Analytic expressions for correlations in coarse-grained simple fluids.

Coarse-graining of fluids is challenging because fluid particles are unbound and diffuse long distances in time. One approach creates coarse-grain variables that group all particles within a region centered on specific points in space and accounts for the movement of particles among such regions. In our previous work, we showed that in many cases, potential interactions for such a scheme adopted a generalized quadratic form, whose parameters depend on means, variances, and correlation coefficients among the coarse-grain variables. In this work, we use statistical mechanics to derive analytic expressions for these parameters, using properties of the fluid, including pair distribution functions. These expressions are compared against simulation-derived values and shown to be in good agreement. This approach can be used to calculate a priori the potential for any homogeneous, simple fluid, without the need for fitting procedures or matching, thus increasing the ease of use of this coarse-grain scheme and creating a foundation for large-scale bottom-up simulations. Furthermore, these expressions provide a quantitative way of studying the boundary between discrete (atomic) and continuum models of fluids.

Conservative Potentials for a Lattice-Mapped, Coarse-Grain Scheme with Fuzzy Switching Functions.

We extend our previous work (Luo, S.; Thachuk, M. J. Phys. Chem. A 2021, 125, 64866497) on determining conservative potentials for lattice-like, coarse-grain (CG) mapping schemes to the case where the boundaries between different spatial regions are not sharply defined but are fuzzy. In other words, the system is divided into interpenetrating "subcells" such that atomistic particles continuously change their memberships as they move through space. This is done by using fuzzy switching functions to define overlapping regions between subcells with fractional particle occupations. In this case, a full mass matrix is required to describe the system, and its off-diagonal elements are nonzero and contribute to the CG potential. As the overlapping region increases in size, we observe the mass distribution transitions from a discrete spectrum, through an intermediate state, and finally to a continuous Gaussian-like function. We interpret this as a quantitative measure for signaling when a continuum-theory description of the system is appropriate. Nonzero correlations among all CG variables are calculated and are found to depend strongly on the degree of overlap. In particular, those for the diagonal mass elements decrease in magnitude, and there exists a specific value of the overlap for which the correlations are zero. Other correlations are strong only when the overlap is quite large, so there is a trade-off between the complexity of the interactions in the system and the degree of fuzziness between the subcells. However, if the number of particles in a subcell is large enough and the overlap is moderate, then the CG potential is found to be well-approximated by a generalized quadratic function. These results demonstrate the transition between atomistic and continuum resolutions in a system and have implications for designing CG schemes with mixed atomistic and continuum character.

Conservative Potentials for a Lattice-Mapped Coarse-Grained Scheme.

The conservative potential, arising from a coarse-grain (CG) mapping scheme for nonbonded atomistic particles, is studied. This is a bottom-up approach from first-principles that maps atomistic particles to fluid element-like subcells whose centers lie on a regular, cubic lattice. Unlike standard CG mapping schemes, the current one uses dynamic labeling which on-the-fly changes the CG labels of the particles. The subcells can also be different sizes and shapes, in principle. Equilibrium atomistic molecular dynamics trajectories for different Lennard-Jones fluids are calculated and converted to CG ones, from which CG probability distribution functions are calculated. Correlation studies show position and mass CG variables are uncoupled in a given subcell, as are different vector components of position. Furthermore, the strongest coupling occurs with neighboring cells in specific directions, and the resulting distribution is well described by a multivariate Gaussian. This implies the CG potential has a generalized quadratic form, whose derivative can be determined analytically. A microscopic rationalization is provided for the signs and relative magnitudes of different correlation coefficients, and in some cases, a connection is made with bulk properties of the fluid. We argue the generalized quadratic form should be robust to changes in the particulars of the CG scheme, as well as the nature of the atomistic intermolecular potential. Only a few potential parameters need to be calculated from the underlying atomistic system. This is significant because it indicates the transferability of this form to other, more complex systems. This transferability will be tested in future work, where mapping schemes with fuzzy boundaries will be considered.

Equations of motion for position-dependent coarse-grain mappings obtained with Mori-Zwanzig theory.

A position-dependent transformation is introduced for mapping a system of atomistic particles to a system of coarse-grained (CG) variables, which under some circumstances might be considered particles. This CG mapping allows atomistic particles to simultaneously contribute to more than a single CG particle and to change in time the CG particle they are associated with. That is, the CG mapping is dynamic. Mori-Zwanzig theory is then used to obtain the equations of motion for this CG mapping, resulting in conservative, dissipative, and random force terms in generalized, non-Markovian Langevin equations. In addition to the usual forces arising from the effective CG potential derived from atomistic interactions, new forces arise from the dynamic changes in the CG mapping itself. These new forces effectively account for changes arising from fluxes of atomistic particles into and out of CG ones as time progresses. Several examples are given showing the range of problems that can be addressed with this new CG mapping. These range from the usual case where atomistic particles are grouped into large molecular-like chunks, with mappings that remain fixed in time and for which an atomistic particle is part of only a single CG one, to the case where CG particles resemble fluid elements, containing many hundreds of independent atomistic particles. The new CG mapping also allows for hybrid descriptions, in which a part of the system remains atomistic or molecular-like and a part is highly coarse-grained to mesoscopic fluid element-like particles, for example. In the latter case, the equations of motion then provide the correct formalism for determining the forces, beyond the usual conservative ones. This provides a theoretical foundation upon which approximate equations of motion can be formulated to thus build numerical algorithms for expanded applications of accurate CG molecular dynamics.

A general, rotating, hard sphere model applied to the transport properties of a low density gas.

A general, spherical, rigid model is introduced for describing rotating and translating particles. The model contains a parameter, which we label γ, that smoothly interpolates between the smooth hard sphere (γ = 0) and rough hard sphere (γ = 1) limits. Analytic expressions for transport coefficients are determined for the general model in the low density limit and compared with those for the smooth and rough hard sphere cases. While the diffusion constant decreases monotonically on moving from the smooth to the rough sphere limits, both the viscosity and thermal conductivity first decrease and then increase, thereby producing a minimum between the two limits. This qualitative change in behaviour is new and suggests translational-rotational coupling acts to decrease the values of the transport coefficients (in contrast to the prediction from the rough sphere model). Although the model still has the (known) deficiencies of rigid models, it is more flexible than either the smooth or rough sphere model and should find use in better representing molecular behaviour. The general model provides a consistent representation of the transport coefficients because it has proper, microscopic collision dynamics obeying conservation laws for total momentum, total angular momentum, and total energy.

Relaxation of hot and massive tracers using numerical solutions of the Boltzmann equation.

A numerical method using B-splines is used to solve the linear Boltzmann equation describing the energy relaxation of massive tracer particles moving through a dilute bath gas. The smooth and rough hard sphere and Maxwell molecule models are used with a variety of mass ratios and initial energies to test the capability of the numerical method. Massive tracers are initialized with energies typically found in energy loss experiments in mass spectrometry using biomolecules. The method is also used to examine the applicability of known expressions for the kinetic energy decay from the Fokker-Planck equation for the Rayleigh gas, where we find that results are generally good provided that the initial energy is properly bounded. Otherwise, the energy decay is not constant and a more complex behaviour occurs. The validity of analytical expressions for drag coefficients for spherical particles under specular and diffuse scattering is also tested. We find such expressions are generally good for hard spheres but cannot account, as expected, for the softer repulsive walls of the Maxwell (and real) molecules. Overall, the numerical method performed well even when tracers more than 400 times as massive as the bath were initialized with energies very far from equilibrium. This is a range of applicability beyond many of the standard methods for solving the Boltzmann equation.

Controlling dissociation channels of gas-phase protein complexes using charge manipulation.

Coarse-grained simulations with charge hopping were performed for a positively charged tetrameric transthyretin (TTR) protein complex with a total charge of +20. Charges were allowed to move among basic amino acid sites as well as N-termini. Charge distributions and radii of gyration were calculated for complexes simulated at two temperatures, 300 and 600 K, under different scenarios. One scenario treated the complex in its normal state allowing charge to move to any basic site. Another scenario blocked protonation of all the N-termini except one. A final scenario used the complex in its normal state but added a basic-site containing tether (charge tag) near the N-terminus of one chain. The differences in monomer unfolding and charging were monitored in all three scenarios and compared. The simulation results show the importance of the N-terminus in leading the unfolding of the monomer units; a process that follows a zipper-like mechanism. Overall, experimentally modifying the complex by adding a tether or blocking the protonation of N-termini may give the potential for controlling the unraveling and subsequent dissociation of protein complexes.

Kernels of the linear Boltzmann equation for spherical particles and rough hard sphere particles.

Kernels for the collision integral of the linear Boltzmann equation are presented for several cases. First, a rigorous and complete derivation of the velocity kernel for spherical particles is given, along with reductions to the smooth, rigid sphere case. This combines and extends various derivations for this kernel which have appeared previously in the literature. In addition, the analogous kernel is derived for the rough hard sphere model, for which a dependence upon both velocity and angular velocity is required. This model can account for exchange between translational and rotational degrees of freedom. Finally, an approximation to the exact rough hard sphere kernel is presented which averages over the rotational degrees of freedom in the system. This results in a kernel depending only upon velocities which retains a memory of the exchange with rotational states. This kernel tends towards the smooth hard sphere kernel in the limit when translational-rotational energy exchange is attenuated. Comparisons are made between the smooth and approximate rough hard sphere kernels, including their dependence upon velocity and their eigenvalues.

A Charge Moving Algorithm for Molecular Dynamics Simulations of Gas-Phase Proteins.

A method for moving charges in a coarse-grained simulation of gas-phase proteins is presented which uses a Monte Carlo approach to move charges between charge sites. The method is used to study the role of charge movement in the dissociation mechanism of protein complexes in order to better understand experimentally observed mass spectra from CID studies. The charge hopping process is analyzed using energy distributions and a pair correlation plot. Hopping rates, charge distributions, and structural parameters (radius of gyration and RMSD) are also calculated. The importance of charge movement for the unfolding of protein complexes is demonstrated. The algorithm is implemented in the GROMACS molecular dynamics software package. In this study, transthyretin (TTR) tetramer is used with the MARTINI force field as a model system, and comparisons to experiments are made. The hopping and unfolding are found to be controlled by the Coulomb repulsion among the charges in the complex.

A numerical solution of the linear Boltzmann equation using cubic B-splines.

A numerical method using cubic B-splines is presented for solving the linear Boltzmann equation. The collision kernel for the system is chosen as the Wigner-Wilkins kernel. A total of three different representations for the distribution function are presented. Eigenvalues and eigenfunctions of the collision matrix are obtained for various mass ratios and compared with known values. Distribution functions, along with first and second moments, are evaluated for different mass and temperature ratios. Overall it is shown that the method is accurate and well behaved. In particular, moments can be predicted with very few points if the representation is chosen well. This method produces sparse matrices, can be easily generalized to higher dimensions, and can be cast into efficient parallel algorithms.

Transport properties of the rough hard sphere fluid.

Results are presented of a systematic study of the transport properties of the rough hard sphere fluid. The rough hard sphere fluid is a simple model consisting of spherical particles that exchange linear and angular momenta, and energy upon collision. This allows a study of the sole effect of particle rotation upon fluid properties. Molecular dynamics simulations have been used to conduct extensive benchmark calculations of self-diffusion, shear and bulk viscosity, and thermal conductivity coefficients. As well, the validity of several kinetic theory equations have been examined at various levels of approximation as a function of density and translational-rotational coupling. In particular, expressions from Enskog theory using different numbers of basis sets in the representation of the distribution function were tested. Generally Enskog theory performs well at low density but deviates at larger densities, as expected. The dependence of these expressions upon translational-rotational coupling was also examined. Interestingly, even at low densities, the agreement with simulation results was sometimes not even qualitatively correct. Compared with smooth hard sphere behaviour, the transport coefficients can change significantly due to translational-rotational coupling and this effect becomes stronger the greater the coupling. Overall, the rough hard sphere fluid provides an excellent model for understanding the effects of translational-rotational coupling upon transport coefficients.

Suitability of the MARTINI Force Field for Use with Gas-Phase Protein Complexes.

The MARTINI coarse-grained force field [Monticelli, L. et al. J. Chem. Theory Comput.2008, 4, 819-834] is examined for use in molecular dynamics simulations of the dissociation of gas-phase protein complexes. Coarse-grained force fields allow longer time scales and larger systems to be treated compared with all-atom force fields. In this work, results for the dissociation of the cytochrome c' dimer using MARTINI are compared with published studies using the OPLS-AA/L all-atom force field. Several structural parameters such as the minimum distance between monomers, radius of gyration, and root-mean-square deviation as well as potential energy contributions (Lennard-Jones and Coulomb) are calculated as a function of the center of mass distance. The MARTINI force field semiquantitatively reproduces the results of previous all-atom studies but appears to be somewhat too attractive.

The effect of rotational and translational energy exchange on tracer diffusion in rough hard sphere fluids.

A study is presented of tracer diffusion in a rough hard sphere fluid. Unlike smooth hard spheres, collisions between rough hard spheres can exchange rotational and translational energy and momentum. It is expected that as tracer particles become larger, their diffusion constants will tend toward the Stokes-Einstein hydrodynamic result. It has already been shown that in this limit, smooth hard spheres adopt "slip" boundary conditions. The current results show that rough hard spheres adopt boundary conditions proportional to the degree of translational-rotational energy exchange. Spheres for which this exchange is the largest adopt "stick" boundary conditions while those with more intermediate exchange adopt values between the "slip" and "stick" limits. This dependence is found to be almost linear. As well, changes in the diffusion constants as a function of this exchange are examined and it is found that the dependence is stronger than that suggested by the low-density, Boltzmann result. Compared with smooth hard spheres, real molecules undergo inelastic collisions and have attractive wells. Rough hard spheres model the effect of inelasticity and show that even without the presence of attractive forces, the boundary conditions for large particles can deviate from "slip" and approach "stick."

Toward an improved understanding of the dissociation mechanism of gas phase protein complexes.

Understanding the dissociation mechanism of multimeric protein complex ions is important for deciphering gas phase dissociation experiments. The dissociation of cytochrome c' dimer ions in the gas phase was investigated in the present study by constrained molecular dynamics simulations. The center of mass (COM) distance between two monomers was selected as the constrained coordinate. The number of intermolecular hydrogen bonds, smallest distance of intermolecular residuals, value of dipole moments, root-mean-square deviations, and potential energy components of the force field as a function of COM distance were examined for different charge partitionings of the +10 total charge state. These data were rationalized with free energy profiles to produce a qualitative description of the dissociation process. When charges are symmetrically distributed between the monomers in the dimer, dissociation occurs at a well-defined distance with only small structural changes in the monomers. There is an elastic type of stretching that initially resists the separation of the monomers but after dissociation the monomers recoil slightly from this and relax. For asymmetrically distributed charges, the dissociation event is not nearly as well-defined because the more highly charged monomer unfolds before dissociation occurs. It is found in almost all cases, a charged N-terminus tethers this unfolding monomer to its dimer partner by binding in a nonspecific manner. This helps encourage monomer unfolding in the dissociation pathway. It is also shown that while the intermolecular Coulomb repulsion between the monomers is not the largest contribution to the overall potential energy, it dominates the potential energy difference between different charge states.

Free energy barrier estimation for the dissociation of charged protein complexes in the gas phase.

Free energies are calculated for the protonated cytochrome c' dimer ion in the gas phase as a function of the center of mass distance between the monomers. A number of different charge partitionings are examined as well as the behavior of the neutral complex. 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 dissociation barrier relative to unfolding, and complexes tend to dissociate promptly with little structural changes occurring in the monomers. Alternatively, asymmetric charge 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 the homodimer considered here, this pathway has a large free energy barrier. The results can be rationalized using schematic two-dimensional free energy surfaces. Additionally, for large multimeric complexes, it is argued that 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.

Gas-phase proton-transfer pathways in protonated histidylglycine.

Pathways for proton transfer in the histidylglycine cation are examined in the gas-phase environment with the goal of understanding the mechanism by which protons may become mobile in proteins with basic amino acid residues. An extensive search of the potential energy surface is performed using density functional theory (DFT) methods. After corrections for zero-point energy are included, it is found that all the lowest energy barriers for proton transfer between the N-terminus and the imidazole ring have heights of only a few kcal/mol, while those between the imidazole ring and the backbone amide oxygen have heights of approximately 15 kcal/mol when the proton is moving from the ring to the backbone and only a few kcal/mol when moving from the backbone to the imidazole ring. In mass spectrometric techniques employing collision-induced dissociation to dissociate protein complex ions or to fragment peptides, these barriers can be overcome, and the protons mobilized.

Theoretical investigations of the dissociation of charged protein complexes in the gas phase.

A series of calculations, varying from simple electrostatic to more detailed semi-empirical based molecular dynamics ones, were carried out on charged gas phase ions of the cytochrome c(') dimer. The energetics of differing charge states, charge partitionings, and charge configurations were examined in both the low and high charge regimes. As well, preliminary free energy calculations of dissociation barriers are presented. 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. All these results also indicate that in the high charge limit, the dissociation of protein complex ions is governed by electrostatic repulsion from the net charges, the consequences of which are enumerated and discussed. There are two main trends deriving from this, namely that charges will move so as to approximately maintain constant surface charge density, and that the lowest barrier to dissociation is the one that produces fragment ions with equal charges. In particular, it is shown that the charge-to-mass ratio of a fragment ion is not the key physical parameter in predicting dissociation products. In fact, from the perspective of the division of total charge, many dissociation pathways reported to be "asymmetric" in the literature should be more properly labelled as "symmetric" or "near-symmetric". The Coulomb repulsion model assumes that the timescale for charge transfer is faster than that for protein structural changes, which in turn is faster than that for complex dissociation.

Affinity capillary electrophoresis using a low-concentration additive with the consideration of relative mobilities.

Most affinity studies in capillary electrophoresis assume that the analyte concentration is much smaller than the additive concentration so that the migration of the analyte has no effect on the concentration of the additive in the capillary. However, in most medium- to high-affinity interactions, the additive concentration has to be kept rather low to observe the changes in analyte mobility before saturation is reached. In this paper, a mathematical model is developed to describe the migration behavior of the analyte in a system where the complex formed becomes concentrated to levels much greater than the original concentration of the additive due to the differences in the mobilities of the analyte, additive, and complex. The analyte is flurbiprofen, the additive is transthyretin, and the stoichiometry of the reaction between the two is 1:2. This study also provides a new algorithm to determine medium- to high-affinity binding interaction constants by CE.