The Total Quasi-Steady-State Approximation for Fully Competitive Enzyme Reactions

The validity of the Michaelis–Menten–Briggs–Haldane approximation for single enzyme reactions has recently been improved by the formalism of the total quasi-steady-state approximation. This approach is here extended to fully competitive systems, and a criterion for its validity is provided. We show that it extends the Michaelis–Menten–Briggs–Haldane approximation for such systems for a wide range of parameters very convincingly, and investigate special cases. It is demonstrated that our method is at least roughly valid in the case of identical affinities. The results presented should be useful for numerical simulations of many in vivo reactions.     

Application of kinetic Monte Carlo method to the vapour–liquid equilibria of associating fluids and their mixtures

Canonical kinetic Monte Carlo (C-kMC) simulations have been carried out to assess their feasibility and potential for calculating the vapour–liquid equilibria of various pure components with increasingly strong electrostatic interactions (carbon dioxide, methanol, ammonia and water) over a wide range of temperatures and for methanol/water mixtures at 298 K. The simulation results show that C-kMC is successful as a method for studying phase equilibria and thermodynamic properties. For all the examples investigated, the performance of the C-kMC method is at least as good as that of the conventional Monte Carlo (MC) methods and is efficient at low temperature where these fail. It also provides a route that is superior to the Widom method for the calculation of chemical potential. We recommend this method for this purpose and as an alternative to conventional MC for simulations of strongly associating fluids and at low temperatures.     

Computational efficiency of numerical approximations of tangent moduli for finite element implementation of a fiber-reinforced hyperelastic material model

In this study, we evaluated computational efficiency of finite element (FE) simulations when a numerical approximation method was used to obtain the tangent moduli. A fiber-reinforced hyperelastic material model for nearly incompressible soft tissues was implemented for 3D solid elements using both the approximation method and the closed-form analytical method, and validated by comparing the components of the tangent modulus tensor (also referred to as the material Jacobian) between the two methods. The computational efficiency of the approximation method was evaluated with different perturbation parameters and approximation schemes, and quantified by the number of iteration steps and CPU time required to complete these simulations. From the simulation results, it can be seen that the overall accuracy of the approximation method is improved by adopting the central difference approximation scheme compared to the forward Euler approximation scheme. For small-scale simulations with about 10,000 DOFs, the approximation schemes could reduce the CPU time substantially compared to the closed-form solution, due to the fact that fewer calculation steps are needed at each integration point. However, for a large-scale simulation with about 300,000 DOFs, the advantages of the approximation schemes diminish because the factorization of the stiffness matrix will dominate the solution time. Overall, as it is material model independent, the approximation method simplifies the FE implementation of a complex constitutive model with comparable accuracy and computational efficiency to the closed-form solution, which makes it attractive in FE simulations with complex material models.     

Molecular Dynamics Simulations of Proteins in Water Without the Truncation of Long-range Coulomb Interactions

A new program package (COSMOS90) for molecular dynamics simulations was developed to simulate large molecular systems consisting of more than tens of thousands of atoms without the truncation of long-range coulomb interactions. This program package was based on a new approximation scheme (PPPC) for calculating efficiently the coulomb interactions without sacrificing accuracy. In this approximation scheme, the group of charges at a long distance from each atom was represented by a total charge and total dipole moment of the group. In order to assess the accuracy of PPPC and the ability of COSMOS90, molecular dynamics simulations were carried out for a large system consisting of 16108 atoms (human lysozyme in water) for 50 ps using this program package. The coulomb energy per solute atom was calculated with only five percent of the error found in the 10 Å cut-off approximation (about 0.9 kcal/mol versus 18 kcal/mol, respectively). The molecular dynamics simulations using COSMOS90 require no more CPU time than the simulations based on the 10 Å cut-off approximation of the conventional programs for macromolecular simulations.     

Hydrodynamics of Sperm Cells near Surfaces

Sperm are propelled by an actively beating tail, and display a wide variety of swimming patterns. When confined between two parallel walls, sperm swim either in circles or on curvilinear trajectories close to the walls. We employ mesoscale hydrodynamics simulations in combination with a mechanical sperm model to study the swimming behavior near walls. The simulations show that sperm become captured at the wall due to the hydrodynamic flow fields which are generated by the flagellar beat. The circular trajectories are determined by the chiral asymmetry of the sperm shape. For strong (weak) chirality, sperm swim in tight (wide) circles, with the beating plane of the flagellum oriented perpendicular (parallel) to the wall. For comparison, we also perform simulations based on a local anisotropic friction of the flagellum. In this resistive force approximation, surface adhesion and circular swimming patterns are obtained as well. However, the adhesion mechanism is now due to steric repulsion, and the orientation of the beating plane is different. Our model provides a theoretical framework that explains several distinct swimming behaviors of sperm near and far from a wall. Moreover, the model suggests a mechanism by which sperm navigate in a chemical gradient via a change of their shape.     

Molecular Dynamics Simulations With First Order Coupling to a Bath of Constant Chemical Potential

Abstract

In molecular dynamics simulations the temperature or pressure can be controlled by applying a weak first-order coupling to a bath of constant temperature or pressure. This weak coupling technique to control system properties using a first-order relaxation equation is analyzed from a statistical mechanics point of view. It is shown, how the weak coupling scheme can be generalized and applied to a bath of contstant chemical potential. The presented method, to which in the following will be referred to as chemical potential weak coupling, is applied and tested on a Lennard-Jones fluid. The thermodynamic quantities known from the literature are accuratly reproduced.

The temperature and chemical potential weak coupling methods aim to sample the canonical and grand canonical ensembles respectively. By analyzing the fluctuations in energy and number of particles, the tight relation between the ensembles and the distributions obtained from the weak coupling simulations is demonstrated. The influence of the choice of the coupling parameters on the quality of the approximation of the ensemble distribution is discussed.     

Simulation of the gel-fluid transition in a membrane composed of lipids with two connected acyl chains: application of a dimer-move step.

Phospholipids have been treated as dimers on a hexagonal lattice, and a move has been introduced that allows the dimers to move and change their orientation on the lattice. Simulations have been performed in which phospholipid chains have been treated as being either independent or infinitely coupled thermodynamically with regard to their conformational state. Both types of simulation have reproduced well experimental heat-capacity curves of dipalmitoyl phosphatidylcholine small unilamellar vesicles. Apart from a different gel-fluid interaction parameter and a different number of unlike nearest-neighbor contacts, most of the averages and thermodynamic quantities were essentially the same in the two types of simulation. These results indicate that the transition is not first order and validate those of previous Monte Carlo simulations that have neglected the dimeric nature of phospholipids in the sense that they show that for the thermotropic transition the approximation of phospholipids as monomers is valid.     

A comparative study on the stability and structure of two different green fluorescent proteins in organic co-solvent systems

Green fluorescent protein (GFP) has been used as a reporter marker in a wide range of biological and bioengineering studies. The expanded use of GFP in the field of biosensors, biochips and bio-conjugations requires the stability of GFP in organic co-solvent systems. This prompted us to examine the kinetic stability of two different GFP sequences, n-GFP and s-GFP, showing different folding robustness and thermodynamic stability, under a range of organic co-solvent systems. n-GFP and s-GFP are variants whose biophysical properties are comparable to wild type and super folder GFPs, respectively. The stability of n-GFP and s-GFP in 50% water-miscible organic solvents showed that s-GFP with higher thermodynamic stability exhibited much higher stability against organic solvents than n-GFP, which has lower thermodynamic stability. s-GFP was quite stable even in 90% organic solvents. Circular dichroism analysis confirmed that s-GFP maintained its native structure in organic co-solvent systems, whereas n-GFP showed structural variations under these conditions. Four highly fluctuating loop regions were identified from molecular dynamic simulations under the organic cosolvent conditions. A structural comparison of n-GFP and s-GFP suggested that the improved kinetic stability of s-GFP was due to its larger number of hydrogen bonds and salt-bridges that were present in four loop regions. This study suggests that thermodynamically stable s-GFP can be a good choice for use under harsh organic co-solvent conditions.     

Sampling efficiency in explicit and implicit membrane environments studied by peptide folding simulations

Efficient exploration of the conformational space of peptides embedded in biological membranes is vital to extract converged thermodynamic and kinetic data from simulation studies. Recently developed implicit membrane models promise vast increases in sampling efficiency compared to explicit membrane simulations, allowing for ab initio structure prediction and functional studies. In this study, a previously developed implicit membrane model, based on the generalized Born method, is compared to an explicit di‐palmitoyl‐phosphatidyl‐choline lipid bilayer and an octane slab membrane mimic. The complete folding process of a synthetic 16‐residue peptide is compared using these three setups. Since the comparison requires the entire folding pathway to be captured, individual simulations ranged up to 3 μs of MD. A quantitative sampling comparison using a wide range of performance metrics reveals that the implicit membrane model is at least 2 orders of magnitude more efficient than the simplest explicit setups. Proteins 2009. © 2008 Wiley‐Liss, Inc.     

Non-Boltzmann thermodynamic integration (NBTI) for macromolecular systems: relative free energy of binding of trypsin to benzamidine and benzylamine

The relative free energies of binding of trypsin to two amine inhibitors, benzamidine (BZD) and benzylamine (BZA), were calculated using non-Boltzmann thermodynamic integration (NBTI). Comparison of the simulations with the crystal structures of both complexes, trypsin-BZD and trypsin-BZA, shows that NBTI simulations better sample conformational space relative to thermodynamic integration (TI) simulations. The relative binding free energy calculated using NBTI was much closer to the experimentally determined value than that obtained using TI. The error in the TI simulation was found to be primarily due to incorrect sampling of BZA's conformation in the binding pocket. In contrast, NBTI produces a smooth mutation from BZD to BZA using a surrogate potential, resulting in a much closer agreement between the inhibitors' conformations and the omit electron density maps. This superior agreement between experiment and simulation, of both relative binding free energy differences and conformational sampling, demonstrates NBTI's usefulness for free energy calculations in macromolecular simulations.     

Finite-size effects of Kirkwood–Buff integrals from molecular simulations

The modelling of thermodynamic properties of liquids from local density fluctuations is relevant to many chemical and biological processes. The Kirkwood–Buff (KB) theory connects the microscopic structure of isotropic liquids with macroscopic properties such as partial derivatives of activity coefficients, partial molar volumes and compressibilities. Originally, KB integrals were formulated for open and infinite systems which are difficult to access with standard Molecular Dynamics (MD) simulations. Recently, KB integrals for finite and open systems were formulated (J Phys Chem Lett. 2013;4:235). From the scaling of KB integrals for finite subvolumes, embedded in larger reservoirs, with the inverse of the size of these subvolumes, estimates for KB integrals in the thermodynamic limit are obtained. Two system size effects are observed in MD simulations: (1) effects due to the size of the simulation box and the size of the finite subvolume embedded in the simulation box, and (2) effects due to computing radial distribution functions (RDF) from a closed and finite system. In this study, we investigate the two effects in detail by computing KB integrals using the following methods: (1) Monte Carlo simulations of finite subvolumes of a liquid with an analytic RDF and (2) MD simulations of a WCA mixture for various simulation box sizes, but at the same thermodynamic state. We investigate the effect of the size of the simulation box and quantify the differences compared to KB integrals computed in the thermodynamic limit. We demonstrate that calculations of KB integrals should not be extended beyond half the size of the simulation box. For finite-size effects related to the RDF, we find that the Van der Vegt correction (J Chem Theory Comput. 2013;9:1347) yields the most accurate results.     

Ranking ligand binding affinities with avidin: a molecular dynamics-based interaction energy study.

The binding of 14 biotin analogues to avidin is examined to evaluate the viability of calculating binding free energy based on molecular dynamics (MD) trajectories. Two approaches were investigated in this work. The first one uses the linear interaction energy approximation, while the other approach utilizes the interaction free energy. The results obtained from these two methods were found to correlate well with the experimental binding free energy data for 10 out of 14 ligands. For the other four ligands, both methods overestimate their binding strength by more than 7 kcal/mol. Free energy calculations using the thermodynamic integration method are employed to understand this overestimation. The effect of protein flexibility on binding free energy calculation and the effect of charged or neutral ligands on the calculated results are discussed. MD simulations are shown to be able to provide insight into the interactions occurring in the active site and the origins of variations in binding free energy.     

Towards understanding the mechanisms of molecular recognition by computer simulations of ligand-protein interactions

The thermodynamic and kinetic aspects of molecular recognition for the methotrexate (MTX)-dihydrofolate reductase (DHFR) ligand-protein system are investigated by the binding energy landscape approach. The impact of 'hot' and 'cold' errors in ligand mutations on the thermodynamic stability of the native MTX-DHFR complex is analyzed, and relationships between the molecular recognition mechanism and the degree of ligand optimization are discussed. The nature and relative stability of intermediates and thermodynamic phases on the ligand-protein association pathway are studied, providing new insights into connections between protein folding and molecular recognition mechanisms, and cooperativity of ligand-protein binding. The results of kinetic docking simulations are rationalized based on the thermodynamic properties determined from equilibrium simulations and the shape of the underlying binding energy landscape. We show how evolutionary ligand selection for a receptor active site can produce well-optimized ligand-protein systems such as MTX-DHFR complex with the thermodynamically stable native structure and a direct transition mechanism of binding from unbound conformations to the unique native structure.     

A stochastic model for a single click of Muller's ratchet

This work presents a new approach to Muller's ratchet, where Haigh's model is approximately mapped into a simpler model that describes the behaviour of a population after a click of the ratchet, i.e., after loss of what was the fittest class. This new model predicts the distribution of times to the next click of the ratchet and is equivalent to a Wright-Fisher model for a population of haploid asexual individuals with one locus and two alleles. Within this model, the fittest members of a population correspond to carriers of one allele, while all other individuals have suboptimal fitness and are represented as carriers of the other allele. In this way, all suboptimal fitness individuals are amalgamated into a single “mutant” class.The approach presented here has some limitations and the potential for improvement. However, it does lead to results for the rate of the ratchet that, over a wide range of parameters, are accurate within one order of magnitude of simulation results. This contrasts with existing approaches, which are designed for only one or other of the two different parameter regimes known for the ratchet and are more accurate only in the parameter regime they were designed for.Numerical results are presented for the mean time between clicks of the ratchet for (i) the Wright-Fisher model, (ii) a diffusion approximation of this model and (iii) individually based simulations of a full model. The diffusion approximation is validated over a wide range of parameters by its close agreement with the Wright-Fisher model.The present work predicts that: (a) the time between clicks of the ratchet is insensitive to the value of the selection coefficient when the genomic mutation rate is large compared with the selection coefficient against a deleterious mutation, (b) the time interval between clicks of the ratchet has, approximately, an exponential distribution (or its discrete analogue). It is thus possible to determine the variance in times between clicks, given the expected time between clicks. Evidence for both (a) and (b) is seen in simulations.