Polymer length

Both Flory [143] and Huggins [144] in 1941 addressed themselves to this problem with the initial aim of describing solutions of linear polymers in low molecular weight solvents. Both used lattice models, and their initial derivations considered only polymer length (rather than shape, i.e. branching, etc.) The derivation given here will also limit itself to differences in molecular size, but will be based on an available volume approach. 

M. S. Turner, M. E. Cates. The relaxation spectrum of polymer length distributions. J Physique 57 307-316, 1990. 

An interesting aspect here is the length of the crystal - in the direction of the parallel channels - in relation to the polymer length. A polymer molecular mass of 10.000 corresponds with a length of 0.05 pm. So some 20 polymer chains are required for a crystal length of 1 pm. 

Figure 8a shows the turbidity measurements for different guest residues in ELP [VgX2l. Lower transition temperatures (Tt) correlate with increased hydrophobicity of the guest residue [24, 25]. This data was extrapolated to other ratios of VahXaa (Fig. 8b). The transition temperature could also be influenced by the molecular weight of the ELP. The Tt was shown to increase with decreasing polymer length (Fig. 8c) [23, 26].

ELPs can be produced via chemical synthesis and biosynthetically. For chemical synthesis via solid phase peptide synthesis, the attainable polymer length is limited, and if long polymers with a defined length are required then the biosynthetic approach is more appropriate. An advantage of chemical synthesis is, however, that it enables the facile introduction of functional residues in the polypeptide [27]. 

Diffusion of flexible macromolecules in solutions and gel media has also been studied extensively [35,97]. The Zimm model for diffusion of flexible chains in polymer melts predicts that the diffusion coefficient of a flexible polymer in solution depends on polymer length to the 1/2 power, D N. This theoretical result has also been confirmed by experimental data [97,122]. The reptation theory for diffusion of flexible polymers in highly restricted environments predicts a dependence D [97,122,127]. Results of various... 

A simple scaling law has been postulated to define the relationship between polymer length and Rg under various solvent conditions (Flory, 1953) ... 

Mann A, Richa R, Ganguli M (2008) DNA condensation by poly-L-lysine at the single molecule level role of DNA concentration and polymer length. J Control Release 125 252-262... 

PEG-Amine Derivative (Jeffamine Series from Texaco Various Polymer Lengths Available)... 

For low-Reynolds-number fluids the second term in the right-hand side of the Navier-Stokes equation can be neglected. Additionally, assuming that the viscous relaxation occurs more rapidly than the change of the order parameter, the acceleration term in Eq. (65) can be also omitted. Such approximations are validated in the case of polymer blends, for which they become exact in the limit of infinite polymer length, N —> oo. After these approximations, the NS equation can be easily solved in the Fourier space [160]. 

Fig. 14. A,B. A living polymerization of mannose-substituted norbornene derivatives can be used to produce materials of defined lengths for biological testing. B The relationship between polymer length and biological activity was explored...

Relationship Between Polymer Length and Binding Enhancement... 

The polymer self-assembly theory of Oosawa and Kasai (1962) provides valuable insights into the nature of the nucleation process. The polymerization nucleus is considered to form by the accretion of protomers, but the process is highly cooperative and unfavorable. Indeed, this is strongly suggested by the observation that thousands of actin or tubulin protomers are found in F-actin and microtubule structures if nucleation of self-assembly were readily accomplished and highly favorable, the consequence would be that many more fibers of shorter polymer length would be observed. The Oosawa kinetic theory for nucleation permits one to obtain information about the size of the polymerization nucleus if two basic assumptions can be satisfied in the experimental system. First, the rate of nuclei formation is assumed to be proportional to the loth power of the protomer concentration with io representing the number of protomers required to create the nucleus. Second, the treat-... 

This theory clearly predicts that the shape of the polymer length distribution curve determines the shape of the time course of depolymerization. For example Kristofferson et al. (1980) were able to show that apparent first-order depolymerization kinetics arise from length distributions which are nearly exponential. It should also be noted that the above theory helps one to gain a better feeling for the time course of cytoskeleton or mitotic apparatus disassembly upon cooling cells to temperatures which destabilize microtubules and effect unidirectional depolymerization. Likewise, the linear depolymerization kinetic model could be applied to the disassembly of bacterial flagella, muscle and nonmuscle F-actin, tobacco mosaic virus, hemoglobin S fibers, and other linear polymers to elucidate important rate parameters and to test the sufficiency of the end-wise depolymerization assumption in such cases. 

The equilibrium distribution of polymer length can be determined by recalling [see Eqs. (25) and (29)] that Kci asymptotically approaches unity from below as co approaches infinity. The equilibrium between MTj and MT,+i is... 

Because Kc is, however, slightly less than unity, and even though no measurable difference is discernible between the concentrations of two adjacent polymer species (e.g., MTj and MT,+i), an overall exponential decline should be observed over a sufficiently broad range of polymer length. This is clearly evident when, for c,+i/c, equal to X (where X is Kc ), one considers the case for X slightly less than unity. From the model for Eq. (26), we then obtain... 

To the left of the peak where the tubules have shorter lengths, c, i is less than Cii so the net flux is from i-mers to (i — l)-mers. To the right of the peak, the distribution of polymer length falls off, and c, i is greater than Cj. Therefore, the net flux will be in the opposite direction. The combined action of these fluxes wiU result in the broadening of the peak distribution. Eventually, the peak will completely disappear due to the relationships among the concentrations of each polymer species. In this respect, the initial polymer-protomer equilibrium is maintained by the balanced rates of protomer addition and loss from polymer ends, and the protomers will scramble or diffuse from one polymer to another. Indeed, even after the polymer length redistribution reaches its thermo-... 

Fig. 8. Experimental length distribution changes for bovine brain microtubule protein at 1 and 4 hours after initiation of assembly. Plots A and B correspond to unsheared samples, and give average length values of 11.3 and 12.1 /um, respectively. Plots C and D correspond to the sheared samples at 1 and 4 hours after assembly was induced, and polymer length values averaged 3.1 and 5.2 jam, respectively. [Reproduced from Kristofferson and Purich (1981). Arch. Biochem. Biaphys. 211, 222-226.]...

Fig. 3 Schematic illustrating a conformations of stuface-anchored polymers, and polymer brush assemblies with a b grafting density gradient, and c a gradient in polymer length. Part d depicts polymer conformations on a substrate comprising grafting density and polymer length orthogonal gradients...

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