Douglas C. Poland

Phase transitions in one dimension and the helix-coil transition in polyamino acids.

Phase transitions in one dimension are discussed from the point of view of order—disorder transitions in linear polymers using the formalism of sequence generating functions due to Lifson. If the statistical weight vj of an ordered sequence of j units has the form (lnvj)/j=a—bj−α, then a phase transition occurs when 0<α<1. As demonstrated by Fisher, this will occur, for example, if certain many‐body interactions of short range are introduced. This behavior is obtained if we consider long‐range pair potentials of the form 1/r1+α, for 0<α<1, occurring between all the units of an ordered sequence. An exponential potential, e−γx, gives a phase transition in the limit γ→0 if interactions are restricted to the units in an ordered sequence. The occurence of a phase transition arises from the convergence of the sequence generating function and its first derivative at the value of the unit partition function equal to the statistical weight of the ordered unit. This gives rise to a bend in the curve of the unit par...

Occurrence of a Phase Transition in Nucleic Acid Models

The double‐stranded helix found in nucleic acids suggests a model where order is represented by specific, one‐to‐one bonding between two infinite chains and where disorder arises from the formation of large loops. The statistical weight per unit in an unordered sequence (i.e., a loop) is given by (1/i) lnui=a—(1/i) (b+clni). This quantity becomes large, as i is increased, slower than 1/i because of the c lni term. From the previous paper, this suggests that a first‐order phase transition may take place. It is found that this happens if c>2, in which case both U(x) (the sequence generating function for loops) and ∂U(x)/∂x converge at the point where x1 (the unit partition function) equals u2 (the statistical weight per unit of an infinite loop, i.e., a free chain). If 2>c>1, then U(x) converges at x1=u2 but ∂U(x)/∂x does not. For a loop in three dimensions, c would be about 32 (½ for each dimension); hence, there is no discontinuity in θ, the fraction of ordered states, for a real system. However, θ does b...

Hydrophobic bonding and micelle stability; the influence of ionic head groups

Abstract Our previous theory of nonionic micelles involved three contributions to the overall free energy: the external free energy of the micelle (i.e., translational and rotational free energy), the free energy due to solvent effects around the hydrocarbon tails, and the internal free energy arising from the “loose” nature of hydrophobic bonds. To extend the theory to ionic micelles it is necessary to introduce an electrostatic free energy and a free energy arising from the solvation of the charged heads. It is argued that the original Debye concept of ionic micelle stability, arising from a balance of the positive free energy due to electrostatic repulsion between the charged heads on the surface of the micelle and the negative free energy of bringing the hydrocarbon tails together, is the major explanation of a stable micelle of large size, thus making the formation of ionic micelles easier to explain than the corresponding nonionic ones. It is shown how the addition of a term, representing the repulsion of charged heads, alters our previous treatment of nonionic micelles and, by means of the graphical procedure of that paper, how charged heads shift the most probable micelle size to smaller values. The Nemethy-Scheraga theory of the stability of hydrophobic bonds predicts that the critical micelle concentration should have a minimum at about 60°C. when one considers only the nonpolar portions of a micelle. In ionic micelles this minimum is observed at about 25°C., from which we argue that there must be a negative enthalpy corresponding to a change in a solvation process around the charged head on entering a micelle.

Kinetics of the Helix—Coil Transition in Polyamino Acids

The matrix theory of the helix—coil transition in polyamino acids is extended to give the initial rate of change of helical content when the system is suddenly perturbed from equilibrium. Using the matrix to treat chains of arbitrary length, it is possible to assign not only the equilibrium statistical weights for the initial probability of occurrence of each species but also the rate constants for all possible initial reactions involving the formation and breakdown of helical states. It is shown that all the rate constants can be related to the equilibrium statistical weights and to only one rate constant. It is found that reactions at helical ends dominate the initial rate, even though there are many more interior sites than there are ends. As a result, the initial rate of change of helical content goes through a maximum at the transition temperature. With the aid of some approximate expressions for the number of helical ends, it is possible to discuss the initial rate for all ranges of chain length. The technique used to compute the initial rate can also be applied to calculate the moments of the equilibrium distribution of lengths of helical sequences; this calculation is given in the Appendix.