Aims and Objectives To determine the health‐related quality of life (HRQoL) of COVID‐19 patients after discharge and its predicting factors. Background COVID‐19 has caused a worldwide pandemic and led a huge impact on the health of human and daily life. It has been demonstrated that physical and psychological conditions of hospitalised COVID‐19 patients are impaired, but the studies focus on physical and psychological conditions of COVID‐19 patients after discharge from hospital are rare. Design A multicentre follow‐up study. Methods This was a multicentre follow‐up study of COVID‐19 patients who had discharged from six designated hospitals. Physical symptoms and HRQoL were surveyed at first follow‐up (the third month after discharge). The latest multiple laboratory findings were collected through medical examination records. This study was performed and reported in accordance with STROBE checklist. Results Three hundred eleven patients (57.6%) were reported with one or more physical symptoms. The scores of HRQoL of COVID‐19 patients at third month after discharge, except for the dimension of general health, were significantly lower than Chinese population norm ( p < .001). Results of logistic regression showed that female (odds ratio (OR): 1.79, 95% confidence interval (CI): 1.04–3.06), older age (≥60 years) (OR: 2.44, 95% CI: 1.33–4.47) and the physical symptom after discharge (OR: 40.15, 95% CI: 9.68–166.49) were risk factors for poor physical component summary; the physical symptom after discharge (OR: 6.68, 95% CI: 4.21–10.59) was a risk factor for poor mental component summary. Conclusions Health‐related quality of life of discharged COVID‐19 patients did not come back to normal at third month after discharge and affected by age, sex and the physical symptom after discharge. Relevance to clinical practice Healthcare workers should pay more attention to the physical and psychological rehabilitation of discharged COVID‐19 patients. Long‐term follow‐up on COVID‐19 patients after discharge is needed to determine the long‐term impact of COVID‐19.

The effects of finite particle size on electrostatics, density profiles, and diffusion have been a long existing topic in the study of ionic solution. The previous size-modified Poisson-Boltzmann and Poisson-Nernst-Planck models are revisited in this article. In contrast to many previous works that can only treat particle species with a single uniform size or two sizes, we generalize the Borukhov model to obtain a size-modified Poisson-Nernst-Planck (SMPNP) model that is able to treat nonuniform particle sizes. The numerical tractability of the model is demonstrated as well. The main contributions of this study are as follows. 1), We show that an (arbitrarily) size-modified PB model is indeed implied by the SMPNP equations under certain boundary/interface conditions, and can be reproduced through numerical solutions of the SMPNP. 2), The size effects in the SMPNP effectively reduce the densities of highly concentrated counterions around the biomolecule. 3), The SMPNP is applied to the diffusion-reaction process for the first time, to our knowledge. In the case of low substrate density near the enzyme reactive site, it is observed that the rate coefficients predicted by SMPNP model are considerably larger than those by the PNP model, suggesting both ions and substrates are subject to finite size effects. 4), An accurate finite element method and a convergent Gummel iteration are developed for the numerical solution of the completely coupled nonlinear system of SMPNP equations.

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Poisson-Boltzmann electrostatics is a well established model in biophysics; however, its application to large-scale biomolecular processes such as protein-protein encounter is still limited by the efficiency and memory constraints of existing numerical techniques. In this article, we present an efficient and accurate scheme that incorporates recently developed numerical techniques to enhance our computational ability. In particular, a boundary integral equation approach is applied to discretize the linearized Poisson-Boltzmann equation; the resulting integral formulas are well conditioned and are extended to systems with arbitrary numbers of biomolecules. The solution process is accelerated by Krylov subspace methods and a new version of the fast multipole method. In addition to the electrostatic energy, fast calculations of the forces and torques are made possible by using an interpolation procedure. Numerical experiments show that the implemented algorithm is asymptotically optimal O(N) in both CPU time and required memory, and application to the acetylcholinesterasefasciculin complex is illustrated. In recent years, because of the rapid advances in biotechnology, both the temporal and spatial scales of biomolecular studies have been increased significantly: from single molecules to interacting molecular networks in a cell, and from the static molecular structures at different resolutions to the dynamical interactions in biophysical processes. In these studies, the electrostatics modeled by the well established Poisson-Boltzmann (PB) equation has been shown to play an important role under physiological solution conditions. Therefore, its accurate and efficient numerical treatment becomes extremely important, especially in the study of large-scale dynamical processes such as protein-protein association and dissociation in which the PB equation has to be solved repetitively during a simulation.Traditional numerical schemes for PB electrostatics include the finite difference methods, where difference approximations are used on structured grids, and finite element methods in which arbitrarily shaped biomolecules are discretized by using elements and the associated basis functions. The resulting algebraic systems for both are commonly solved by using multigrid or domain decomposition accelerations for optimal efficiency. However, as the grid number (and thus the storage, number of operations, and condition number of the system) increases proportionally to the volume size, finite difference and finite element methods become less efficient and accurate for systems with large spatial sizes, e.g., as encountered in protein association and dissociation. Alternative methods include the boundary element method (BEM) and the boundary integral equation (BIE) method. In these methods, only the surfaces of the molecules are discretized; hence, the number of unknowns is greatly reduced. Unfortunately, in earlier versions of BEM, the matrix is stored explicitly and the resulting dense linear system is solved by using Gauss eli...

a b s t r a c tIn this paper we developed accurate finite element methods for solving 3-D PoissonNernst-Planck (PNP) equations with singular permanent charges for simulating electrodiffusion in solvated biomolecular systems. The electrostatic Poisson equation was defined in the biomolecules and in the solvent, while the Nernst-Planck equation was defined only in the solvent. We applied a stable regularization scheme to remove the singular component of the electrostatic potential induced by the permanent charges inside biomolecules, and formulated regular, well-posed PNP equations. An inexact-Newton method was used to solve the coupled nonlinear elliptic equations for the steady problems; while an AdamsBashforth-Crank-Nicolson method was devised for time integration for the unsteady electrodiffusion. We numerically investigated the conditioning of the stiffness matrices for the finite element approximations of the two formulations of the Nernst-Planck equation, and theoretically proved that the transformed formulation is always associated with an ill-conditioned stiffness matrix. We also studied the electroneutrality of the solution and its relation with the boundary conditions on the molecular surface, and concluded that a large net charge concentration is always present near the molecular surface due to the presence of multiple species of charged particles in the solution. The numerical methods are shown to be accurate and stable by various test problems, and are applicable to real large-scale biophysical electrodiffusion problems.

A computational framework is presented for the continuum modeling of cellular biomolecular diffusion influenced by electrostatic driving forces. This framework is developed from a combination of state-of-the-art numerical methods, geometric meshing, and computer visualization tools. In particular, a hybrid of ͑adaptive͒ finite element and boundary element methods is adopted to solve the Smoluchowski equation ͑SE͒, the Poisson equation ͑PE͒, and the Poisson-Nernst-Planck equation ͑PNPE͒ in order to describe electrodiffusion processes. The finite element method is used because of its flexibility in modeling irregular geometries and complex boundary conditions. The boundary element method is used due to the convenience of treating the singularities in the source charge distribution and its accurate solution to electrostatic problems on molecular boundaries. Nonsteady-state diffusion can be studied using this framework, with the electric field computed using the densities of charged small molecules and mobile ions in the solvent. A solution for mesh generation for biomolecular systems is supplied, which is an essential component for the finite element and boundary element computations. The uncoupled Smoluchowski equation and Poisson-Boltzmann equation are considered as special cases of the PNPE in the numerical algorithm, and therefore can be solved in this framework as well. Two types of computations are reported in the results: stationary PNPE and time-dependent SE or Nernst-Planck equations solutions. A biological application of the first type is the ionic density distribution around a fragment of DNA determined by the equilibrium PNPE. The stationary PNPE with nonzero flux is also studied for a simple model system, and leads to an observation that the interference on electrostatic field of the substrate charges strongly affects the reaction rate coefficient. The second is a time-dependent diffusion process: the consumption of the neurotransmitter acetylcholine by acetylcholinesterase, determined by the SE and a single uncoupled solution of the Poisson-Boltzmann equation. The electrostatic effects, counterion compensation, spatiotemporal distribution, and diffusion-controlled reaction kinetics are analyzed and different methods are compared.

Understanding the mechanism of ion permeation across lipid bilayers is key to controlling osmotic pressure and developing new ways of delivering charged, drug-like molecules inside cells. Recent reports suggest ion-pairing as the mechanism to lower the free energy barrier for the ion permeation in disagreement with predictions from the simple electrostatic models. In this paper we quantify the effect of ion-pairing or charge quenching on the permeation of Na+ and Cl− ions across DMPC lipid bilayer by computing the corresponding potentials of mean force (PMFs) using fully atomistic molecular dynamics simulations. We find that the free energy barrier to permeation reduces in the order Na+−Cl− ion-pair (27.6 kcal/mol) > Cl− (23.6 kcal/mol) > Na+ (21.9 kcal/mol). Furthermore, with the help of these PMFs we derive the change in the binding free energy between the Na+ and Cl− with respect to that in water as a function of the bilayer permeation depth. Despite the fact that the bilayer boosts the Na+−Cl− ion binding free energy by as high as 17.9 kcal/mol near its center, ion-pairing between such hydrophilic ions as Na+ and Cl− does not assist their permeation. However, based on a simple thermodynamic cycle, we suggest that ion-pairing between ions of opposite charge and solvent philicity could enhance ion permeation. Comparison of the computed permeation barriers for Na+ and Cl− ions with available experimental data supports this notion. This work establishes general computational methodology to address ion-pairing in fluid anisotropic media and details the ion permeation mechanism on atomic level.

Protein-protein interactions, particularly weak and transient ones, are often mediated by peptide recognition domains, such as Src Homology 2 and 3 (SH2 and SH3) domains, which bind to specific sequence and structural motifs. It is important but challenging to determine the binding specificity of these domains accurately and to predict their physiological interacting partners. In this study, the interactions between 35 peptide ligands (15 binders and 20 non-binders) and the Abl SH3 domain were analyzed using molecular dynamics simulation and the Molecular Mechanics/Poisson-Boltzmann Solvent Area method. The calculated binding free energies correlated well with the rank order of the binding peptides and clearly distinguished binders from non-binders. Free energy component analysis revealed that the van der Waals interactions dictate the binding strength of peptides, whereas the binding specificity is determined by the electrostatic interaction and the polar contribution of desolvation. The binding motif of the Abl SH3 domain was then determined by a virtual mutagenesis method, which mutates the residue at each position of the template peptide relative to all other 19 amino acids and calculates the binding free energy difference between the template and the mutated peptides using the Molecular Mechanics/Poisson-Boltzmann Solvent Area method. A single position mutation free energy profile was thus established and used as a scoring matrix to search peptides recognized by the Abl SH3 domain in the human genome. Our approach successfully picked ten out of 13 experimentally determined binding partners of the Abl SH3 domain among the top 600 candidates from the 218,540 decapeptides with the PXXP motif in the SWISS-PROT database. We expect that this physical-principle based method can be applied to other protein domains as well.