Stuart A. Allison

Bending and twisting dynamics of short linear DNAs. Analysis of the triplet anisotropy decay of a 209 base pair fragment by Brownian simulation

Brownian dynamics is used to simulate the decay anisotropy of short linear DNA fragments modeled as a string of beads. The model is sufficiently general to allow for static bends, anisotropic bending, and elastic constants for bending and twisting which can vary along the chain. In limiting cases, simulations are found to be in excellent agreement with analytic theory down to a correlation length of at least 500 A. This model is then used to analyze the 0–2.5 μs triplet depletion anisotropy decay of a 209 base pair sea urchin DNA fragment. It is concluded that the conventional worm‐like chain model, in which bending is isotropic and/or there are no static bends along the chain, is unable to account for the experimental results unless a correlation length of 1000 A is assumed. A worm‐like chain with anisotropic bending requires a similar but slightly larger correlation length.

Brownian dynamics with rotation–translation coupling

A generalized algorithm is proposed for simulating a system of interacting spherical particles simultaneously executing both rotational and translational Brownian motion. Rotation–translation couplings are obtained numerically for (a) a pair of rigid spheres using the generalized algorithm and (b) a pair of rigid cubic octamer particles using a translational algorithm with rigid constraints. The likely importance of rotation–translation coupling in the Brownian–dynamics context is discussed.

Diffusion-controlled enzymatic reactions

Publisher Summary The rate of diffusional encounter among reactant molecules in solution sets the ultimate limit on the speed of enzymatic and other reactions. If the reactant molecules are such that subsequent events develop very rapidly when the reactants come into contact, the net rate of the reaction will be equal to the rate of diffusional encounter. The reaction is then said to be diffusion-controlled. This chapter describes the way computer simulations may be used to highlight the nature of diffusion-controlled reactions. Such simulations can, in principle, aid in the detailed interpretation of experimental results or in the design of molecules with prescribed kinetic properties. The chapter also describes methods for calculating electrostatic interactions among solute molecules in solution. The long range of such interactions makes them particularly important in the consideration of diffusional encounters. Diffusional encounter involves the interaction of molecules at large separations. Electrostatic interactions are, therefore, particularly important because of the long-range nature of the Coulombic potential. At small separations, the effects of other interactions can often be represented by suitable boundary conditions. As a result, most attention has been paid to the calculation of electrostatic forces. The methods described in the chapter can also be used to study the role of electrostatics in determining equilibrium properties, such as the stability of molecular folding and association.

Modeling the electrophoresis of rigid polyions: application to lysozyme

An algorithm is developed to determine the electrophoretic mobility of a rigid polyion modeled as a low dielectric volume element of arbitrary shape containing an arbitrary charge distribution. The solvent is modeled as a high dielectric continuum with salt distributed according to the linearized Poisson Boltzmann equation. Account is also taken of a Stern layer that separates the molecular surface and the surface of hydrodynamic shear, or Stern surface. Relaxation of the ion atmosphere because of the presence of the external field is ignored. The electrostatic and hydrodynamic problems are both solved by boundary element methods. The procedure is first applied to spherical polyions containing monopolar, dipolar, and quadrupolar charge distributions, and calculated mobilities are found to be in excellent agreement with the theory of Yoon and Kim. It is then applied to lysozyme by using models that account for the detailed shape and charge distribution of the enzyme. For reasonable choices of the molecular and Stern surfaces, calculated and experimental mobilities are found to be in fair agreement with each other. However, if a pH independent Stern layer (or, equivalently, translational diffusion constant, Dt) is assumed, the calculated mobilities exhibit a stronger pH dependence than is observed experimentally. A small increase in Dt with increasing pH could correct this discrepancy.

Extended Brownian dynamics of diffusion controlled reactions

The Brownian dynamics simulation method of Northrup et al. is extended so that dynamical trajectories can be initiated with the reactants in close proximity to one another. A more general analysis is presented which shows that this procedure is exact in cases where the first‐time encounter flux to the more proximal starting surface is isotropic, such as cases where interparticle forces are centrosymmetric, but is approximate otherwise. Diffusion controlled rate constants for three model systems obtained by this procedure are compared with analytic results or with exact rate constants derived from simulations following the original Northrup procedure. Agreement is good to excellent in all cases considered. The extended method is expected to be of considerable practical importance in systems with highly anisotropic reactivity where it is computationally inefficient to obtain rate constants by the original method.

Optimization of Brownian dynamics methods for diffusion‐influenced rate constant calculations

One‐dimensional Brownian dynamics algorithms for reaction and reflection recently developed by Lamm and Schulten are adapted into the special boundary topology necessary to compute diffusion‐influenced rate constants of arbitrarily complicated bimolecular reactions in three dimensions. Performance of these relative to the primitive free diffusion algorithm commonly employed is discussed. Remaining sources of error arising from boundaries are (1) boundary curvature effects and (2) reactive discontinuity effects in cases where orientational criteria for reaction exist. The magnitudes of these errors are calculated as a function of simulation time step size. In addition, a special statistical sampling procedure is developed which allows the simultaneous treatment of a large class of reactive boundary problems in one simulation. This procedure is illustrated by the treatment of reactive patch size effects on the rate constant in the model of Solc and Stockmayer.

Use of T4 lysozyme charge mutants to examine electrophoretic models.

The electrophoretic mobility of a macro-ion is affected in a complex manner by a variety of forces that arise from the applied field. Coupling of the macro-ion and small-ion flows gives rise to non-conserved forces that are greater than those expected from ordinary hydrodynamic considerations. It is difficult to separate the steady-state hydrodynamic and electrodynamic contributions to the macro-ion mobility. Membrane-confined electrophoresis (MCE), a free solution technique, provides an experimental means by which to gain insight into these contributions. In this work we used MCE steady-state electrophoresis (SSE) of a series of T4 lysozyme charge mutants to investigate these effects and to examine the existing theoretical descriptions. These experiments isolate the effects of charge on electrophoretic mobility and permit a unique test of theories by Debye-Hückel-Henry, Booth and Allison. Our results show that for wild type (WT) T4, where divergence is expected to be greatest, the predicted results are within 15, 8 and 1%, respectively, of experimental SSE results. Parallel experiments using another free-solution technique, capillary electrophoresis, were in good agreement with MCE results. The theoretical predictions were within 20, 13 and 5% of CE mobilities for WT. Boundary element modeling by Allison and co-workers, using continuum hydrodynamics based on detailed structural information, provides predictions in excellent agreement with experimental results at ionic strengths of 0.11 M.

Boundary element modeling of biomolecular transport

The boundary element (BE) methodology has emerged as a powerful tool in modeling a broad range of different transport phenomena of biomolecules in dilute solution. These include: sedimentation, diffusion (translational and rotational), intrinsic viscosity, and free solution electrophoresis. Modeling is carried out in the framework of the continuum primitive model where the biomolecule is modeled as an arbitrary array of solid platelets that contains fixed charges within. The surrounding fluid is modeled as a electrodynamic/hydrodynamic continuum which obeys the Poisson and low Reynolds number Navier-Stokes equations. Ion relaxation (the distortion of the ion atmosphere from equilibrium) can also be accounted for by solving the coupled ion transport equation (for each mobile ion species present), Poisson, and Navier-Stokes equations in tandem. Several examples are presented in this work. It is first applied to a detailed model of 20 bp DNA and it is concluded that it is not necessary to include a layer of bound water to reconcile experimental and model translational diffusion constants. With regards to diffusion, the BE approach is also applied to a 375-bp supercoiled DNA model (without ion relaxation), and also 20-60-bp DNA fragments with ion relaxation included in order to assess the magnitude of the electrolyte friction effect under a number of different salt/buffer conditions. Attention is then turned to modeling the electrophoretic mobility of three different cases. First of all, we consider a sphere with a central charge large enough in magnitude to insure that ion relaxation is significant. Excellent agreement with independent theory is obtained. Finally, it is applied to modeling short DNA fragments in KCl and Tris acetate salts. Quantitative agreement is achieved when the salt is KCl, but the calculated (absolute) mobility in Tris acetate is substantially higher than the experimental value. The interpretation of this is that there is an association between Tris(+) and DNA (perhaps hydrogen bonding) not accounted for in our modeling that is responsible for this discrepancy.

The free solution electrophoretic mobility of peptides by a bead modeling methodology.

A bead modeling methodology, BMM, discussed previously to compute the free solution electrophoretic mobility of peptides [H. Pei, Y. Xin, S.A. Allison, J. Sep. Sci. 31 (2008) 554-564], is generalized to avoid the approximation of orientationally preaveraging hydrodynamic interaction. In general, peptide mobilities computed without preaveraging are lower by about 2%. The BMM is then used to study the free solution electrophoretic mobility of several insect oostatic peptides reported previously in a variety of different buffer systems ranging in pH from 2.25 to 8.1 [V. Solinova, V. Kasicka, D. Koval, J. Hlavacek, Electrophoresis, 25 (2004) 2299-2308]. With minor adjustment of the intrinsic pK(a0) of the N-terminal peptide, good agreement between modeling and experiment is achieved for peptide models with random secondary structures in the entire pH range. Model mobilities of these peptides appear to be relatively insensitive to the assumed secondary structure.

Modeling the electrophoresis of lysozyme. II. Inclusion of ion relaxation

In this work, boundary element methods are used to model the electrophoretic mobility of lysozyme over the pH range 2-6. The model treats the protein as a rigid body of arbitrary shape and charge distribution derived from the crystal structure. Extending earlier studies, the present work treats the equilibrium electrostatic potential at the level of the full Poisson-Boltzmann (PB) equation and accounts for ion relaxation. This is achieved by solving simultaneously the Poisson, ion transport, and Navier-Stokes equations by an iterative boundary element procedure. Treating the equilibrium electrostatics at the level of the full rather than the linear PB equation, but leaving relaxation out, does improve agreement between experimental and simulated mobilities, including ion relaxation improves it even more. The effects of nonlinear electrostatics and ion relaxation are greatest at low pH, where the net charge on lysozyme is greatest. In the absence of relaxation, a linear dependence of mobility and average polyion surface potential, (lambda zero)s, is observed, and the mobility is well described by the equation [formula: see text] where epsilon 0 is the dielectric constant of the solvent, and eta is the solvent viscosity. This breaks down, however, when ion relaxation is included and the mobility is less than predicted by the above equation. Whether or not ion relaxation is included, the mobility is found to be fairly insensitive to the charge distribution within the lysozyme model or the internal dielectric constant.