Mark Thachuk

Coarse-grained (CG) models reduce the number of degrees of freedom in a system, allowing the dynamics of large systems to be studied for longer times. Many biological simulations today are performed using CG potentials. However, the use of Newtonian equations of motion (EOM) for mesoscopic variables only yields correct equilibrium properties but with the wrong dynamics. Conventional CG mapping schemes such as the center-of-mass mapping are also not suitable for coarse-graining nonbonded fluid systems.The conservative terms in the CG EOM derived using Mori-Zwanzig theory are studied. The fluid systems are divided into cubic subcells with equal volumes. Atomistic particles associated with a subcell are mapped to a set of position-dependent CG variables using either a Heaviside function or a fuzzy function. A diffusion blob model is developed to qualitatively understand the correlation between two subcells. The distribution of CG mass is found to change from symmetric and discrete to skewed and continuous. The form of the CG potential can be approximated as a multivariate Gaussian.Distribution function theory is used to derive the parameters of the CG potential analytically. The behaviour of the potential parameters as a function of different geometric relationships, the size of the subcell or the fuzziness of the subcell boundary, is discussed.A density-based expansion method is developed to quantitatively understand the behaviour of the one-dimensional distribution of CG variables. The origin of the skewed mass distribution comes from the asymmetry in the variance of CG mass distribution conditioned on a fixed number of atoms. The projected fluxes are studied with distribution function theory and Gaussian process regression.This work provides a basis for correctly simulating complex fluid systems at a mesoscopic scale without any ad-hoc assumptions. The Gaussian-like CG potential is general for single-component, atomic fluids. Parameters of a CG potential are, for the first time, computed from analytical theories. Understanding the source of the skewed mass gives a complete solution to finding the correct fluctuation for densities. This solves a long-standing problem in fluctuating hydrodynamics. The density-based expansion formula gives a complete solution to the back-mapping problem in performing multiscale simulations.

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With the invention of gentle ionization methods such as electrospray ionization, mass spectrometry has been used as a tool for studying large protein complexes. Simulations of these gas-phase protein complexes will allow for better understanding of the dissociation mechanism and lead to methods for controlling the dissociation. Controlling the dissociation will help obtain structural information from the mass spectrometry experiments.In this work, the suitability of the MARTINI coarse-grained force field for gas-phase simulations is studied and a charge hopping algorithm is developed.Using a coarse-grained force field makes the simulations faster so that longer simulation times can be accessed.This is important because protein motions can take place on time scales of nanoseconds to milliseconds and these long times are not practical with all-atom simulations.Most molecular dynamics simulations use fixed charges, but including charge motion allows for better simulation of mobile protons.Two protein complexes are studied here, one dimer and one tetramer. Hopping rates, energies, radii of gyration, and distances within the complexes are calculated. Simulations with the cytochrome c$^{\prime}$ dimer (no charge hopping) are compared to published all-atom results. The MARTINI force field is found to be good for qualitative results, but slightly more attractive than the OPLS all-atom (for the isolated protein complex).The transthyretin tetramer is used to study the hopping algorithm. Modifications of the protein (blocking N-termini from accepting charges and adding basic sites with a tether) are also explored. The dissociation behavior of the protein complexes is controlled by the Coulomb repulsion model. Protein modifications near the N-termini show potential for controlling the dissociation.

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A numerical method for solving a one-dimensional linear Boltzmann equation is developed using cubic B-splines. Collision kernels are derived for smooth and rough hard spheres. A complete velocity kernel for spherical particles is shown that is reduced to the smooth, rigid sphere case. Similarly, a collision kernel for the rough hard sphere is derived that depends upon velocity and angular velocity. The exact expression for the rough sphere collision kernel is reduced to an approximate expression that averages over the rotational degrees of freedom in the system. The rough sphere collision kernel tends to the smooth sphere collision kernel in the limit when translational-rotational energy exchange is attenuated. Comparisons between the smooth sphere and approximate rough sphere kernel are made. Four different representations for the distribution function are presented. The eigenvalues and eigenfunctions of the collision matrix are obtained for various mass ratios and compared with known values. The distribution functions, first and second moments are also evaluated for different mass and temperature ratios. This is done to validate the numerical method and it is shown that this method is accurate and well-behaved. In addition to smooth and rough hard spheres, the collision kernels are used to model the Maxwell molecule. Here, a variety of mass ratios and initial energies are used to test the capability of the numerical method. Massive tracers are set to high initial energies, representing kinetic energy loss experiments with biomolecules in experimental mass spectrometry. The validity of the Fokker-Planck expression for the Rayleigh gas is also tested. Drag coefficients are calculated and compared to analytic expressions. It is shown that these values are well predicted for massive tracers but show a more complex behaviour for small mass ratios especially at higher energies. The numerical method produced well converged values, even when the tracers were initialized far from equilibrium. In general this numerical method produces sparse matrices and can be easily generalized to higher dimensions that can be cast into efficient parallel algorithms. Future work has been planned that involves the use of this numerical method for a multi-dimension linear Boltzmann equation.

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This thesis examines the dynamics and physical properties of the rough hard sphere fluid (RHSF). The RHSF model consists of spherical particles with well-defined radii that exchange linear and angular momenta upon collision. The simplicity of this model allows for a precise theoretical description that provides a basis for studying fluid properties on the most fundamental level.Extensive molecular dynamics calculations were made of transport properties as functions of density, tracer particle size, and degree of rotational-translational coupling. Results were compared with the smooth hard sphere case and it was found that transport coefficients change significantly due to rotational-translational coupling, becoming stronger with an increase in coupling. Tracer diffusion coefficients were examined for a range of tracer sizes and at various densities. As tracer particles become larger, their diffusion coefficient moves from an Enskog (molecular) to a Stokes-Einstein (hydrodynamic) functional form; the latter depends upon the boundary condition at the surface of the tracer. These boundary conditions for the RHSF are directly proportional to the degree of rotational-translational energy exchange, and can be tuned from "slip" to "stick" values. The validity of several kinetic theory equations have been examined as functions of density and translational-rotational coupling. At very low densities, Boltzmann theory was accurate even at low order except for describing the dependence upon rotational-translational coupling, where low order expansions are less accurate. Enskog theory performed well at low and moderate densities but deviated at larger densities, as expected. For thermal conductivity as a function of translational-rotational coupling even the qualitative behavior was incorrect. The Enskog predictions for diffusion were also found to be quite poor at low order. Finally, motivated by the results of the thesis, experimental diffusion coefficient data were analyzed, especially for nanoparticles. It was shown that defining the correct radius is crucial for describing such systems. In addition, a new formula for predicting tracer diffusion was tested.

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Understanding the dissociation mechanism of multimeric protein complex ions is importantfor interpreting gas-phase experiments. The aim of this thesis work was to study thedissociation of charged protein complexes. To pursue this, a number of model and molecular dynamics calculations were conducted using the cytochrome c′ dimer.The energetics of differing charge states, partitionings, and configurations were examined in both the low and high charge regimes. It is shown that one must always consider distributions of charge configurations, once protein relaxation effects are taken into account, and that no single configuration dominates. These results also indicate that in the high charge limit, the dissociation is governed by electrostatic repulsion from the net charges. This causes two main trends: i) charges will move so as to approximately maintain constant surface charge density, and ii) the lowest barrier to dissociation is the one that produces fragment ions with equal charges.Free energies are also calculated for the protonated dimer ion as a function of the center of mass distance between the monomers.In addition, the change of intermolecular properties such as intermolecular hydrogen bonds and the smallest separation of intermolecular residues were analyzed. It is found that monomer unfolding competes with complex dissociation, and that the relative importance of these two factors depends upon the charge partitioning in the complex. Symmetric charge partitionings preferentially suppress the dissociationbarrier relative to unfolding, and complexes tend to dissociate promptly with little structural change occurring in the monomers. Alternatively, asymmetriccharge partitionings preferentially lower the barrier for monomer unfolding relative to the dissociation barrier. In this case, the monomer with the higher charge unfolds before the complex dissociates. For large multimeric proteins, the unfolding and subsequent charging of a single monomer is a favorable process, cooperatively lowering both the unfolding and dissociation barriers at the same time. For the homodimer considered here, this pathway has a large free energy barrier. Overall, the work presented herein demonstrates that molecular dynamics simulation can be useful for understanding the dissociation mechanism of protein complexes.

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