Brownian Molecular Dynamics

There are many problems that would require so much computer time that their study by the previous method would not be possible. For example polyelectrolyte solutions, or motions of particles in membranes would not be susceptible to study because of wide separations in the time scales for different dynamic processes characterizing solute and solvent or because the property of interest evolves so slowly that an excessively long trajectory would be required. The study of these systems requires a different approach. A beginning was made many years ago by Simon,who studied the melting of DNA by solving the coupled set of stochastic Langevin equations on a computer. This required an assumption about the statistical distribution of random forces. The precise values of the forces were then sampled from this distribution. 

The molecular theory of fluctuations and Brownian motion offers a generalization of the Langevin equations. These equations provide a set of equations of motion that are stochastic in nature and that can be modeled on the basis of phenomenology. For example the velocity of the yth Brownian particle in solution is described by the equation of motion 

We will not mention ail the assumptions that are required to establish Eq. (30). 

Equation (30) can be solved if Fy is sampled at each time step in some specified way. If Fy is assumed to be a Gaussian Markov process it follows from Doob s theorem that Kj(t) is an exponential function of time. Then only two parameters need be specified before Fy can be sampled from the Gaussian two-time probability distribution and these are the mean square value (Fy) and the correlation time of Fy, say xy. Equation (30) then forms a set of coupled stochastic differential equations that can be solved by methods similar to those already mentioned. 

A modification of this very convenient procedure has recently been adopted by Lantelme They applied it to the study of a small system, but 

---

Molecular dynamics is a simulation of the time-dependent behavior of a molecular system, such as vibrational motion or Brownian motion. It requires a way to compute the energy of the system, most often using a molecular mechanics calculation. This energy expression is used to compute the forces on the atoms for any given geometry. The steps in a molecular dynamics simulation of an equilibrium system are as follows ... 

Btamp/e Conformations of molecules like n-decane can be globally characterized by the end-to-end distance, R. In a comparison of single-molecule Brownian (Langevin) dynamics to molecular dynamics, the average end-to-end distance for n-decane from a 600 ps single-molecule Langevin dynamics run was almost identical to results from 19 ps of a 27-molecule molecular dynamics run. Both simulations were at 481K the time step and friction coeffi-... 

Further support for this approach is provided by modern computer studies of molecular dynamics, which show that much smaller translations than the average inter-nuclear distance play an important role in liquid state atom movement. These observations have conhrmed Swalin s approach to liquid state diffusion as being very similar to the calculation of the Brownian motion of suspended particles in a liquid. The classical analysis for this phenomenon was based on the assumption that the resistance to movement of suspended particles in a liquid could be calculated by using the viscosity as the frictional force in the Stokes equation... 

As with Newtonian molecular dynamics, a number of different algorithms have been developed to calculate the diffusional trajectories. An efficient algorithm for solving the Brownian equation of motion was introduced by Ermak and McCammon [21]. A detailed survey of this and other algorithms as well as their application can be found in Ref. 2. 

But a computer simulation is more than a few clever data structures. We need algorithms to manipulate our system. In some way, we have to invent ways to let the big computer in our hands do things with the model that is useful for our needs. There are a number of ways for such a time evolution of the system the most prominent is the Monte Carlo procedure that follows an appropriate random path through configuration space in order to investigate equilibrium properties. Then there is molecular dynamics, which follows classical mechanical trajectories. There is a variety of dissipative dynamical methods, such as Brownian dynamics. All these techniques operate on the fundamental degrees of freedom of what we define to be our model. This is the common feature of computer simulations as opposed to other numerical approaches. 

F. Ould-Kaddour and D. Levesque, Determination of the friction coefficient of a Brownian particle by molecular-dynamics simulation, J. Chem. Phys. 118, 7888 (2003). 

Vallverdu G, Demachy I, Ridard J, Levy B (2009) Using biased molecular dynamics and Brownian dynamics in the study of fluorescent proteins. Theochem-J Mol Struct 898 73-81... 

Sometimes the theoretical or computational approach to description of molecular structure, properties, and reactivity cannot be based on deterministic equations that can be solved by analytical or computational methods. The properties of a molecule or assembly of molecules may be known or describable only in a statistical sense. Molecules and assemblies of molecules exist in distributions of configuration, composition, momentum, and energy. Sometimes, this statistical character is best captured and studied by computer experiments molecular dynamics, Brownian dynamics, Stokesian dynamics, and Monte Carlo methods. Interaction potentials based on quantum mechanics, classical particle mechanics, continuum mechanics, or empiricism are specified and the evolution of the system is then followed in time by simulation of motions resulting from these direct... 

Once the boundary conditions have been implemented, the calculation of solution molecular dynamics proceeds in essentially the same manner as do vacuum calculations. While the total energy and volume in a microcanonical ensemble calculation remain constant, the temperature and pressure need not remain fixed. A variant of the periodic boundary condition calculation method keeps the system pressure constant by adjusting the box length of the primary box at each step by the amount necessary to keep the pressure calculated from the system second virial at a fixed value (46). Such a procedure may be necessary in simulations of processes which involve large volume changes or fluctuations. Techniques are also available, by coupling the system to a Brownian heat bath, for performing simulations directly in the canonical, or constant T,N, and V, ensemble (2,46). 

Molecular motions in low molecular weight molecules are rather complex, involving different types of motion such as rotational diffusion (isotropic or anisotropic torsional oscillations or reorientations), translational diffusion and random Brownian motion. The basic NMR theory concerning relaxation phenomena (spin-spin and spin-lattice relaxation times) and molecular dynamics, was derived assuming Brownian motion by Bloembergen, Purcell and Pound (BPP theory) 46). This theory was later modified by Solomon 46) and Kubo and Tomita48 an additional theory for spin-lattice relaxation times in the rotating frame was also developed 49>. 

NMR measurements is performed by numerical simulations with a multi-scale modelling [11] of the structure of the clay dispersions and the diffusion of the water molecules or the sodium counterions, by using Brownian Dynamics in order to bridge the gap between the time scale accessible by Molecular Dynamics (typically a few ps) and that explored by the NMR measurements (from ns to ms). 

Kramers idea was to give a more realistic description of the dynamics in the reaction coordinate by including dynamical effects of the solvent. Instead of giving a deterministic description, which is only possible in a large-scale molecular dynamics simulation, he proposed to give a stochastic description of the motion similar to that of the Brownian motion of a heavy particle in a solvent. From the normal coordinate analysis of the activated complex, a reduced mass pi has been associated with the motion in the reaction coordinate, so the proposal is to describe the motion in that coordinate as that of a Brownian particle of mass g in the solvent. 

A number of methodologies have been developed and generalized in recent years to quantitatively describe the ion atmosphere around nucleic acids [11, 12, 17, 28, 29]. These include models based on Poisson-Boltzmann equation [11, 12], counterion condensation [17], and simulation methods, such as Monte Carlo, molecular dynamics, and Brownian dynamics [28, 29]. 

What is the mechanism of spins dropping down from the state to the a state and fanning out around the two cones, and what determines the rates (R = HT and/ 2 = 1/72) of NMR relaxation These processes are intimately tied to the motion of molecules as they tumble ( reorient ) in solution in their rapid Brownian motion, and measurement of the NMR relaxation parameters T and T2 can even give us detailed information about molecular dynamics (motion) from the point of view of each spin in the molecule. A simplified model... 

In order to determine a system thermodynamically, one has to specify some independent parameters (e.g. N, T, P or V) besides the composition of the system. The most common choice in MC simulation is to specify N, V and T resulting in the canonical ensemble, where the Helmholtz free energy A is the natural thermodynamical potential. However, MC calculations can be performed in any ensemble, where the suitable choice depends on the application. It is straightforward to apply the Metropolis MC algorithm to a simple electric double layer in the iVFT ensemble. It is however, not so efficient for polymers composed of more than a few tens of monomers. For long polymers other algorithms should be considered and the Pivot algorithm [21] offers an efficient alternative. MC simulations provide thermodynamic and structural information, but time-dependent properties are not accessible. If kinetic or time-dependent properties are of interest one has to use molecular dynamic or brownian dynamic simulations. 

Another illustration of the power of molecular dynamics simulation can be drawn from the sphere of enzyme catalysis. Many enzyme-catalyzed reactions proceed at a rate that depends on the diffusion-limited association of the substrate with the active site. Sharp et al. [28] have carried out Brownian dynamics simulations of the association of superoxide anions with superoxide dismutase (SOD). The active center in SOD is a positively charged copper atom. The distribution of charge over the enzyme is not uniform, and so an electric field is produced. Using their model, Sharp et al. [28] have shown that the electric field enhances the association of the substrate with the enzyme by a factor of 30 or more. Their calculations also predict correctly the response of the association rate to changes in ionic strength and amino... 

Now that we have settled on a model, one needs to choose the appropriate algorithm. Three methods have been used to study polymers in the continuum Monte Carlo, molecular dynamics, and Brownian dynamics. Because the distance between beads is not fixed in the bead-spring model, one can use a very simple set of moves in a Monte Carlo simulation, namely choose a monomer at random and attempt to displace it a random amount in a random direction. The move is then accepted or rejected based on a Boltzmann weight. Although this method works very well for static and dynamic properties in equilibrium, it is not appropriate for studying polymers in a shear flow. This is because the method is purely stochastic and the velocity of a mer is undefined. In a molecular dynamics simulation one can follow the dynamics of each mer since one simply solves Newton s equations of motion for mer i,... 

Simulation techniques suitable for the description of phenomena at each length-scale are now relatively well established Monte Carlo (MC) and Molecular Dynamics (MD) methods at the molecular length-scale, various mesoscopic simulation methods such as Dissipative Particle Dynamics (Groot and Warren, 1997), Brownian Dynamics, or Lattice Boltzmann in the colloidal domain, Computational Fluid Dynamics at the continuum length-scale, and sequential-modular or equation-based methods at the unit operation/process-systems level. 

An interesting application of the molecular dynamics technique on single chains is found in the work of Mattice et al. One paper by these authors is cited here because it is relevant to both RIS and DRIS studies and deals with the isomerization kinetics of alkane chains. The authors have computed the trajectories for linear polyethylene chains of sizes C,o to Cioo- The simulation was fully atomistic, with bond lengths, bond angles, and rotational states all being variable. Analysis of the results shows that for very short times, correlations between rotational isomeric transitions at bonds i and i 2 exist, which is something a Brownian dynamics simulation had shown earlier. 

---