6580
J. Phys. Chem. 1984,88, 6580-6586
Real-Space Renormalization Group Treatment of the Helix-Coil Transition in a Homopolyamino Acid Chain Zhenqin Li and Harold A. Scheraga*
Baker Laboratory of Chemistry, Cornell University, Ithaca, New York 14853 (Received: May 30, 1984)
A real-space renormalization group (RG) transformation is used to treat the Lifson-Roig model for the helix-coil transition in a homopolyamino acid. By utilizing the matrix formalism, we can obtain the eigenvalues of the statistical weight matrix with the RG method, without having to solve the secular equation. In the treatment of the finite chain, the availability of the eigenvalues leads directly to the partition function after carrying out the usual routine calculations of the eigenvectors and transformation matrix. The application of the RG method to the infinite chain is particularly straightforward. In this case, the relationship between the helix-coil transition and critical phenomena is established by recognizing that the transition point of this one-dimensionalsystem corresponds to a fixed point of the renormalization group transformation of the statistical weight parameters in the complex plane near the real axis (real part of In w,where w is a "growth" parameter), while true critical points in critical phenomena lead to fixed points on the real axis. This relationship provides a clear picture of the behavior of the system in the transition region. It can be generalized to define the cooperativity of a complex system, with a true phase transition being an extreme case of a cooperative phenomenon (in which the fixed points in the complex plane approach the real axis). The behavior of the thermodynamic functions at the transition point can be obtained by expansion of the RG equations around the transition point.
I. Introduction Conformational transitions in biopolymers, e.g., the helix-coil transitions in homopolyamino acids and polynucleotides, may be treated as a one-dimensional problem in which the interactions are of short range.'-3 As a consequence, matrix which treat nearest-neighbor interactions, have been applied to this problem. More complex systems, with long-range interactions, e.g., globular proteins, can no longer be considered to be onedimensional, and the use of a matrix method would involve massive calculations with a high-speed computer. It is therefore of interest to explore procedures that might be applicable to such complex systems. Recent advances in applying renormalization group (RG) theory to critical phenomena and other physical problems (for a review see ref 7) suggest that it may provide a powerful theoretical tool for the study of polymer systems as well. In fact, the excluded volume effect in polymers has already been treated successfully with RG and it is of interest to consider the application of this methodology to transitions involving a-helices, @-sheets, statistical coils, and higher-order conformations in proteins. The cooperativities exhibited in these transitions are reminiscent of those in true phase transitions in critical phenomena, and it is of interest to examine the connection a n d distinction between cooperativity and a phase transition. Before exploring the possibility of treating such conformational transitions in complex three-dimensional systems, we make an initial application of R G theory to a simple problem, the helix-coil transition in a homopolyamino acid. In this paper, we shall apply RG theory to the Lifson-RoigS model and show that the problem can be solved exactly for both the finite and infinite chains. This model was selected because of its explicit treatment of the hydrogen-bonding structure of the a-helix. The original Lifson-Roig theory for finite chains,5 and a sequence-generating function treatment thereof for infinite chains,I0 required the solution of a cubic secular equation. This Poland, D.; Scheraga, H. A. J . Chem. Phys. 1966, 45, 1456. Poland, D.; Scheraga, H.A. J . Chem. Phys. 1966, 45, 1464. Fisher, M. E. J. Chern. Phys. 1966, 45, 1469. Zimm, B. H.; Bragg, J. K. J . Chem. Phys. 1959,32, 526. Lifson, S.; Roig, A. J. Chem. Phys. 1961, 34, 1963. (6) Poland, D.; Scheraga, H. A. "Theory of Helix-Coil Transitions in Biopolymers"; Academic Press: New York, 1970; pp 25, 41, 171, 298-299, (1) (2) (3) (4) (5)
3 12-3 13. (7) Wilson, K. Reo. Mod. Phys. 1983, 55, 583. (8) Oono, Y.; Freed, K. F. J . Chem. Phys. 1981, 74, 6458; 75, 993. (9) Oono, Y.; Freed, K. F. J. Phys. A: Math. Gen. 1982, 25, 1931.
0022-3654/84/2088-6580$01.50/0
difficulty is avoided by the RG method. 11. RG Treatment of the Lifson-Roig Model
Lifson-Roig The Lifson-Roig model involves a Hamiltonian (analogous to the king model) of the form % = -JCn,-ln,ni+l I
+ HCni i
(1)
with
w = exp[(J - H ) / k T ] v = exp[-H/kT]
(2) (3)
where -J is the energy of formation of a hydrogen bond, and H (>O) arises from the decrease in entropy in converting a coil (c) to a helical (h) state; ni (= 0 or 1) indicates that residue i is in a c or h state, respectively. Lifson and Roig used a matrix method to obtain the following expression for the partition function in a chain of N residues: ZN = uwN-2u+
(4)
u = (u, u, 1) = (0, 0, l)W = eW u+ = (u, 1, u 1)' = W(0, 1, 1)' = We+
(5)
where
+
(with the statistical weight of a coil state taken as 1) and
t :3
w=o
0
1
By computing the eigenvalues and eigenvectors of the matrix W, Lifson and Roig obtained an expression for Z, in terms of the three eigenvalues of W. This requires the solution of a cubic secular equation. RG Approach. The real-space RG approach involves recursive grouping of the residues into blocks so that each block serves as an effective residue with effective statistical weights. Such groupings in real space represent a particular example of the general RG transformation. Of course, the physics of the problem must not be changed by such groupings. Various approximate methods, making use of such groupings, have been used in the past. For example, the "coarse-graining" procedure6," was used to treat the helix-coil transition in poly(10) Lifson, S. J . Chem. Phys. 1964, 40, 3705. (11) Crothers, D. M.;Kallenbach, N. R. J . Chem. Phys. 1966, 45, 917.
0 1984 American Chemical Society
R G Treatment of Helix-Coil Transition
The Journal of Physical Chemistry, Vol. 88, No. 26, 1984 6581
peptides and polynucleotides by grouping successive residues into blocks. However, such blocks were not assumed to increase in size, iteratively, and hence this procedure does not have the utility of the R G method in statistical mechanical calculations. The power of the RG approach lies in iterative RG transformations, or in iterative groupings in our context, thereby dramatically reducing the number of degrees of freedom in the problem. Many real-space RG methods have been developed in obtaining approximate RG transformations; I 2 , I 3 these are sometimes also designated as “coarse-graining”, although they are of quite a different nature from that of ref 6 and 1 1 . The dedecoration procedure constitutes one of the early exact RG treatments of one-dimensional systems (in particular, the king model),14 but its application to systems with longer-range interactions becomes increasingly difficult. The method developed below exploits the matrix formalism to obtain the RG transformations. It is, in principle, applicable to any one-dimensional statistical-mechanical two-state problem, e.g., the h and c states of each residue of a homopolyamino acid. As we shall see later, there are many ways of carrying out the R G transformation with the matrix formalism, and all give the same result for the free energy. The particular method of choice depends on the nature of the given problem, with the selection criterion being based on simplicity and ease of analysis. In the remainder of this section, we shall present one procedure to solve this problem. Other (more general) methods to construct the RG transformation for one-dimensional systems are discussed in section 111; these can be applied to models with longer-range interactions. In this problem, we group three residues into a block, with the block being described by an effective statistical weight matrix W’.
2
1
. e I -
o
Y
3 C
-J
-0.50v
I
I
I
I
1
I I
J
I
0.15
0
-0.15
Ln w Figure 1. Plot of In w’vs. In w ,after one iteration, for a given initial value of v = 0.0141. The upper and lower curves correspond to the positive and negative sign, respectively, in eq 12. It can be seen that w N 1 is close to being a fixed point, Le., close to being a true critical point. Here, and in Figure 2, the transition occurs in the region where w N 1 u or (since u
