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 partition function as a function of temperature and, hence, a discontinuity in the population of ordered states. End effects in an ordered sequence in one dimension (the analog of surface effects in higher dimensions) are equivalent to the case of α=1; hence, as in polypeptides, one-dimensional systems with end effects show no true discontinuities.

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 (Iii) lnu;=a-(Iii) (b+clni). This quantity becomes large, as i is increased, slower than Iii because of the c Ini 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 aU(x)lax converge at the point where Xl (the unit partition function) equals u 2 (the statistical weight per unit of an infinite loop, i.e., a free chain). If 2>c> 1, then U(x) converges at Xl =u 2 but au(x) lax does not. For a loop in three dimensions, c would be about! (1 for each dimension) j hence, there is no discontinuity in (J, the fraction of ordered states, for a real system. However, (J does behave nonanalytically at Xl = u 2 , giving a continuous or higher-order transition. Specifically, (J does go exactly to zero in the transition region. This is a qualitative difference from infinite polypeptides where the sigmoidal transition curve goes asymptotically to 1 and 0 at the extremes of temperature. Explicit calculations are given for various values of c to illustrate the behavior of the model.

SynopsisThe problem of calculating detailed probability profiles giving the probability of each unit in the chain to be in the ordered state (and all other average quantities as well including the fraction of strand association) for specific-sequence macromolecules requiring statistical weights that correlate up to the total number of units in the chain (e.g., DNA, collagen) is formulated in terms of recursion relations for appropriate a priori and conditional probabilities, thus generalizing the approach of Lacombe and Simha for nearest-neighbor correlations in specific sequence macromolecules. The technique allows the probability profiles for chains of thousands of units to be calculated in minutes making no approximations in the basic model. 1859

synopsisThe theories of the helix-coil transition of Lifson, Lifson and Allegra, and Allegra for random copolymers are compared with the exact results of Lehman and McTague, as was done by Fink and Crothen in an earlier test of the approximate theories of Fixman and Zeroka; Reiss et al, and Montroll and Goel. The Fixman-Zeroka theory is the best approximation to the exact result (under the conditions used), and the Allegra theory is next best, the calculated slope of the transition curve in the Allegra theory being too small by a factor of two. A n alternative derivation of the Allegra approximation, using the method of sequence generating functions, is given, and the resulting equations are of very simple form. A hybrid method, based on the work of Lifson and Allegra, is also developed here; it involves a technique of successive approximations, and reduces to the Allegra theory in first order. The fourth-order approximation gives a slope that is too small by 10%; however, the value of the slope, extrapolated to infinite order of approximation, converges to the exact result of Lehman and McTague. By using illustrative calculations of helix-coil transition curves, some physical insight into the behavior of a copolymer in the transition region is provided; it is found that an important feature determining the shape of the transition curve is the variation in composition over the correlation length. The question of the application of copolymer theory to DNA is discussed.

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