Beads-on-a-string model

The nucleosome core particle NCP (Fig. 4) is the first organisation level of chromatin, constituent of chromosomes. It is the "bead" in the "beads on a string" model of chromatine. 

Quadruplexes could also act purely as a structural feature, enabling easy recognition by proteins or other species. This model has been proposed for telomeric repeats, perhaps as a beads-on-a-string model of sequential folded quadruplexes. Other regions could also have such structural behaviours. 

Fig. 3.13. (Top) An electron micrograph of an artificial chromatin model composed of T4 DNA and cationic nanoparticles of diameter 15nm. (Bottom) Typical snapshots of a model DNA (semiflexible polyelectrolyte) complexed with cationic nanoparticles. At low salt concentration (Debye screening length m/a = 1), a beads-on-a-string nucleosome-like structure is observed (left), while locally segregated clusters are formed at higher salt concentrations (rn/a = 0.3) (right) (See [46] for more details)...

It has recently been reported that surfactant molecules can associate with carbon nanotubes (CNTs) to form supramolecular assemblies [70,71]. In the two models that have been proposed, the hydrophobic portion of the surfactants associate with the nanotube to form either a uniformly covered cylindrical super molecule, or a beads-on-a-string assembly of disks encapsulating the nanotube (Fig. 11). As such, amphiphiles are frequently used in the disaggregation of CNTs into discreet nanotubes in aqueous solution [72]. 

Fig. 11 Two proposed models for the supramolecular assembly of surfactants with CNTs a cylindrical assembly, and b beads-on-a-string assembly. Reproduced with permission from [74]. 2007 by Wiley-VCH...

To perform activities, the protein units need a definite and stable 3D structure. When a protein folds to form a well-defined 3D structure, it exhibits primary, secondary, tertiary, and quaternary levels of structures. The genetically determined sequence of amino acids is the primary structure. The primary structure is often modeled as beads on a string, where each bead represents one amino acid unit. The intermediate level of protein structure is called secondary structure. This includes the a-heUces, -sheets, and turns that allow the amides to hydrogen bond very efhciently with one another. The tertiary structure might be modeled as a tightly packed snowball to form the well-defined 3D structure, where each atom in the protein has a well-defined... 

Proteins can be modeled as beads on a string, where each bead represents an amino acid residue (see Conformational Search Proteins Molecular Dynamics Techniques and Applications to Proteins and Protein Modeling Folding... 

Coarse graining may be combined with a confinement of the conformational space by placing the coarse grain particles on a cubic lattice. For example, Kurcinski and Kolinski (76) (CABS-REMC) represented the polypeptide chains by strings of Ca beads on a cubic lattice. Side chains were represented by up to two interaction centers, which correspond to Cp and the centers of mass of the remaining portions of the side groups, respectively. The conformational space of this model is sampled by means of a multicopy Monte Carlo method. 

The model that is formed by Eqs. (24) and (25), with the extra assumption that wall effects can be neglected (i.e., that the reactor is built up solely of central subchannel elementary cells such as the one depicted in Fig. 10), is called the catalyst bead (CB) model. Because of the assumption in the CB model that the value of the mass transfer coefficient at the flat ends of the particle equals the value at the cylindrical surface, it can be expected that this model overestimates the conversion obtained in a BSR, especially when the particle length-over-diameter ratio is much smaller than 0.5 and when the gap between consecutive particles on a string is small compared to the particle diameter. However, whether the CB model over- or underestimates the BSR performance also depends on the error made in the calculation of the particle effectiveness factor. 

The focus of this chapter is on an intermediate class of models, a picture of which is shown in Fig. 1. The polymer molecule is a string of beads that interact via simple site-site interaction potentials. The simplest model is the freely jointed hard-sphere chain model where each molecule consists of a pearl necklace of tangent hard spheres of diameter a. There are no additional bending or torsional potentials. The next level of complexity is when a stiffness is introduced that is a function of the bond angle. In the semiflexible chain model, each molecule consists of a string of hard spheres with an additional bending potential, EB = kBTe( 1 + cos 0), where kB is Boltzmann s constant, T is... 

This agrees with all previous results as far as the first term is concerned. The second constant term 0.3863 is very close to Broersma s 0.38 (73) and to the value 0.392 obtained by Broomfield et al. (76) from their shell-model theory, which is essentially a limiting case of the Kikwood-Riseman theory (77) for the bead model of flexible chains. However, these values are about 0.3 smaller than the corresponding term in Eq. (D-5). This implies that if the ellipsoid model and the continuous string model are applied to the same experimental data for as a function of M, the former should lead to a d value which is about i.35 times larger than that obtained by the latter. On the other hand, both models should give an identical value for ML. 

To assess the feasibility of the BSR as a competitor of the monolithic reactor, the parallel-passage reactor, and the lateral-flow reactor, it is necessary to do case studies in which the performance and price of these reactors are compared, for certain applications. To allow such case studies, two tools are needed (1) mathematical models of the reactors that predict the reactor performance, and (2) an optimization routine that, given a mathematical reactor model and a set of process specifications, finds the optimum reactor configuration. Furthermore, data are needed on costs, safety, availability, etc. In this section, five mathematical models of different complexity for the bead-string reactor (BSR) are presented that can be numerically solved on a personal computer within a few hours down to a few minutes. The implementation of the reactor models in an optimization routine, as well as detailed cost analyses of the reactor, are beyond the scope of this text. 

One may visualize a tube as the continuum limit of a discrete chain of tethered disks or coins [21] of fixed radius separated from one another by a distance a in the limit of a -> 0. The inherent anisotropy associated with a coin (the heads-to-tails direction being different from the other two) reflects the fact that a special local direction at each position is defined by the locations of the adjacent objects along the chain. An alternative description of a discrete chain molecule is a string-and-beads model in which the tethered objects are spheres. The key difference between these two descriptions is the different symmetry of the tethered objects. On compaction, spheres tend to surround themselves isotropically with other spheres, unlike the tube situation in which nearby tube segments need to be placed parallel to one another. Even for unconstrained particles, deviations from spherical symmetry (replacing a system of hard spheres with one of hard rods, for example) lead to rich new liquid crystal phases [22, 23]. Likewise, the tube and a chain of tethered spheres exhibit quite distinct behaviors. 

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