Sample algorithms

Fig. 22. Male peripheral blood mononuclear cells labeled with biotin- l chromosome probe/avidin (F1TC green) and counterstained with propidium iodide (red). Seventeen optical sections (27 x 27/zm each) were imaged at 0.2

FIGURE 8.19 Proportion of compounds sampled by the GAO algorithm (blue) compared to the fiiU dataset (red), binned by the primary assay. It can be seen that the algorithm samples eompounds fiom aU parts of the activity spectrum, but is very efficient at sampling the most potent compounds. Reprinted with permission from Pickett et al. [40], 2011 American Chemical Society. For color details, please see color plate section. 

Table 4. Sequential simulation algorithm, sample size 10 ... 

In Ref. 13 it is demonstrated that this algorithm samples the desired distribution. 

Three other combinations of variables are possible (Table 1), but the corresponding ensembles [45] are of more limited practical relevance. The last combination (generalized ensemble) is not physical, because its size is not specified (no independent extensive variable). Note that although MD samples the microcanonical ensemble by default, the basic Monte Carlo (MC [82, 83, 84]) and stochastic dynamics (SD [22, 23, 24, 25, 2]) algorithms sample the canonical ensemble. 

In traditional Fan-Beam CT the radiation emitted from the X-ray tube is collimated to a planar fan, and so most of the intensity is wasted in the collimator blades (Fig. 2a). Cone-Beam CT, where the X-rays not only diverge in the horizontal, but also in the vertical direction, allows to use nearly the whole emitted beam-profile and so makes best use of the available LINAC photon flux (Fig. 2b). So fast scanning of the samples three-dimensional structure is possible. For Cone-Beam 3D-reconstruction special algorithms, taking in consideration the vertical beam divergence of the rays, were developed. 

Using the theorem that the sufficiency condition for mathematical correctness in 3D-reconstruction is fulfilled if all planes intersecting the object have to intersect the source-trajectory at least in one point [8], it is possible to generalise Feldkamp s method. Using projection data measured after changing the sotuce-trajectory from circular to spiral focus orbit it is possible to reconstruct the sample volume in a better way with the Wang algorithm [9]. 

Shortcomings of Wang s method like limited pitch of the spiral and blurring in the vertical direction can be improved by the CFBP-algorithm [10], where gaps in the spiral sampling pattern are filled using X-rays measured from the opposite side. 

A weight factor WCF) may be introduced in the MC sampling algorithm, to generate a modified, or weighted , distribution. 

Lee J 1993 New Monte Carlo algorithm—entropic sampling Phys. Rev.L 71 211-14... 

Metzler W J, Hare D R and Pardi A 1989 Limited sampling of conformational space by the distance geometry algorithm implications for structures generated from NMR data Bioohemistry 2S 7045-52... 

For many applications, it may be reasonable to assume that the system behaves classically, that is, the trajectories are real particle trajectories. It is then not necessary to use a quantum distribution, and the appropriate ensemble of classical thermodynamics can be taken. A typical approach is to use a rnicrocanonical ensemble to distribute energy into the internal modes of the system. The normal-mode sampling algorithm [142-144], for example, assigns a desired energy to each normal mode, as a harmonic amplitude... 

The simplest way to add a non-adiabatic correction to the classical BO dynamics method outlined above in Section n.B is to use what is known as surface hopping. First introduced on an intuitive basis by Bjerre and Nikitin [200] and Tully and Preston [201], a number of variations have been developed [202-205], and are reviewed in [28,206]. Reference [204] also includes technical details of practical algorithms. These methods all use standard classical trajectories that use the hopping procedure to sample the different states, and so add non-adiabatic effects. A different scheme was introduced by Miller and George [207] which, although based on the same ideas, uses complex coordinates and momenta. 

We have developed Monte Carlo algorithms based on sampling Tsallisian distributions. Using a uniform random trial move and the acceptance probability... 

A similar algorithm has been used to sample the equilibrium distribution [p,(r )] in the conformational optimization of a tetrapeptide[5] and atomic clusters at low temperature.[6] It was found that when g > 1 the search of conformational space was greatly enhanced over standard Metropolis Monte Carlo methods. In this form, the velocity distribution can be thought to be Maxwellian. 

Given this effective potential, it is possible to define a constant temperature molecular dynamics algorithm such that the trajectory samples the distribution Pg(r ). The equation of motion then takes on a simple and suggestive form... 

We recently received a preprint from Dellago et al. [9] that proposed an algorithm for path sampling, which is based on the Langevin equation (and is therefore in the spirit of approach (A) [8]). They further derive formulas to compute rate constants that are based on correlation functions. Their method of computing rate constants is an alternative approach to the formula for the state conditional probability derived in the present manuscript. 

A related algorithm can be written also for the Brownian trajectory [10]. However, the essential difference between an algorithm for a Brownian trajectory and equation (4) is that the Brownian algorithm is not deterministic. Due to the existence of the random force, we cannot be satisfied with a single trajectory, even with pre-specified coordinates (and velocities, if relevant). It is necessary to generate an ensemble of trajectories (sampled with different values of the random force) to obtain a complete picture. Instead of working with an ensemble of trajectories we prefer to work with the conditional probability. I.e., we ask what is the probability that a trajectory being at... 

A molecular dynamics simulation samples the phase space of a molecule (defined by the position of the atoms and their velocities) by integrating Newton s equations of motion. Because MD accounts for thermal motion, the molecules simulated may possess enough thermal energy to overcome potential barriers, which makes the technique suitable in principle for conformational analysis of especially large molecules. In the case of small molecules, other techniques such as systematic, random. Genetic Algorithm-based, or Monte Carlo searches may be better suited for effectively sampling conformational space. 

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