Efficient coarse-grained brownian dynamics simulations for dna and lipid bilayer membrane with hydrodynamic interactions

The coarse-grained molecular dynamics (CGMD) or Brownian dynamics (BD) simulation is a particle-based approach that has been applied to a wide range of biological problems that involve interactions with surrounding fluid molecules or the so-called hydrodynamic interactions (HIs). From simple biological systems such as a single DNA macromolecule to large and complicated systems, for instances, vesicles and red blood cells (RBCs), the numerical results have shown outstanding agreements with experiments and continuum modeling by adopting Stokesian dynamics and explicit solvent model. Finally, when combined with fast algorithms such as the fast multipole method (FMM) which has nearly optimal complexity in the total number of CG particles, the resulting method is parallelizable, scalable to large systems, and stable for large time step size, thus making the long-time large-scale BD simulation within practical reach. This will be useful for the study of a large collection of molecules or cells immersed in the fluids. This dissertation can be divided into three main subjects: (1) An efficient algorithm is proposed to simulate the motion of a single DNA molecule in linear flows. The algorithm utilizes the integrating factor method to cope with the effect of the linear flow of the surrounding fluid and applies the Metropolis method (MM) in [N. Bou-Rabee, A. Donev, and E. Vanden-Eijnden, Multiscale Model. Simul. 12, 781 (2014)] to achieve more efficient BD simulation. More importantly, this proposed method permits much larger time step size than methods in previous literature while still maintaining the stability of the BD simulation, which is advantageous for long-time BD simulation. The numerical results on λ-DNA agree very well with both experimental data and previous simulation results. (2) Lipid bilayer membranes have been extensively studied by CGMD simulations. Numerical efficiencies have been reported in the cases of aggressive coarse-graining, where several lipids are coarse-grained into a particle of size 4 ~ 6 nm so that there is only one particle in the thickness direction. In [H. Yuan et al., Phys. Rev. E, 82, 011905 (2010)], Yuan et al. proposed a pair-potential between these one-particle-thick coarse-grained lipid particles to capture the mechanical properties of a lipid bilayer membrane, such as gel-fluid-gas phase transitions of lipids, diffusion, and bending rigidity. This dissertation provides a detailed implementation of this interaction potential in LAMMPS to simulate large-scale lipid systems such as a giant unilamellar vesicle (GUV) and RBCs. Moreover, this work also considers the effect of cytoskeleton on the lipid membrane dynamics as a model for RBC dynamics, and incorporates coarse-grained water molecules to account for hydrodynamic interactions. (3) An action field method for lipid bilayer membrane model is introduced where several lipid molecules are represented by a Janus particle with corresponding orientation pointing from lipid head to lipid tail. With this level of coarse-grained modeling, as the preliminary setup, the lipid tails occupy a half sphere and the lipid heads take the other half. An action field is induced from lipid-lipid interactions and exists everywhere in the computational domain. Therefore, a hydrophobic attraction energy can be described from utilizing the variational approach and its minimizer with respect to the action field is the so-called screened Laplace equation. For the numerical method, the well-known integral equation method (IEM) has great capability to solve exterior screened Laplace equation with Dirichlet boundary conditions. Finally, one then can obtain the lipid dynamics to validate the self-assembly property and other physical properties of lipid bilayer membrane. This approach combines continuum modeling with CGMD and gives a different perspective to the membrane energy model from the traditional Helfrich membrane free energy

A numerical method for fluid-structure interactions of slender rods in turbulent flow

This thesis presents a numerical method for the simulation of fluid-structure interaction (FSI) problems on high-performance computers. The proposed method is specifically tailored to interactions between Newtonian fluids and a large number of slender viscoelastic structures, the latter being modeled as Cosserat rods. From a numerical point of view, such kind of FSI requires special techniques to reach numerical stability. When using a partitioned fluid-structure coupling approach this is usually achieved by an iterative procedure, which drastically increases the computational effort. In the present work, an alternative coupling approach is developed based on an immersed boundary method (IBM). It is unconditionally stable and exempt from any global iteration between the fluid part and the structure part. The proposed FSI solver is employed to simulate the flow over a dense layer of vegetation elements, usually designated as canopy flow. The abstracted canopy model used in the simulation consists of 800 strip-shaped blades, which is the largest canopy-resolving simulation of this type done so far. To gain a deeper understanding of the physics of aquatic canopy flows the simulation data obtained are analyzed, e.g., concerning the existence and shape of coherent structures

Computational modelling of cellular blood flow in complex vascular networks.

Microcirculatory disorders are associated with some of the most prevailing medical conditions in modern society, e.g., cancer and cardiovascular disease (CVD). Early detection and effective treatment of these diseases require an in-depth knowledge of the changes in the haemodynamic environment preceding fatal deteriorating conditions. However, such a knowledge is difficult to obtain merely relying on experiments, on account of the overwhelming complexity of blood flow at the microscale that is sometimes beyond the capability of contemporary experimental techniques. Alternatively, computational modelling provides a potent tool to uncover the missing details of haemodynamics at the microcirculation level. Thanks to the advent of information era which fosters growingly powerful computing facilities and architectures, the progress that has been made on blood flow modelling over recent years is unprecedented. Notwithstanding, exhaustive modelling of blood flow at the microcirculation level incorporating all blood constituents remains daunting. Existing studies employing a range of models are only possible via invoking simplifications justified under different assumptions. However, one important assumption for many modelling studies, namely that blood in the microcirculation can be approximated by a homogeneous non-Newtonian fluid, has been increasingly challenged. The reason is that the microscopic behaviour of red blood cells (RBCs) as the primary blood constituent is found to non-trivially modify key rheological properties of blood flow at the microscale, such as its effective viscosity, cell-free layer (CFL) and wall shear stress (WSS). To ultimately facilitate the translation of scientific investigations to real medical applications, the cellular character of microcirculatory blood flow has to be properly considered by computational models. Bearing the above challenges in mind, the present PhD embarks on a venture to research the complex behaviour of cellular blood flow under microcirculatory conditions, capitalising on a recently developed computational tool equipped with high-level parallelisation. This computational thesis sets out to answer several important questions, ranging from the rich dynamics of individual RBCs to the collective phenomena of RBC suspensions in either microvascular networks or microfluidic mimicries. The current three-dimensional (3D) computational model is based on the lattice Boltzmann method (LBM) coupled with the immersed boundary method (IBM) for high-level resolution of discrete RBCs, which are modelled as Lagrangian membranes using the finite-element method (FEM). In the thesis, an concise introduction of the computational model is given in Chapter 4. Before applied to research projects, the model has been systematically validated against existing numerical or experimental observations. Three benchmark tests of close relevance to the scope of microscale blood flow are selected for demonstration and discussion in Chapter 5. The main body of this thesis (Chapters 6–8) reports several novel aspects of blood flow at the microscale including, but not limited to, the non-inertial focusing of RBCs under low Reynolds number as revealed in Chapter 6, the excessive haemodilution induced by CFL asymmetry as revealed in Chapter 7, and the strong association between RBC perfusion and vascular patterning as revealed in Chapter 8. Some confusion about or misinterpretation of well-known effects in the community has also been clarified, such as the spatial scaling of hydrodynamic lift in non-circular channels, the development length of CFL in typical microfluidic flows, and the existence of high- /low-flow attraction near bifurcating geometries. Quantitative or qualitative agreement has been achieved through elaborated comparison with supplementary experiments from my collaborators or with established empirical models in the literature. Starting from blood flow in a single microchannel, Chapter 6 highlights an exceedingly large CFL development length even under low inertia, which is greater than 28 times channel hydraulic diameter (Dh) in simulation and 46Dh in experiment (experimental data from my collaborator in Glasgow, UK). This finding suggests that microfluidic designs need to be longer if their purpose is to investigate localised microscopic behaviour of a dilute suspension without interference from entrance effects or upstream disturbances. On a network level where the RBCs flow through bifurcating microchannels arranged biomimetically following Murray’s law, Chapter 7 identifies ideal partitioning of RBCs at symmetric bifurcations (agreeing with predictions of a classic empirical model derived from in vivo data), but biased partitioning when significant CFL asymmetry arises in inter-bifurcation branches. Furthermore, the breakdown of CFL symmetry leads to severe haemo-dilution/concentration in the bifurcating network. In Chapter 8, the computational framework is applied to model blood flow in realistic microvasculatures of developmental mouse retina and demonstrates an unreported highly heterogeneous distribution of RBCs in the post-sprouting vascular network. Remarkably, a strong association between vessel regression and RBC depletion is uncovered, driven by the effect of plasma skimming. The association is further confirmed by in vivo observation of simultaneous vascular remodelling alongside blood perfusion using a developmental zebrafish model (experimental data from my collaborator in Berlin, Germany). In summary, this thesis provides insights for the design of improved microfluidic devices and the conception of haemodynamic mechanisms governing the onset and progression of microcirculatory disorders. Additionally, the computational model successfully applied to various biological or biomimetic scenarios in this thesis justifies itself as a feasible and reliable tool for practical simulation of microcirculatory blood flows and may seek wider applications of its own accord

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described