Electrostatics

静电
  • 文章类型: Journal Article
    SARS-CoV-2是有史以来最具传染性的病毒之一。尽管过去几年进行了大量的研究,病毒的生命周期仍然没有得到很好的理解,特别是膜融合。此过程由融合域(FD)启动,由融合肽(FP)和融合环(FL)组成的高度保守的氨基酸段,协同作用会干扰靶细胞的脂质膜,以降低融合所需的能量成本。在这项研究中,通过基于诱变的方法,我们研究了FD(K825,K835,R847,K854)内的碱性残基利用体外融合测定和19FNMR,通过传统的13C15N技术验证。丙氨酸和电荷守恒突变体揭示了每个基本残基在启动融合的机制中起着高度特异性的作用。有趣的是,K825A导致融合性增加,这被发现与螺旋一内的氨基酸数量相关,进一步暗示了这种特定螺旋在FD融合机制中的作用。这项工作发现碱性残基在FD融合机制中很重要,并强调了K825A,在SARS-CoV-2刺突蛋白的FD内产生的特定突变,由于其可能导致SARS-CoV-2的更强毒株,因此需要进一步研究。
    SARS-CoV-2 is one of the most infectious viruses ever recorded. Despite a plethora of research over the last several years, the viral life cycle is still not well understood, particularly membrane fusion. This process is initiated by the fusion domain (FD), a highly conserved stretch of amino acids consisting of a fusion peptide (FP) and fusion loop (FL), which in synergy perturbs the target cells\' lipid membrane to lower the energetic cost necessary for fusion. In this study, through a mutagenesis-based approach, we have investigated the basic residues within the FD (K825, K835, R847, K854) utilizing an in vitro fusion assay and 19F NMR, validated by traditional 13C 15N techniques. Alanine and charge-conserving mutants revealed every basic residue plays a highly specific role within the mechanism of initiating fusion. Intriguingly, K825A led to increased fusogenecity which was found to be correlated to the number of amino acids within helix one, further implicating the role of this specific helix within the FD\'s fusion mechanism. This work has found basic residues to be important within the FDs fusion mechanism and highlights K825A, a specific mutation made within the FD of the SARS-CoV-2 spike protein, as requiring further investigation due to its potential to contribute to a more virulent strain of SARS-CoV-2.
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  • 文章类型: Journal Article
    背景:泛素阳性包涵体的额颞叶变性(FTLD-TDP),肌萎缩侧索硬化(ALS)和边缘型年龄相关性TDP-43脑病(LATE)与神经元中TARDNA结合蛋白43(TDP-43)的细胞质内含物沉积有关。该过程的一个复杂性在于TDP-43在细胞中形成液相无膜细胞器的能力。以前的工作表明,重组体,纯化,朊病毒样结构域(PrLD)在体外形成液滴,但是互补片段的行为是不确定的。
    方法:我们在没有PrLD的情况下纯化了这种构建体(无PrLD的TDP-43),并使用溶液跳跃方法和一系列生物物理技术诱导了其相分离,以研究形态学,TDP-43组件的物质状态和结构。
    结果:荧光TMR标记的蛋白质构建体,使用共聚焦荧光成像,快速形成(<1分钟)圆形,均质和0.5-1.0µm宽的组件,然后合并成更大的组件,然而,圆形,种。当用AlexaFluor488标记时,它们最初在光漂白(FRAP)后表现出荧光恢复,显示与全长TDP-43不同的液体行为,与PrLD相似。蛋白质分子没有发生重大的结构变化,用圆二色性和固有荧光光谱法测定。该过程具有与全长TDP-43及其PrLD不同的pH和盐依赖性,这可以在静电力的基础上合理化。
    结论:类似于PrLD,无PrLD的TDP-43通过液-液相分离(LLPS)在体外形成液滴,与全长蛋白质不同,后者经历液-固相分离(LSPS)。这些结果提供了控制全长TDP-43及其片段相分离的复杂静电力的基本原理。一方面,无PrLD的TDP-43具有低pI和带正电荷的结构域,LLPS被盐抑制,削弱域间静电吸引力。另一方面,由于高的等离子点(pi),PrLD带正电,因此LLPS被盐促进,并且pH增加,因为它们都减少静电排斥。相比之下,全长TDP-43在其pI下最有利地经受LSPS,在较低和较高的pH值下具有正盐和负盐依赖性,分别,取决于排斥力还是吸引力占主导地位,分别。
    BACKGROUND: Frontotemporal lobar degeneration with ubiquitin-positive inclusions (FTLD-TDP), amyotrophic lateral sclerosis (ALS) and limbic-predominant age-related TDP-43 encephalopathy (LATE) are associated with deposition of cytoplasmic inclusions of TAR DNA-binding protein 43 (TDP-43) in neurons. One complexity of this process lies in the ability of TDP-43 to form liquid-phase membraneless organelles in cells. Previous work has shown that the recombinant, purified, prion-like domain (PrLD) forms liquid droplets in vitro, but the behaviour of the complementary fragment is uncertain.
    METHODS: We have purified such a construct without the PrLD (PrLD-less TDP-43) and have induced its phase separation using a solution-jump method and an array of biophysical techniques to study the morphology, state of matter and structure of the TDP-43 assemblies.
    RESULTS: The fluorescent TMR-labelled protein construct, imaged using confocal fluorescence, formed rapidly (< 1 min) round, homogeneous and 0.5-1.0 µm wide assemblies which then coalesced into larger, yet round, species. When labelled with AlexaFluor488, they initially exhibited fluorescence recovery after photobleaching (FRAP), showing a liquid behaviour distinct from full-length TDP-43 and similar to PrLD. The protein molecules did not undergo major structural changes, as determined with circular dichroism and intrinsic fluorescence spectroscopies. This process had a pH and salt dependence distinct from those of full-length TDP-43 and its PrLD, which can be rationalized on the grounds of electrostatic forces.
    CONCLUSIONS: Similarly to PrLD, PrLD-less TDP-43 forms liquid droplets in vitro through liquid-liquid phase separation (LLPS), unlike the full-length protein that rather undergoes liquid-solid phase separation (LSPS). These results offer a rationale of the complex electrostatic forces governing phase separation of full-length TDP-43 and its fragments. On the one hand, PrLD-less TDP-43 has a low pI and oppositively charged domains, and LLPS is inhibited by salts, which attenuate inter-domain electrostatic attractions. On the other hand, PrLD is positively charged due to a high isoionic point (pI) and LLPS is therefore promoted by salts and pH increases as they both reduce electrostatic repulsions. By contrast, full-length TDP-43 undergoes LSPS most favourably at its pI, with positive and negative salt dependences at lower and higher pH, respectively, depending on whether repulsive or attractive forces dominate, respectively.
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  • 文章类型: Journal Article
    含铜蛋白质在生物系统中起着至关重要的作用。天青蛋白是一种含铜蛋白,具有1型铜位点,可促进细胞色素链中的电子转移。先前的研究已经强调了铜位点的轴向Met121中的突变对还原电位的显著影响。然而,该法规的机制尚未完全建立。在这项研究中,我们采用理论模型来研究1型铜站点的还原,专注于Met121的非天然氨基酸取代如何影响其行为。我们的研究结果表明,静电相互作用与铜位点的还原电位之间存在很强的线性相关性,这表明还原电势的扰动主要受金属离子与连接原子之间的静电相互作用的影响。此外,我们发现CF/π和CF...H相互作用会引起几何形状的细微变化,从而影响所研究系统的电子性质。此外,我们的计算表明,配位模式和离子-配体距离可以显着影响铜位点的还原电位。总的来说,这项研究为1型铜位点的结构和电子特性提供了有价值的见解,这可能会指导未来人造催化剂的设计。
    Copper-containing proteins play crucial roles in biological systems. Azurin is a copper-containing protein which has a Type 1 copper site that facilitates electron transfer in the cytochrome chain. Previous research has highlighted the significant impact of mutations in the axial Met121 of the copper site on the reduction potential. However, the mechanism of this regulation has not been fully established. In this study, we employed theoretical modeling to investigate the reduction of the Type 1 copper site, focusing on how unnatural amino acid substitutions at Met121 influence its behavior. Our findings demonstrated a strong linear correlation between electrostatic interactions and the reduction potential of the copper site, which indicates that the perturbation of the reduction potential is primarily influenced by electrostatic interactions between the metal ion and the ligating atom. Furthermore, we found that CF/π and CF…H interactions could induce subtle changes in geometry and hence impact the electronic properties of the systems under study. In addition, our calculations suggest the coordination mode and ion-ligand distance could significantly impact the reduction potential of a copper site. Overall, this study offers valuable insights into the structural and electronic properties of the Type 1 copper site, which could potentially guide the design of future artificial catalysts.
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  • 文章类型: Journal Article
    在早期候选铅选择和优化过程中,对抗体开发能力的计算机评估至关重要。提供快速和无材料的筛选方法。然而,这些方法的预测能力和可重复性在很大程度上取决于分子描述符的选择,模型参数,预测结构模型的准确性,和构象采样技术。这里,我们提出了一组专门设计用于预测抗体发展的分子表面描述符。我们通过将它们的相关性与大量实验确定的生物物理特性进行基准测试来评估这些描述符的性能,包括粘度,聚合,疏水相互作用色谱,人体药代动力学清除率,肝素保留时间,和多特异性。Further,我们研究了这些表面描述符对方法学细微差别的敏感性,例如内部介电常数的选择,疏水性尺度,结构预测方法,以及构象抽样的影响。值得注意的是,根据所使用的结构预测方法,我们观察到表面描述符分布的系统变化,驱动表面描述符在结构模型中的弱相关性。在来自分子动力学的构象分布上对描述符值进行平均可以减轻系统偏移,并提高不同结构预测方法之间的一致性。尽管与生物物理数据的相关性改善不一致。根据我们的基准分析,我们提出了六个计算机可开发性风险标志,并评估了它们在预测一组案例研究分子的潜在可开发性问题方面的有效性.
    In silico assessment of antibody developability during early lead candidate selection and optimization is of paramount importance, offering a rapid and material-free screening approach. However, the predictive power and reproducibility of such methods depend heavily on the selection of molecular descriptors, model parameters, accuracy of predicted structure models, and conformational sampling techniques. Here, we present a set of molecular surface descriptors specifically designed for predicting antibody developability. We assess the performance of these descriptors by benchmarking their correlations with an extensive array of experimentally determined biophysical properties, including viscosity, aggregation, hydrophobic interaction chromatography, human pharmacokinetic clearance, heparin retention time, and polyspecificity. Further, we investigate the sensitivity of these surface descriptors to methodological nuances, such as the choice of interior dielectric constant, hydrophobicity scales, structure prediction methods, and the impact of conformational sampling. Notably, we observe systematic shifts in the distribution of surface descriptors depending on the structure prediction method used, driving weak correlations of surface descriptors across structure models. Averaging the descriptor values over conformational distributions from molecular dynamics mitigates the systematic shifts and improves the consistency across different structure prediction methods, albeit with inconsistent improvements in correlations with biophysical data. Based on our benchmarking analysis, we propose six in silico developability risk flags and assess their effectiveness in predicting potential developability issues for a set of case study molecules.
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  • 文章类型: Journal Article
    静电纺丝(ES)的应用范围从生物医学设备和组织再生支架的制造到光操纵和能量转换,甚至沉积作为纳米级催化生长平台的材料。广泛采用ES的一个主要限制是由于聚合物流接近沉积表面时的混沌运动导致的随机纤维沉积。在过去,具有精确确定的中尺度形态的结构或材料的制造已通过电极形状的修改完成,使用多维电极或引脚,沉积在织布机上,手持式静电纺丝装置,允许用户引导沉积,或通过透镜元件或孔径的电场操纵。在这项工作中,我们演示了一个ES系统,该系统包含多个高压电源,这些高压电源通过LabVIEW中实现的控制算法独立控制。最终结果是我们所说的“多路复用ES”,其中ES光纤将多个独立控制的高压信号组合在一起,从而实现独特的沉积控制。COMSOLMultiphysics®软件用于对该新型ES系统中产生的电场进行建模。采用多电源供电系统,我们展示了编织纤维材料的制造,不需要复杂的沉积表面。时变正弦波输入用于创建电纺圆环形状。托里的外径被发现了,通过参数分析,对沉积过程中使用的频率相当不敏感,而内径与频率成反比,导致圆环的总宽度随频率增加。多路复用ES具有基于高压电路的时间响应的高频截止。通过添加并联电阻器来测量并最小化这些时间常数,从而降低了系统的阻抗并将高频截止提高了63%。
    Applications of electrospinning (ES) range from fabrication of biomedical devices and tissue regeneration scaffolds to light manipulation and energy conversion, and even to deposition of materials that act as growth platforms for nanoscale catalysis. One major limitation to wide adoption of ES is stochastic fiber deposition resulting from the chaotic motion of the polymer stream as is approaches the deposition surface. In the past, fabrication of structures or materials with precisely determined mesoscale morphology has been accomplished through modification of electrode shape, use of multi-dimensional electrodes or pins, deposition onto weaving looms, hand-held electrospinning devices that allow the user to guide deposition, or electric field manipulation by lensing elements or apertures. In this work, we demonstrate an ES system that contains multiple high voltage power supplies that are independently controlled through a control algorithm implemented in LabVIEW. The end result is what we term \"multiplex ES\" where multiple independently controlled high-voltage signals are combined by the ES fiber to result in unique deposition control. COMSOL Multiphysics® software was used to model the electric field produced in this novel ES system. Using the multi-power supply system, we demonstrate fabrication of woven fiber materials that do not require complex deposition surfaces. Time-varied sinusoidal wave inputs were used to create electrospun torus shapes. The outer diameter of the tori was found, through parametric analysis, to be rather insensitive to frequency used during deposition, while inner diameter was inversely related to frequency, resulting in overall width of the tori increasing with frequency. Multiplex ES has a high-frequency cutoff based on the time response of the high voltage electrical circuit. These time constants were measured and minimized through the addition of parallel resistors that decreased impedance of the system and improved the high-frequency cutoff by up to 63%.
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  • 文章类型: Journal Article
    捕食者和猎物受益于检测彼此存在的感官线索。当他们在他们的环境中移动时,陆生动物积累静电荷。因为电荷在一定距离处施加力,可以想象,被捕食的动物可以感应到电力来探测接近的捕食者。这里,我们报告了这样一个案例,一个陆地动物通过电接收来检测它的捕食者。我们证明掠夺性黄蜂被起诉,从而发出电场,毛虫以防御行为回应这些领域。此外,毛虫的机械感觉刚毛被这些静电力偏转,并被调整到它们的昆虫捕食者的翼拍频率。这种能力揭示了猎物和捕食者之间的感官相互作用的一个维度,并且可能在陆生动物中普遍存在。
    Predators and prey benefit from detecting sensory cues of each other\'s presence. As they move through their environment, terrestrial animals accumulate electrostatic charge. Because electric charges exert forces at a distance, a prey animal could conceivably sense electrical forces to detect an approaching predator. Here, we report such a case of a terrestrial animal detecting its predators by electroreception. We show that predatory wasps are charged, thus emit electric fields, and that caterpillars respond to such fields with defensive behaviors. Furthermore, the mechanosensory setae of caterpillars are deflected by these electrostatic forces and are tuned to the wingbeat frequency of their insect predators. This ability unveils a dimension of the sensory interactions between prey and predators and is likely widespread among terrestrial animals.
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  • 文章类型: Journal Article
    极端微生物已经适应了极端的物理化学条件。嗜盐生物,特别是,在非常高的盐浓度下生存。为了实现这一点,他们设计了蛋白质的表面,以增加短,极性和酸性氨基酸,虽然减少很大,疏水性和碱性残基。虽然这些适应最初在没有盐的情况下降低了蛋白质的稳定性,它们赋予嗜盐蛋白在极高盐浓度的环境中显著的稳定性,其中非适应蛋白质展开和聚集。嗜盐蛋白实现这一目标的分子机制,然而,还不清楚。这里,我们检验了嗜盐氨基酸组成使蛋白质表面不稳定的假设,但在交换中提高了盐存在下的稳定性。要做到这一点,我们已经测量了在不存在和存在不同盐的情况下具有不同亲盐程度的各种蛋白质变体的折叠热力学,在不同的pH值下调节酸性氨基酸的电离状态。我们的结果表明,嗜盐氨基酸降低嗜盐蛋白在嗜温条件下的稳定性,但在交换中提高盐诱导的稳定性和溶解度。我们还发现,与传统假设相反,与静电相比,疏水效应和优先离子排斥产生的贡献与光环适应更相关。总的来说,我们的发现表明,在折叠热力学和嗜盐适应性之间进行权衡,以优化蛋白质以适应高盐环境。
    Extremophile organisms have adapted to extreme physicochemical conditions. Halophilic organisms, in particular, survive at very high salt concentrations. To achieve this, they have engineered the surface of their proteins to increase the number of short, polar and acidic amino acids, while decreasing large, hydrophobic and basic residues. While these adaptations initially decrease protein stability in the absence of salt, they grant halophilic proteins remarkable stability in environments with extremely high salt concentrations, where non-adapted proteins unfold and aggregate. The molecular mechanisms by which halophilic proteins achieve this, however, are not yet clear. Here, we test the hypothesis that the halophilic amino acid composition destabilizes the surface of the protein, but in exchange improves the stability in the presence of salts. To do that, we have measured the folding thermodynamics of various protein variants with different degrees of halophilicity in the absence and presence of different salts, and at different pH values to tune the ionization state of the acidic amino acids. Our results show that halophilic amino acids decrease the stability of halophilic proteins under mesophilic conditions, but in exchange improve salt-induced stabilization and solubility. We also find that, in contrast to traditional assumptions, contributions arising from hydrophobic effect and preferential ion exclusion are more relevant for haloadaptation than electrostatics. Overall, our findings suggest a trade-off between folding thermodynamics and halophilic adaptation to optimize proteins for hypersaline environments.
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  • 文章类型: Journal Article
    PHPT1是组氨酸磷酸酶,其通过其催化活性调节真核生物中的信号传导。这里,我们通过溶液NMR的组合来分析PHPT1的结构和动力学,分子动力学,和生化实验。我们确定了在R78胍部分和Y125上的C末端羧基之间形成的盐桥,这对于配体结合至关重要。通过在C末端(G126)附加甘氨酸残基来破坏盐桥,会导致催化活性和对假底物的结合亲和力降低。对硝基苯磷酸酯(pNPP),以及活性位点抑制剂,苯基膦酸(PPA)。我们通过核磁共振显示化学位移,15N弛豫测量,和分子动力学轨迹分析,该盐桥的去除导致活性位点的结构和动态改变,从而显著影响酶的功能并证实该静电相互作用的重要性。
    PHPT1 is a histidine phosphatase that modulates signaling in eukaryotes through its catalytic activity. Here, we present an analysis of the structure and dynamics of PHPT1 through a combination of solution NMR, molecular dynamics, and biochemical experiments. We identify a salt bridge formed between the R78 guanidinium moiety and the C-terminal carboxyl group on Y125 that is critical for ligand binding. Disruption of the salt bridge by appending a glycine residue at the C-terminus (G126) leads to a decrease in catalytic activity and binding affinity for the pseudo substrate, para-nitrophenylphosphate (pNPP), as well as the active site inhibitor, phenylphosphonic acid (PPA). We show through NMR chemical shift, 15N relaxation measurements, and analysis of molecular dynamics trajectories, that removal of this salt bridge results in an active site that is altered both structurally and dynamically thereby significantly impacting enzymatic function and confirming the importance of this electrostatic interaction.
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  • 文章类型: Journal Article
    脂质是调节膜融合的关键因素。脂质不仅是形成膜的结构成分,而且是囊泡融合和神经递质释放的活性催化剂,由可溶性N-乙基马来酰亚胺敏感因子附着蛋白受体(SNARE)蛋白驱动。SNARE蛋白似乎在融合前部分组装,但是在Ca2流入之前阻止囊泡融合的机制仍然不清楚。这里,我们表明,磷脂酰肌醇4,5-二磷酸(PIP2)静电触发囊泡融合作为一种静电催化剂,通过降低水合能和肉豆蔻酰化的富含丙氨酸的C激酶底物(MARCKS),PIP2结合蛋白,在SNARE复合物部分组装的囊泡对接状态下阻止囊泡融合。囊泡模拟脂质体无法通过掩蔽PIP2来再现囊泡融合停滞,这表明天然囊泡对于生理囊泡融合的重建至关重要。PIP2吸引阳离子以排斥膜中的水分子,从而降低水合能量屏障。
    Lipids are key factors in regulating membrane fusion. Lipids are not only structural components to form membranes but also active catalysts for vesicle fusion and neurotransmitter release, which are driven by soluble N-ethylmaleimide-sensitive factor attachment protein receptor (SNARE) proteins. SNARE proteins seem to be partially assembled before fusion, but the mechanisms that arrest vesicle fusion before Ca2+ influx are still not clear. Here, we show that phosphatidylinositol 4,5-bisphosphate (PIP2) electrostatically triggers vesicle fusion as an electrostatic catalyst by lowering the hydration energy and that a myristoylated alanine-rich C-kinase substrate (MARCKS), a PIP2-binding protein, arrests vesicle fusion in a vesicle docking state where the SNARE complex is partially assembled. Vesicle-mimicking liposomes fail to reproduce vesicle fusion arrest by masking PIP2, indicating that native vesicles are essential for the reconstitution of physiological vesicle fusion. PIP2 attracts cations to repel water molecules from membranes, thus lowering the hydration energy barrier.
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  • 文章类型: Journal Article
    对纯净水定量评估了有针对性地替换经验力场中单个术语的影响,二氯甲烷(CH2$${}_2$$Cl2$${}_2$$),和溶剂化K+$${}^{+}$$和Cl-$${}^{-}$$离子。对于静电相互作用,点电荷(PC)和基于机器学习(ML)的最小分布电荷(MDCM)与基于库仑积分的静电一起评估。通过将碎片分子轨道(FMO)衍生的极化能量添加到现有力场来研究明确包含二阶项的影响,在这种情况下CHARMM。证明了各向异性静电降低了水的RMSE(1.4kcal/mol),CH2$${}_2$$Cl2$${}_2$$(0.8kcal/mol)和溶剂化Cl-$${}^{-}$$$簇(0.4kcal/mol)。对于CH2$${}_2$$Cl2$${}_2$${}_2$$,可以忽略额外的极化项,但进一步改进了纯净水(〜$$$\\sim$$$1.0kcal/mol)和水合Cl-$${}^{-}$$(0.4kcal/mol)的模型,并且是溶剂化K+$${}^{+}$$的关键,将RMSE降低2.3kcal/mol。12-6Lennard-Jones函数形式在PC和MDCM静电下表现令人满意,但不适用于说明静电穿透能量的描述。通过将严格的2体方法与自洽的参考数据进行比较来评估多体贡献的重要性。两体相互作用足以满足CH2$$${}_2$Cl2$${}_2$$,而水和溶剂化K$${}^{}$$和Cl-$${}^{-}$$离子需要明确的多体校正。最后,多体校正的二聚体势能面超过使用常规经验力场获得的精度,可能达到FMO计算的水平。本工作系统地量化了哪些术语可以改善现有力场的性能,以及使用哪些参考数据以易于处理的方式对这些术语进行参数化,以用于纯系统和异构系统的ML拟合。
    The impact of targeted replacement of individual terms in empirical force fields is quantitatively assessed for pure water, dichloromethane (CH   2 Cl   2 ), and solvated K   + and Cl   - ions. For the electrostatic interactions, point charges (PCs) and machine learning (ML)-based minimally distributed charges (MDCM) fitted to the molecular electrostatic potential are evaluated together with electrostatics based on the Coulomb integral. The impact of explicitly including second-order terms is investigated by adding a fragment molecular orbital (FMO)-derived polarization energy to an existing force field, in this case CHARMM. It is demonstrated that anisotropic electrostatics reduce the RMSE for water (by 1.4 kcal/mol), CH   2 Cl   2 (by 0.8 kcal/mol) and for solvated Cl   - clusters (by 0.4 kcal/mol). An additional polarization term can be neglected for CH   2 Cl   2 but further improves the models for pure water (by ∼ 1.0 kcal/mol) and hydrated Cl   - (by 0.4 kcal/mol), and is key for solvated K   + , reducing the RMSE by 2.3 kcal/mol. A 12-6 Lennard-Jones functional form performs satisfactorily with PC and MDCM electrostatics, but is not appropriate for descriptions that account for the electrostatic penetration energy. The importance of many-body contributions is assessed by comparing a strictly 2-body approach with self-consistent reference data. Two-body interactions suffice for CH   2 Cl   2 whereas water and solvated K   + and Cl   - ions require explicit many-body corrections. Finally, a many-body-corrected dimer potential energy surface exceeds the accuracy attained using a conventional empirical force field, potentially reaching that of an FMO calculation. The present work systematically quantifies which terms improve the performance of an existing force field and what reference data to use for parametrizing these terms in a tractable fashion for ML fitting of pure and heterogeneous systems.
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