Biasing potentials

The silicon diode (photodiode) detector consists of a strip of p-type silicon on the surface of a silicon chip (n-type silicon). By application of a biasing potential with the silicon chip connected to the positive pole of the biasing source, electrons and holes are caused to move away from the p-n junction. This creates a depletion region in the neighbourhood of the junction which in effect becomes a capacitor. When light strikes the surface of the chip, free... 

The second term in Eq. (10-14) represents the kinetic energy of 0j, which has a fictitious mass similar to that of a heavy atom. The last term represents the sum of the biasing potential for each titrating group, defined as... 

For the sake of convenience, we will express the biasing potentials as a function of X below. /bar(0 ) is a harmonic potential,... 

Four years later, Christian Bartels and Martin Karplus [55] used the WHAM equations as the core of their adaptive US approach, in which the efficiency of free energy calculations was improved through refinement of the biasing potentials as the simulation progressed. Efforts to develop adaptive US techniques had, however, started even before WHAM was developed. They were pioneered by Mihaly Mezei [56], who used a self-consistent procedure to refine non-Boltzmann biases. 

It is clear, however, that in complex situations it is not possible to make an educated guess of the biasing potential. In particular the position of the transition regions... 

Guo et al. [57] used this approach to calculate the binding affinities of different inhibitors of trypsin, shown in Fig. 4.16. They proposed to improve the sampling by adding the following biasing potential ... 

For the calculation of the free energy difference AA between two states 0 and 1, the umbrella sampling utilizes the biasing potential W to ensuring sampling of phase space important to both 0 and 1 (cf. Fig. 6.Id). Then, F in (6.80) is taken as... 

Note that for both FEP and TI, the umbrella restraint introduces a term that depends on e+VblaJRT, which may (since Vbias is always > 0) fluctuate widely, especially if the biasing function is attempting to restrain the system to a conformation far from a local minimum. As a result, use of the umbrella term Equations 19 and 20 is often problematic. This has led to the development of alternate (but related) approaches to Umbrella Sampling.41,51,52 Many of these derive from the following equation, which relates the work function W to the probability of states, corrected for use of the biasing potential ... 

Here p (I) is the distribution of conformational states that arises from a simulation using the biased potential. The tricky point with this method comes from the fact that we ultimately need to integrate the work function over a series of windows, and the integration constant for each window is undefined. In practice, this problem is addressed using clever approaches that attempt to match up the probability distributions on consecutive intervals. 

In many applications, a single biasing potential is not sufficient to cover the whole range of and simultaneously produce good sampling. Thus a set of restraining potentials, C/ ( ), are used to shift the local minima in the desired direction. In this windowing approach, the potential of mean force, ), in each window takes the form... 

Tx and Tx are the kinetic energies of the atomic coordinates and X variables, respectively. The As are treated as volumeless particles with mass mx. Since the X variables are associated with the chemical reaction coordinates , the A-dynamics method can utilize the power of specific biasing potentials in the umbrella sampling method to overcome sampling problems that require conventional FEP calculations to be performed in multiple steps. 

The summation is taken over the bound state (X2=l) of the A-dynamics trajectory, including the restraining potential. Unfortunately, with this biasing potential the effect of the restraint (/ , ) becomes too large to yield reasonable convergence as the number of the unbound ligands increases. 

In the situation where the transformation involved barrier crossing, e.g., associated with a nonpolar to polar transformation, the computational time was substantially reduced using the X-dynamics formalism, compared with a standard FEP method. This is because X-dynamics searches for alternative lower free energy pathways the coupling parameters (A/ and A2) evolve in the canonical ensemble independently and find a smoother path then when constrained to move as A = A2. Furthermore, a biasing potential in the form... 

Figure 1 Schematic illustration of the hyperdynamics method. A bias potential (AV(r)) is added to the original potential (V(r), solid line). Provided that AV(r) meets certain conditions, primarily that it be zero at the dividing surfaces between states, a trajectory on the biased potential surface (V(r) + AV(r), dashed line) escapes more rapidly from each state without corrupting the relative escape probabilities. The accelerated time is estimated as the simulation proceeds.

Thus, the state-to-state dynamics on the biased potential is equivalent to that on the original potential as long as the time is renormalized to account for the uniform relative increase of all the rates introduced by the biased potential. This renormalization is in practice obtained by multiplying the MD timestep AImd by the inverse Boltzmann factor for the bias potential, so that n MD timesteps on the biased potential are equivalent to an elapsed time of... 

Indeed, as discussed above, the applicability of hyperdynamics is often hampered by the difficulty in building bias potentials that satisfy all the formal requirements — namely that (i) the bias potential should vanish at any dividing surface between different states and (ii) the kinetics on the biased potential obeys TST — while providing substantial acceleration of the dynamics. Both requirements are very challenging to meet in practice. Indeed, condition... 

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