improves computational efficiency.
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6286cbd1d555504c5fa060fd
| 12 |
where Lk is a Lindblad jump operator that imparts the impact of the environment onto the system with interaction strength Γ k (with a unit of rate or inverse time) and { Â, B} = Â B + B Â represents the anti-commutator. The superoperator L Ĥ [•] is the part of the Liouvillian that describes the Hermitian dynamics of the system governed by ĤS defined as
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6286cbd1d555504c5fa060fd
| 13 |
governed by the approximations of ĤE and ĤI during the derivation of the Lindblad master equation. The sum k L Lk [ρ S ] is sometimes referred to as the "dissipator". The Lindblad master equation provides a dynamical map for ρS that forms a dynamical semigroup with generator L Ĥ + k L Lk . The Lindblad master equation in Eq. 9 can also be derived through the full quantum dynamics of ĤT by assuming weak system-environment interactions, the Born-Markov approximation, and the secular approximation.
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6286cbd1d555504c5fa060fd
| 14 |
(12) One of the most commonly used jump operators corresponds to a transition from one state to another state in the quantum subsystem. For example, for a transition from state |ψ 1 ⟩ of the quantum subsystem to state |ψ 0 ⟩ of the quantum subsystem with interaction strength Γ, the system Lindblad jump operator is
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6286cbd1d555504c5fa060fd
| 15 |
and the jump operator L only acts on the quantum DOFs in the subspace H Q . The above jump operator corresponds to a transition in the quantum subspace H Q with no impact on the classical (such as nuclear) subspace H C . We can represent this jump operator LS as well as the system Hamiltonian ĤS in a convenient basis |ψ a ⟩ ⊗ |χ ν ⟩, where |ψ a ⟩ is the diabatic basis for the quantum subsystem and |χ ν ⟩ is the discrete variable representation (DVR) basis for the classical subsystem.
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6286cbd1d555504c5fa060fd
| 16 |
The mean-field Ehrenfest (MFE) approach is a semiclassical, mixed quantum-classical dynamics approach that simultaneously evolves a quantum subsystem and a classical subsystem. Multiple classical trajectories are evolved such that their distribution through time can be tracked to estimate e.g. the time evolution of the probability density of the wavefunction that the classical subsystem is approximating. For the electronic-nuclear dynamics of molecules, the electrons (or electrons and photons) are treated quantum mechanically while the nuclei are treated classically such that the distribution of the classical nuclear trajectories estimates the probability density of the nuclear wavefunction through time. The trajectory average of the electronic wavefunction is used to compute quantum estimators such as the elements of the electronic density matrix.
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6286cbd1d555504c5fa060fd
| 17 |
The MFE approach treats the electronic-nuclear dynamics such that each nuclear trajectory has a corresponding electronic wavefunction assigned to it. The nuclear coordinates parameterize the electronic Hamiltonian which evolves the electronic wavefunction at each time step, while the coefficients of the electronic wavefunction generate a mean-field potential that exerts a classical force on the corresponding nuclear trajectory at each time step.
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6286cbd1d555504c5fa060fd
| 18 |
and the nuclear coordinates evolve according to classical equations of motion with the above force. More details on the derivation and implementation of the MFE approach can be found in the literature. The MFE approach can also be equivalently written down in terms of the reduced density matrix of the quantum subsystem (Eq. 8) as follows
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6286cbd1d555504c5fa060fd
| 19 |
Eq. 21 can be viewed as taking the mixed quantumclassical approximation on -i ℏ [ ĤS , ρS ] in Eq. 12 then tracing out the classical DOFs R. Eq. 21 can be viewed as a special case of the recently developed density matrix hybrid method that combines MQC methods and master equation approaches, where the master equation is chosen to be Lindblad dynamics.
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6286cbd1d555504c5fa060fd
| 20 |
From Eq. 24, it can be seen that the jump operator L = |0⟩⟨1| causes the population of state |1⟩ to decay with a rate of Γ, state |0⟩ to gain the population lost by state |1⟩, and state |1⟩ to decohere from every other state with a rate of Γ 2 . This is an important feature of the Lindblad dynamics.
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6286cbd1d555504c5fa060fd
| 21 |
In this simplified picture, considering state |1⟩ to be the excited state and state |0⟩ to be the ground state, the jump operator L = |0⟩⟨1| causes the excited state to decay to the ground state and both excited and ground states to decohere from each other. When ĤQ = ĤQC = 0 and no other jump operators have non-zero interaction strength, the time evolution of the reduced density matrix (in Eq. 25) can be analytically expressed as e L Lt ρ 00 ρ 01 ρ 10 ρ 11 = ρ 00 + 1 -e -Γt ρ 11 e -Γt/2 ρ 01 e -Γt/2 ρ 10 e -Γt ρ 11 .
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6286cbd1d555504c5fa060fd
| 22 |
(26) From Eq. 26, it can be seen that the jump operator L = |0⟩⟨1| causes exponential decay of the excited state to the ground state with a rate of Γ and exponential decay of the excited-ground coherence with a rate of Γ/2. This dynamical picture approximately corresponds to many physical phenomena such as spontaneous emission and photonic cavity loss when a single decay channel dominates. Expressing the MFE approach in the density matrix form (Eq. 17) allows for Lindblad dynamics to be implemented in a straightforward manner. This involves performing MFE dynamics in the Liouville space where each element |ψ a ⟩⟨ψ b | is now treated as a basis vector. Eq. 19 and Eq. 21 describe the nuclear force and equation of motion of the density matrix elements, respectively. However, this density matrix formalism of MFE requires K 2 density matrix elements to be dynamically updated for a dimension K of the original quantum subsystem Hilbert space H Q , with a formal scaling in the range of N • K 3 to N • K 4 in terms of computational cost, where N is the number of trajectories used to perform the MFE part of the Liouvillian in Eq. 21. The coefficient formalism MFE (Eq. 14), on the other hand, only requires K coefficients to be dynamically updated, with a formal cost of N • K 2 . Due to the numerically favorable scaling of a wavefunction based approach, together with the consideration that the trajectory average is required anyway in performing the MQC dynamics governed by Eq. 15, we choose to incorporate Lindblad dynamics into the regular coefficient-based approach of MFE. This requires a formulation of Lindblad dynamics in a coefficient-based formalism, which is derived in the next section.
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