Computational Estimates of Passive Permeability in Drug Discovery: History
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Contributor:
Austen Bernardi
,
W. F. Drew Bennett
,
Stewart He
,
Derek Jones
,
Dan Kirshner
,
Brian J. Bennion
,
Timothy S. Carpenter

Passive permeation of cellular membranes is a key feature of many therapeutics. The relevance of passive permeability spans all biological systems as they all employ biomembranes for compartmentalization.

  • passive permeability
  • biomembrane
  • molecular dynamics
  • machine learning
  • lipophilicity

1. Introduction

The transport of small molecules such as drugs, toxins, and nutrients to and from various biological subsystems within living things is generally mediated through lipid membranes [1]. This includes transport in the skin [2][3], the lungs [4], the placenta [5], the intestines [6], the renal system [7], and the brain across the “blood brain barrier” (BBB) [8][9][10]. Interested readers are directed to Di et al. [11], which provides a general, comprehensive review of the biological aspects of passive permeability in drug design. Transport across biological lipid membranes may be characterized as either passive [11], where transport occurs mainly via diffusion across the membrane, or active [12], where transport is actively facilitated through transmembrane proteins.

2. Passive Permeability Studies Using Atomistic Molecular Dynamics

Sampling systems with MD simulations produce time-evolved trajectories at atomistic resolution. MD simulations can be directly applied to studying passive permeability when combined with permeability theory [13]. MD simulations have the potential to enable much greater fidelity than the well-established lipophilicity relations since MD can explicitly model complex system features, such as heterogeneous membrane compositions and nano-scale interfacial effects [13]. Classical MD can also be further enhanced with more intricate sampling techniques, such as constant-pH simulations [14], polarizable MD [15], or ab initio MD [16]. However, it is worth noting that as the methodological complexity of sampling increases, the computation cost and opportunities for incidental systematic errors also increase.

2.1. Inhomogeneous Solubility-Diffusion

ISD approaches have been robustly implemented during the last decade [17][18][19][20][21][22][23][24]. While ISD has several validated successes at predicting passive permeability, it can be computationally intensive and is contingent on the accuracy and reliability of underlying collective variables and parameters.
Sugita M. et al., 2021 [20] performed a comprehensive study computing passive permeabilities of 156 cyclic peptides using ISD. Only one protonation state for each side chain was considered. Also, only cyclic peptides with an AlogP value less than 4.0 were modeled, as the experimental results for those with values greater than 4.0 are thought to be unreliable due to low peptide solubility [25]. Permeabilities and energy barriers were quantified and compared to experimental and AlogP data. Performance was assessed with standard statistical metrics and generally yielded middling results in precision and accuracy. This study exemplifies that, even with restraints applied to simplify the quantification of permeability using ISD, accurate quantification and relation to experiments remain a challenge.
Yue Z. et al., 2019 [21] studied pH-dependent membrane permeation of propanolol using continuous constant-pH MD (CPMD), which enables smooth dynamic protonation of molecules during simulation. CPMD uses a continuous variable ranging from zero to one that represents the (non-physical) partial protonation state, combined with a hybrid explicit/implicit solvent model. The permeability calculated from CPMD was found to be similar in value to the permeability calculated from the minimum of the PMFs of the individual protonated states (without CPMD). The orientational effects on permeation were also studied using 2D PMFs. This study showcases the latest computational techniques to study membrane permeation using dynamic protonation in detail.

2.2. Permeant Counting Studies

Permeant counting methods calculate passive membrane permeability using unbiased simulations that explicitly count permeant transport events [26][27][28][29][30]. These methods are generally best applied to high-permeability permeants since the methods’ accuracy is directly related to the number of permeation events sampled during the simulation. Permeant counting methods are attractive since they do not require any artificial bias to be applied to the simulated system. However, this can also be detrimental to sampling since artificial biases accelerate the sampling of conformational space. Due to these sampling concerns, studies using these methods often also provide comparisons to ISD results.
Mahdi G. et al., 2020 [27] studied concentration-dependent passive permeation of ethanol using flux-based and transition-based permeant counting methods and compared these to a Bayesian inference ISD method. Interestingly, with respect to experimentally calculated permeabilities, it was found that permeabilities were significantly overestimated using the ISD method, while permeabilities were overestimated to a lesser degree using permeant counting methods. Various possible explanations for these discrepancies are provided, such as the unstirred layer effect [31] and force-field inaccuracy.
Krämer A. et al., 2020 [28] studied permeabilities of oxygen, water, and ethanol using permeant counting and also compared to Bayesian-inferenced ISD. They were able to obtain more precise results using permeant counting compared to ISD, while there were issues with Markovian sampling at the membrane-water interface for permeant counting. In general, the authors note that standard additive force fields for these molecules are generally insufficient to yield agreement with experimental results, and polarizable force fields might be required. 

3. Applications Using Coarse-Grained Molecular Dynamics

The high computational cost of atomistic simulations has motivated the development and utilization of coarse-grained (CG) models. Consequently, there have been a large number of passive permeability studies that have employed CG MD [32][33][34][35][36][37][38]. CG models reduce the degrees of freedom of the simulation, enabling larger simulations with longer timesteps and greater computational efficiency. These sampling enhancements come at the cost of atomic resolution detail, accuracy, and dynamics. Due to the expense of these trade-offs, it is especially important to employ experimental comparison for CG MD studies. The popular Martini model, which roughly maps three to four heavy atoms to a single interaction site or “bead”, was parameterized to reproduce the free energies of transfer between water and organic liquids for a variety of chemical building blocks [39]. The speed of the Martini model allows for higher energy barriers to be crossed with relatively low computational cost, allowing permeation to be directly observed in silico, such as for cholesterol [40]. It is also possible to simulate larger molecules permeating membranes with the Martini model, with notable examples including antimicrobial peptides [41], gold nano-particles [42], and DNA-encapsulated nano-particles [43]. The large number of parameters available in the Martini force field also allows simulations of large and complex lipid mixtures, such as realistic bacterial membranes [44], viral membranes [45], and plasma membrane models [46].
In a series of works, Bereau and co-workers ran extensive MD campaigns for Martini small molecules crossing lipid membranes [33][35][47]. ML models were then built on these calculations, extrapolating the potential chemical space covered within the model [48]. 

4. Applications of Machine Learning

Traditional machine-learning or non-deep-learning methods have been applied to various problem domains, including molecular information and modeling, with no exception to passive permeability [49][50][51][52][53][54]. These include traditional ML models such as support vector machines (SVMs) as well as k-nearest neighbors (KNN). Deep-learning techniques, which involve some form of sophisticated neural network, have recently become a key subject of computational research concerning lipophilicity and, by extension, passive permeability [48][53][55][56][57][58]. With increasingly larger collections of available data in recent years, both labeled and unlabeled, as well as advancements in data processing capabilities, the adoption of ML techniques has increased at a seemingly exponential pace. MoleculeNet, a prominent benchmark that aggregates numerous molecular datasets and ML methods, predicts various molecular descriptors, including passive permeability [59]. MoleculeNet provides useful benchmark comparisons to compare state-of-the-art ML methods to standard, validated methods. Additionally, MoleculeNet aggregates a representative set of not only traditional ML methods but state-of-the-art deep-learning techniques such as graph convolution neural networks (GNNs) across 17 distinct molecular datasets.
There are a number of recent studies that combine ML models with MD simulations to study small molecule permeation [48][50][55][60][61][62]. Riniker and co-workers have developed molecular dynamics fingerprints (MDFPs), where short MD simulations of a molecule were performed and analyzed to give time-averaged molecular features that improved ML predictions [62]. Recently, high-throughput atomistic MD free energy calculations for small-molecule transfer from water to cyclo-hexane were used to train different ML models [55]. After training, the ML model had a similar error in predicting the MD free energy compared to the average MD sampling error for new molecules. The computational cost of the ML model was a fraction of running the MD free energies; training on thousands of small molecules could lead to predictions for millions.
Another active area of research is using ML methods to analyze trajectories from MD simulations, which is an attractive application for ML, given the multi-dimensional nature of molecular simulation data. Unsupervised approaches can learn a latent representation of a reaction that allows scientists to better identify rare or interesting states [63]. Other ML-based methods that isolate collective variables are also under development, with the potential for application to membrane permeation [64][65].
Additionally, neural network-based potential energy functions (NNPs) have demonstrated the possibility of extrapolating the more expensive quantum mechanical (QM) potential energy functions. NNPs show competitive accuracy when compared to other popular approximation methods [66]. Simultaneously, open-source tools are being developed using the popular deep-learning framework PyTorch to aid in the development of novel ML force fields that can be trained with backpropagation [67][68].

This entry is adapted from the peer-reviewed paper 10.3390/membranes13110851

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