End-distance distribution function

Ideal behaviour for chain conformation is represented by a Gaussian distribution W(r) of end-to-end distances. Excluded volume effects give a much different end distance distribution function. Figure 1, with the biggest differences operating on chains which happen to have their chain ends in proximity (4). 

Figure 1. A plot of the end-distance distribution function W(0) vs. the ratio of the end separation divided by Rg for a...

If this assumption is accepted, it turns out that S(m) can be obtained from the end distance distribution function IT(R) for an isolated central chain. Barrett chose the Domb-GilHs-W51mers function, i.e., the Mazur function for the case s = t, for IL (R). This choice gives... 

On the other hand, microscopic theories originate from a model of the system under study, which involves microscopic pMameters. For example, the model of a continuous molecular chain uses the contour chain length No No = tiol), the excluded volume parameter Wo, and the minimal scale length , which can be identified with the length of a chain segment. The microscopic model of a polymer chain with excluded volume i.s defined by a bare (i.e. non-renormalized) dimensionless Hamiltonian to be derived from the end-to-end distance distribution function (Fquations 3.1 20d,-205)... 

Using a series of Monte-Carlo ealeulations, the elastic properties of the network are derived from the network ehain end-to-end distance distribution, and are assumed to arise solely as a result of allowed conformational changes in individual network chains. Figure 3.12 shows the calculation for the probability density functions p f) calculated from the simulated radial end-to-end distance distribution functions P f), where... 

Any polymer can be represented as a series of n bonds of length 1. Each conformation has a characteristic end-to-end length r. A population of chains can be described by various distribution functions and ensenble averages. Equilibrium chain properties are usually described in terms of the end-to-end distance distribution function W(r), as well as the mean-squared end-to-end distance Rf and the mean-squared radius of gyration Rg. ... 

Figure 2 A simulation of the autocorrelation function, AC(x), of the donor fluorescence calculated for different diffusion coefficients of one molecular end relative to the other end. The calculations were performed for a model oligopeptide whose end-to-end distance distribution function is given in Fig. 5, Ref. 15 (curve 8). R was assumed to be 25A. The time scale is given in units of 10 sec and each curve is marked by the value assumed for the intramolecular diffusion coefficients in units of 10 1( cm /sec. AC() is given in arbitrary units. (Reprinted with permission from Ref. 17).

Adopting a continuum wormlike chain model Wilhelm and Frey [27] have calculated analytically the end-to-end distance distribution function of a semiflexible polymer. They obtained the following expression ... 

Freed et al. [42,43], among others [44,45] have performed RG perturbation calculations of conformational properties of star chains. The results are mainly valid for low functionality stars. A general conclusion of these calculations is that the EV dependence of the mean size can be expressed as the contribution of two terms. One of them contains much of the chain length dependence but does not depend on the polymer architecture. The other term changes with different architectures but varies weakly with EV. Kosmas et al. [5] have also performed similar perturbation calculations for combs with branching points of different functionalities (that they denoted as brushes). Ohno and Binder [46] also employed RG calculations to evaluate the form of the bead density and center-to-end distance distribution of stars in the bulk and adsorbed in a surface. These calculations are consistent with their scaling theory [27]. 

Fig. 4. Backfolding in dendrimers as predicted by analytical theory [12]. Free end probability distribution function of the radial distance for generations 2-7. All data has been calculated assuming a realistic excluded volume parameter of the segments of the dendrimer (see [12] for further details). Reproduced with permission from [12]...

Fig. 5 The end-to-end distance distribution G is plotted for chains in the surface layer (upper two plots a and c) and for chains in the bulk (lower part b,d) for N = 24. Chains belong to the surface layers if at least one monomer is in contact to a surface. In our simulation box, about 2/3 of the chains have no surface contacts and are counted as bulk chains. Since the distributions in the surface layer and also in the ordered state are anisotropic we distinguish between the spatial components = -direction, + = x-direction, and O =j-direction. The distribution in the surface layer is clearly anisotropic even in the disordered state e = 0. The sharp (non-Gaussian) peak indicates a preferential location of end-points close to the surface which is expected in dense polymer systems. For an interaction constant close but still below the ODT (c,d) the surface chains already display long-range order in the x-direction while the distribution of the z-component is practically not influenced. By contrast, the bulk chains, (part d), are still isotropic but stretching (without inducing any long-range order) is displayed by a broadening of the distribution functions...

The function P r,N) has a peak because it is a product of two quantities the probability P(r, N) diminishes monotonically with distance from the origin, but the number of volume elements, Trrr-, increases with distance from the origin. Figure 32.7 shows that the twm-dimensional end-to-end distance distribution of T3 DNA molecules adsorbed on surfaces and counted in electron micrographs is well predicted by two-dimensional random-walk theory. 

This expression was derived for describing the end-to-end distance distributions of flexible chain molecules (26). This function is... 

Tanford (30), which pertains to the unperturbed end-to-end distance of a polypeptide chain gives a value of 59 +7A for a polypeptide of 50 residues, which is quite close to the above result. However, the latter is not the unperturbed distance (since guanidine hydrochloride is not a 0 solvent) and the actual distance would be expected to be larger than the unperturbed distance. It should be kept in mind that in the reduced state, transfer efficiency is very low and the uncertainty in the distances distribution function is increased. Hence, this result should be discussed with caution. 

For the chain (homogenous) consisting of one con-former, osmotic forces are similar to the ones stretching the molecule by the ends. Then, labor of the distance being estimated at constant temperature T , one can estimate 5ch value from the condition = F AR = T ASch)- If a more accurate estimation of the distance change valRe between the ends is required, one may calculate the R value, taking into account the distribution function of the distances between the ends R. The value of the mean-square distances between the ends of the chain, being stretched by forces, applied to the ends equals [14] ... 

The average force f(r) in the chain when the ends are held a distance r apart could then be obtained from Eq. (10) providing the appropriate configuration distribution function p(r) is known. In the limit of a small extension ratio, p(r) is approximately proportional to peq(r) ... 

Plotting U as a function of L (or equivalently, to the end-to-end distance r of the modeled coil) permits us to predict the coil stretching behavior at different values of the parameter et, where t is the relaxation time of the dumbbell (Fig. 10). When et < 0.15, the only minimum in the potential curve is at r = 0 and all the dumbbell configurations are in the coil state. As et increases (to 0.20 in the Fig. 10), a second minimum appears which corresponds to a stretched state. Since the potential barrier (AU) between the two minima can be large compared to kBT, coiled molecules require a very long time, to the order of t exp (AU/kBT), to diffuse by Brownian motion over the barrier to the stretched state at any stage, there will be a distribution of long-lived metastable states with different chain conformations. With further increases in et, the second minimum deepens. The barrier decreases then disappears at et = 0.5. At this critical strain rate denoted by ecs, the transition from the coiled to the stretched state should occur instantaneously. 

A second simplihcation results from introducing the Born-Oppenheimer separation of electronic and nuclear motions for convenience, the latter is most often considered to be classical. Each excited electronic state of the molecule can then be considered as a distinct molecular species, and the laser-excited system can be viewed as a mixture of them. The local structure of such a system is generally described in terms of atom-atom distribution functions t) [22, 24, 25]. These functions are proportional to the probability of Ending the nuclei p and v at the distance r at time t. Building this information into Eq. (4) and considering the isotropy of a liquid system simplifies the theory considerably. 

An intermolecular pair distribution function evaluated at the end of Step 2 would consist of delta functions at those distances allowed on the 2nnd lattice. After completion of reverse mapping, which moves the system from the discrete space of the lattice to a continuum, the carbon-carbon intermolecular pair distribution function becomes continuous, as depicted in Fig. 4.7 [144]. 

The equations of motion (75) can also be solved for polymers in good solvents. Averaging the Oseen tensor over the equilibrium segment distribution then gives = l/ n — m Y t 1 = p3v/rz and Dz kBT/r sNY are obtained for the relaxation times and the diffusion constant. The same relations as (80) and (82) follow as a function of the end-to-end distance with slightly altered numerical factors. In the same way, a solution of equations of motion (75), without any orientational averaging of the hydrodynamic field, merely leads to slightly modified numerical factors [35], In conclusion, Table 4 summarizes the essential assertions for the Zimm and Rouse model and compares them. 

A classical description of such a structure is of no real use. That is, if we attempt to describe the structure using the same tools we would use to describe a box or a sphere we miss the nature of this object. Since the structure is composed of a series of random steps we expect the features of the structure to be described by statistics and to follow random statistics. For example, the distribution of the end-to-end distance, R, follows a Gaussian distribution function if counted over a number of time intervals or over a number of different structures in space,... 

One of the most powerful tools molecular simulation affords is that of measuring distribution functions and sampling probabilities. That is, we can easily measure the frequencies with which various macroscopic states of a system are visited at a given set of conditions - e.g., composition, temperature, density. We may, for example, be interested in the distribution of densities sampled by a liquid at fixed pressure or that of the end-to-end distance explored by a long polymer chain. Such investigations are concerned with fluctuations in the thermodynamic ensemble of interest, and are fundamentally connected with the underlying statistical-mechanical properties of a system. 

Fig.9a-c. Scaled distribution function for the center-to-end distances of stars of f=3,10 and 50 arms (a is the repulsive distance range of the intramolecular potential) T=4 /kg corresponds to a good solvent T=3e/kg corresponds to a theta solvent T=2e/kg (lower temperatures correspond to the curves on the left). Solid curves Simulation data dashed lines Gaussian functions. Reprinted with permission from [131]. Copyright (1994) American Chemical Society... 

In terms of Gc and Gh the distribution function (unnormalized) for the end-to-end distance, R, of a polypeptide chain consisting of N residues can be written... 

A method is developed for calculating even moments of the end-to-end distance r of polymeric chains, on the basis of the RIS approximation for rotations about skeletal bonds. Expressions are obtained in a form which is applicable in principle to arbitrary k, but practical applications are limited by a tremendous increase in the order of the matrices to be treated, with increasing k. An application is made to the PE chain by using the familiar three-state model. Approximate values of the distribution function Wn (r) of the end-to-end vector r, Wn (0), and , are calculated from these even moments. 

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