Geometric and potential driving formation and evolution of biomolecular surfaces

This paper presents new geometrical flow equations for the theoretical modeling of biomolecular surfaces in the context of multiscale implicit solvent models. To account for the local variations near the biomolecular surfaces due to interactions between solvent molecules, and between solvent and solute molecules, we propose potential driven geometric flows, which balance the intrinsic geometric forces that would occur for a surface separating two homogeneous materials with the potential forces induced by the atomic interactions. Stochastic geometric flows are introduced to account for the random fluctuation and dissipation in density and pressure near the solvent–solute interface. Physical properties, such as free energy minimization (area decreasing) and incompressibility (volume preserving), are realized by some of our geometric flow equations. The proposed approach for geometric and potential forces driving the formation and evolution of biological surfaces is illustrated by extensive numerical experiments and compared with established minimal molecular surfaces and molecular surfaces. Local modification of biomolecular surfaces is demonstrated with potential driven geometric flows. High order geometric flows are also considered and tested in the present work for surface generation. Biomolecular surfaces generated by these approaches are typically free of geometric singularities. As the speed of surface generation is crucial to implicit solvent model based molecular dynamics, four numerical algorithms, a semi-implicit scheme, a Crank–Nicolson scheme, and two alternating direction implicit (ADI) schemes, are constructed and tested. Being either stable or conditionally stable but admitting a large critical time step size, these schemes overcome the stability constraint of the earlier forward Euler scheme. Aided with the Thomas algorithm, one of the ADI schemes is found to be very efficient as it balances the speed and accuracy.      

A 3D Motile Rod-Shaped Monotrichous Bacterial Model

We introduce a 3D model for a motile rod-shaped bacterial cell with a single polar flagellum which is based on the configuration of a monotrichous type of bacteria such as Pseudomonas aeruginosa. The structure of the model bacterial cell consists of a cylindrical body together with the flagellar forces produced by the rotation of a helical flagellum. The rod-shaped cell body is composed of a set of immersed boundary points and elastic links. The helical flagellum is assumed to be rigid and modeled as a set of discrete points along the helical flagellum and flagellar hook. A set of flagellar forces are applied along this helical curve as the flagellum rotates. An additional set of torque balance forces are applied on the cell body to induce counter-rotation of the body and provide torque balance. The three-dimensional Navier–Stokes equations for incompressible fluid are used to describe the fluid dynamics of the coupled fluid–microorganism system using Peskin’s immersed boundary method. A study of numerical convergence is presented along with simulations of a single swimming cell, the hydrodynamic interaction of two cells, and the interaction of a small cluster of cells.     

Improved 3D continuum calculations of ion flux through membrane channels

A continuum model, based on the Poisson–Nernst–Planck (PNP) theory, is applied to simulate steady-state ion flux through protein channels. The PNP equations are modified to explicitly account (1) for the desolvation of mobile ions in the membrane pore and (2) for effects related to ion sizes. The proposed algorithm for a three-dimensional self-consistent solution of PNP equations, in which final results are refined by a focusing technique, is shown to be suitable for arbitrary channel geometry and arbitrary protein charge distribution. The role of the pore shape and protein charge distribution in formation of basic electrodiffusion properties, such as channel conductivity and selectivity, as well as concentration distributions of mobile ions in the pore region, are illustrated by simulations on model channels. The influence of the ionic strength in the bulk solution and of the externally applied electric field on channel properties are also discussed.     

A Lagrangian particle method for reaction–diffusion systems on deforming surfaces

Reaction–diffusion processes on complex deforming surfaces are fundamental to a number of biological processes ranging from embryonic development to cancer tumor growth and angiogenesis. The simulation of these processes using continuum reaction–diffusion models requires computational methods capable of accurately tracking the geometric deformations and discretizing on them the governing equations. We employ a Lagrangian level-set formulation to capture the deformation of the geometry and use an embedding formulation and an adaptive particle method to discretize both the level-set equations and the corresponding reaction–diffusion. We validate the proposed method and discuss its advantages and drawbacks through simulations of reaction–diffusion equations on complex and deforming geometries.     

A multiscale model for red blood cell mechanics

The objective of this article is the derivation of a continuum model for mechanics of red blood cells via multiscale analysis. On the microscopic level, we consider realistic discrete models in terms of energy functionals defined on networks/lattices. Using concepts of Γ-convergence, convergence results as well as explicit homogenisation formulae are derived. Based on a characterisation via energy functionals, appropriate macroscopic stress–strain relationships (constitutive equations) can be determined. Further, mechanical moduli of the derived macroscopic continuum model are directly related to microscopic moduli. As a test case we consider optical tweezers experiments, one of the most common experiments to study mechanical properties of cells. Our simulations of the derived continuum model are based on finite element methods and account explicitly for membrane mechanics and its coupling with bulk mechanics. Since the discretisation of the continuum model can be chosen freely, rather than it is given by the topology of the microscopic cytoskeletal network, the approach allows a significant reduction of computational efforts. Our approach is highly flexible and can be generalised to many other cell models, also including biochemical control.     

Interstitial fluid flow in the osteon with spatial gradients of mechanical properties: a finite element study

Bone remodelling is the process that maintains bone structure and strength through adaptation of bone tissue mechanical properties to applied loads. Bone can be modelled as a porous deformable material whose pores are filled with cells, organic material and interstitial fluid. Fluid flow is believed to play a role in the mechanotransduction of signals for bone remodelling. In this work, an osteon, the elementary unit of cortical bone, is idealized as a hollow cylinder made of a deformable porous matrix saturated with an interstitial fluid. We use Biot’s poroelasticity theory to model the mechanical behaviour of bone tissue taking into account transverse isotropic mechanical properties. A finite element poroelastic model is developed in the COMSOL Multiphysics software. Elasticity equations and Darcy’s law are implemented in this software; they are coupled through the introduction of an interaction term to obtain poroelasticity equations. Using numerical simulations, the investigation of the effect of spatial gradients of permeability or Poisson’s ratio is performed. Results are discussed for their implication on fluid flow in osteons: (i) a permeability gradient affects more the fluid pressure than the velocity profile; (ii) focusing on the fluid flow, the key element of loading is the strain rate; (iii) a Poisson’s ratio gradient affects both fluid pressure and fluid velocity. The influence of textural and mechanical properties of bone on mechanotransduction signals for bone remodelling is also discussed.     

Electrostatic interactions across a charged lipid bilayer

We present theoretical work in which the degree of electrostatic coupling across a charged lipid bilayer in aqueous solution is analyzed on the basis of nonlinear Poisson–Boltzmann theory. In particular, we consider the electrostatic interaction of a single, large macroion with the two apposed leaflets of an oppositely charged lipid bilayer where the macroion is allowed to optimize its distance to the membrane. Three regimes are identified: a weak and a high macroion charge regime, separated by a regime of close macroion–membrane contact for intermediate charge densities. The corresponding free energies are used to estimate the degree of electrostatic coupling in a lamellar cationic lipid–DNA complex. That is, we calculate to what extent the one-dimensional DNA arrays in a sandwich-like lipoplex interact across the cationic membranes. We find that, in spite of the low dielectric constant inside a lipid membranes, there can be a significant electrostatic contribution to the experimentally observed cross-bilayer orientational ordering of the DNA arrays. Our approximate analytical model is complemented and supported by numerical calculations of the electrostatic potentials and free energies of the lamellar lipoplex geometry. To this end, we solve the nonlinear Poisson–Boltzmann equation within a unit cell of the lamellar lipoplex using a new lattice Boltzmann method. Dedicated to Prof. K. Arnold on the occasion of his 65th birthday.     

Multiscale Modelling of Fluid and Drug Transport in Vascular Tumours

A model for fluid and drug transport through the leaky neovasculature and porous interstitium of a solid tumour is developed. The transport problems are posed on a micro-scale characterized by the inter-capillary distance, and the method of multiple scales is used to derive the continuum equations describing fluid and drug transport on the length scale of the tumour (under the assumption of a spatially periodic microstructure). The fluid equations comprise a double porous medium, with coupled Darcy flow through the interstitium and vasculature, whereas the drug equations comprise advection–reaction equations; in each case the dependence of the transport coefficients on the vascular geometry is determined by solving micro-scale cell problems.     

Parametric and non-parametric modeling of short-term synaptic plasticity. Part II: Experimental study

This paper presents a synergistic parametric and non-parametric modeling study of short-term plasticity (STP) in the Schaffer collateral to hippocampal CA1 pyramidal neuron (SC) synapse. Parametric models in the form of sets of differential and algebraic equations have been proposed on the basis of the current understanding of biological mechanisms active within the system. Non-parametric Poisson–Volterra models are obtained herein from broadband experimental input–output data. The non-parametric model is shown to provide better prediction of the experimental output than a parametric model with a single set of facilitation/depression (FD) process. The parametric model is then validated in terms of its input–output transformational properties using the non-parametric model since the latter constitutes a canonical and more complete representation of the synaptic nonlinear dynamics. Furthermore, discrepancies between the experimentally-derived non-parametric model and the equivalent non-parametric model of the parametric model suggest the presence of multiple FD processes in the SC synapses. Inclusion of an additional set of FD process in the parametric model makes it replicate better the characteristics of the experimentally-derived non-parametric model. This improved parametric model in turn provides the requisite biological interpretability that the non-parametric model lacks.     

Shape optimization in unsteady blood flow: a numerical study of non-Newtonian effects

This paper presents a numerical study of non-Newtonian effects on the solution of shape optimization problems involving unsteady pulsatile blood flow. We consider an idealized two dimensional arterial graft geometry. Our computations are based on the Navier-Stokes equations generalized to non-Newtonian fluid, with the modified Cross model employed to account for the shear-thinning behavior of blood. Using a gradient-based optimization algorithm, we compare the optimal shapes obtained using both the Newtonian and generalized Newtonian constitutive equations. Depending on the shear rate prevalent in the domain, substantial differences in the flow as well as in the computed optimal shape are observed when the Newtonian constitutive equation is replaced by the modified Cross model. By varying a geometric parameter in our test case, we investigate the influence of the shear rate on the solution.     

Estimating the dielectric constant of the channel protein and pore

When modelling biological ion channels using Brownian dynamics (BD) or Poisson–Nernst–Planck theory, the force encountered by permeant ions is calculated by solving Poisson’s equation. Two free parameters needed to solve this equation are the dielectric constant of water in the pore and the dielectric constant of the protein forming the channel. Although these values can in theory be deduced by various methods, they do not give a reliable answer when applied to channel-like geometries that contain charged particles. To determine the appropriate values of the dielectric constants, here we solve the inverse problem. Given the structure of the MthK channel, we attempt to determine the values of the protein and pore dielectric constants that minimize the discrepancies between the experimentally-determined current–voltage curve and the curve obtained from BD simulations. Two different methods have been applied to determine these values. First, we use all possible pairs of the pore dielectric constant of water, ranging from 20 to 80 in steps of 10, and the protein dielectric constant of 2–10 in steps of 2, and compare the simulated results with the experimental values. We find that the best agreement is obtained with experiment when a protein dielectric constant of 2 and a pore water dielectric constant of 60 is used. Second, we employ a learning-based stochastic optimization algorithm to pick out the optimum combination of the two dielectric constants. From the algorithm we obtain an optimum value of 2 for the protein dielectric constant and 64 for the pore dielectric constant.     

Shape optimization in unsteady blood flow: A numerical study of non-Newtonian effects

This paper presents a numerical study of non-Newtonian effects on the solution of shape optimization problems involving unsteady pulsatile blood flow. We consider an idealized two dimensional arterial graft geometry. Our computations are based on the Navier–Stokes equations generalized to non-Newtonian fluid, with the modified Cross model employed to account for the shear-thinning behavior of blood. Using a gradient-based optimization algorithm, we compare the optimal shapes obtained using both the Newtonian and generalized Newtonian constitutive equations. Depending on the shear rate prevalent in the domain, substantial differences in the flow as well as in the computed optimal shape are observed when the Newtonian constitutive equation is replaced by the modified Cross model. By varying a geometric parameter in our test case, we investigate the influence of the shear rate on the solution.     

In silico approach to cisplatin toxicity. Quantum chemical studies on platinum(II)–cysteine systems

The behaviour of cisplatin in serum, and the drastic differences between the properties of this drug and its trans-isomer were the main motivations for this work. In a search for model “thiol–platin(II)” interactions, the first steps of the following reaction systems were evaluated: (1) cisplatin–thiomethanol; (2) transplatin–thiomethanol; (3) cisplatin–cysteine; and (4) transplatin–cysteine. In each case, calculations for the associative mode of reactions were performed. The electronic structure of these molecular systems was studied at the non-empirical all-electron level using density functional theory (DFT) within the Huzinaga and WTBS basis sets including polarisation Gaussian functions and full geometry optimisation. B3LYP or EPBO density functionals were applied throughout. The calculated molecular electrostatic potentials are presented graphically. Assuming that electrostatic effects are dominant, cisplatin should interact more strongly with the sulfur atom of CH₃S⁻ and deprotonated CYS-S⁻ than transplatin. This fact has been documented in the supermolecule model of the relevant interaction energies in both gas phase as well as within the solvent polarisable continuum model. The opposite relationship was observed when we compared values of energy differences between products and substrates for both isomers. The data obtained here could be applied to search for correlation between the biological activity of platinum complexes and their properties as estimated by various physico-chemical and in silico methodologies.     

Reconstructing parameters of the FitzHugh–Nagumo system from boundary potential measurements

We consider distributed parameter identification problems for the FitzHugh–Nagumo model of electrocardiology. The model describes the evolution of electrical potentials in heart tissues. The mathematical problem is to reconstruct physical parameters of the system through partial knowledge of its solutions on the boundary of the domain. We present a parallel algorithm of Newton–Krylov type that combines Newton’s method for numerical optimization with Krylov subspace solvers for the resulting Karush–Kuhn–Tucker system. We show by numerical simulations that parameter reconstruction can be performed from measurements taken on the boundary of the domain only. We discuss the effects of various model parameters on the quality of reconstructions. Action Editor: David Terman     

Effects of Exogenous Electromagnetic Fields on a Simplified Ion Channel Model

In this paper, we calculate the effect of an exogenous perturbation (an electromagnetic field [EMF] oscillating in the range of microwave frequencies in the range of 1 GHz) on the flux of two ion species through a cylindrical ion channel, implementing a continuous model, the Poisson–Smoluchowski system of equations, to study the dynamics of charged particle density inside the channel. The method was validated through comparison with Brownian dynamics simulations, supposed to be more accurate but computationally more demanding, obtaining a very good agreement. No EMF effects were observed for low field intensities below the level for thermal effects, as the highly viscous regime and the simplicity of the channel do not exhibit resonance phenomena. For high intensities of the external field (>10⁵ V/m), we observed slightly different behavior of ion concentration oscillations and ion currents as a function of EMF orientation with respect to the channel axis.     

Pressure-induced phase transition in wurtzite ZnTe: an ab initio study

A constant pressure ab initio MD technique and density functional theory with a generalized gradient approximation (GGA) was used to study the pressure-induced phase transition in wurtzite ZnTe. A first-order phase transition from the wurtzite structure to a Cmcm structure was successfully observed in a constant-pressure molecular dynamics simulation. This phase transformation was also analyzed using enthalpy calculations. We also investigated the stability of wurtzite (WZ) and zinc-blende (ZB) phases from energy–volume calculations, and found that both structures show quite similar equations of state and transform into a Cmcm structure at 16 GPa using enthalpy calculations, in agreement with experimental observations. The transition phase, lattice parameters and bulk properties we obtained are comparable with experimental and theoretical data.