The transition between the helical and randomly coiled forms of a polypeptide chain is discussed by reference to a simple model that allows bonding only between each group and the third preceding Two principal parameters are introduced, a statistical parameter that is essentially an equilibrium constant for the bonding of segments to a portion of the chain that is already in helical form, and a special correction factor for the initiation of a helix A third parameter which specifies the minimum number of segments in a random section between two helical portions has only a minor effect on the results The partition function for this model is handled in two alternative ways, either as a summation suitable for short chains, or in terms of the eigenvalues and eigenvectors of a characteristic matrix; the latter is more suitable for long chains A transition from the random to the helical form is encountered as either the bonding parameter or the chain length is increased The critical value of the bonding pa

Theory of the Phase Transition between Helix and Random Coil in Polypeptide Chains

The two-dimensional Ising model for a system of interacting spins (or for the ordering of an AB alloy) on a square lattice is one of the very few nontrivial many-body problems that is exactly soluble and shows a phase transition. Although the exact solution in the absence of an external magnetic field was first given almost twenty years ago in a famous paper by Onsager1 using the theory of Lie algebras, the flow of papers on both approximate and exact methods has remained strong to this day.2 One reason for this has been the interest in testing approximate methods on an exactly soluble problem. A second reason, no doubt, has been the considerable formidability of the Onsager method. The simplification achieved by Bruria Kaufman3 using the theory of spinor representations has diminished, but not removed, the reputation of the Onsager approach for incomprehensibility, while the subsequent application of this method by Yang4 to the calculation of the spontaneous magnetization has, if anything, helped to restore this reputation.

Two-Dimensional Ising Model as a Soluble Problem of Many Fermions

History of the Lenz-Ising Model

(1960). On the theory of cooperative phenomena in crystals. Advances in Physics: Vol. 9, No. 35, pp. 245-361.

On the theory of cooperative phenomena in crystals

We show that given certain plausible assumptions the existence of persistent states in a neural network can occur only if a certain transfer matrix has degenerate maximum eigenvalues. The existence of such states of persistent order is directly analogous to the existence of long range order in an Ising spin system; while the transition to the state of persistent order is analogous to the transition to the ordered phase of the spin system. It is shown that the persistent state is also characterized by correlations between neurons throughout the brain. It is suggested that these persistent states are associated with short term memory while the eigenvectors of the transfer matrix are a representation of long term memory. A numerical example is given that illustrates certain of these features.

The existence of persistent states in the brain

An approximate method of computing the partition function of a binary solid solution is formulated. If the only constraints are on the mean energy and mean composition, it is shown that no metastable phase is predicted. The partition function is shown to be obtainable from the largest eigenvalue of a quadratic form and the condition for a phase transition is shown to be related to the degeneracy of the largest eigenvalue. The physical interpretation of the eigenfunction is shown to be related to the probability of surface configuration, while the square of the eigenfunction is related to the probability of a configuration on the interior of the crystal. Some simple examples are discussed which are related to the effect of coordination number on phase transitions.

Thermodynamic Properties of Binary Solid Solutions on the Basis of the Nearest Neighbor Approximation

The ultraviolet absorption spectrum of benzonitrile vapor has been investigated, and many of the observed bands have been accounted for in terms of ground and excited state frequencies. The data obtained were compared with the data available from infra‐red absorption spectra, the Raman effect, and early work on ultraviolet absorption spectra. The spectrum of benzonitrile vapor was shown to resemble closely the spectra of other monosubstituted benzenes, particularly in the spacing of the excited state bands from the zero‐zero band.

The Ultraviolet Absorption Spectrum of Benzonitrile Vapor

The absorption spectrum of methane from 1450A to 850A is found to be entirely continuous and all excited states up to the first ionization potential are repulsive

The Ultraviolet Absorption of Methane

A previously developed method of calculating the thermodynamic functions of a mixture is further examined. For a lattice which is large in at least two dimensions it is shown that the largest eigenvalue of the fundamental matrix is doubly degenerate over a finite temperature range, and within this range a separation into phases takes place. An approximation to the phase boundary which is valid at low temperatures is obtained.

Thermodynamic Properties of Binary Solid Solutions. Phase Separation

A method is given for integrating the equations for the rate of polymerization reactions without the assumption of steady states for the concentrations of the various free radicals. The main reaction steps used are: free radical production, chain propagation, chain transfer, and chain termination. Explicit expressions are found for the total amount of polymer, the number average, and the weight average as a function of time for the particular case of free radicals produced initially by a pulse of light. Other modes of dependence on time for the primary free radicals may also be treated.

John P. Howe

Papers

Thermodynamic Properties of Binary Solid Solutions on the Basis of the Nearest Neighbor Approximation

The Ultraviolet Absorption Spectrum of Benzonitrile Vapor

The Ultraviolet Absorption of Methane

Thermodynamic Properties of Binary Solid Solutions. Phase Separation

Method of Integrating the Rate Equations for Free Radical Initiated Polymerizations