J. Phys. Chem. B 2001, 105, 10111-10114
10111
A Simple Method for Evaluating Partition Functions of Linear Polymers Yong Kong* CuraGen Corporation, 555 Long Wharf DriVe, New HaVen, Connecticut 06511 ReceiVed: May 8, 2001; In Final Form: July 17, 2001
A simple method for evaluating partition functions of linear systems is presented. The method produces the generating function of the partition functions of systems of all sizes. The method keeps the simplicity of Lifson’s sequence-generating function method. Unlike the sequence-generating function method, which only applies to infinite systems, the method presented here applies to both finite and infinite systems. Explicit expression of the partition functions can be readily derived directly from the generating function. The method provides a simple protocol to handle a large range of models that come out in studying cooperative phenomena in physical chemistry and biophysics.
I. Introduction The properties of a macroscopic system, which is made of a collection of independent small systems, can be obtained directly by studying the small systems themselves.1,2 These small systems, such as individual polymer molecules, in turn may consist of many repeating units themselves. It is usually difficult to calculate the properties of such systems directly from the first principle by using quantum mechanics methods; instead, a small number of accessible states, which are the averages of many quantum eigenstates, are specified for each unit in the system. Different combinations of the states of each unit determine the configurations of the system. The observable properties of the system are then calculated by statistical mechanics methods, which assert that the system fluctuates around all possible configurations, and the experimental observables are the averages over all these configurations. The basic step in these calculations is the evaluation of the grand partition function (PF) of the small system, which is the sum of all the enumerations of the statistical weights of the configurations. When both the number of the units in the small system and the number of accessible states for each unit are small, combinatorial arguments can usually lead directly to the desired PF; when the system becomes complex, lengthy and ingenious methods have to be used to handle the combinatorial problems involved. The transfer matrix method provides a systematic approach for obtaining the PFs, and is especially useful for finite systems, but the order of the matrix is high for complicated models.1,2 The sequence-generating function (SGF) method,3 which uses one “sequence generating function” for each accessible state of the units, uses smaller matrixes than those used in the transfer matrix method. The SGF method was also derived through a different route by using sequence conditional probabilities.4 The drawback of the SGF method is that it only applies to systems containing an infinitely large number of units. In this article, we present a simple method that keeps the simplicity of SGF method, but is applicable to both finite and infinite systems. Instead of studying one PF of a system with a * E-mail:
[email protected].
given size at one time, the method creates a generating function (GF) that contains the PFs for systems of any size. Hence, when the number of the units in the system is small, its PF can be obtained simply from the series expansion of the GF; for long systems composed of many units, standard asymptotic methods exist to obtain approximations to the PFs from the GF. The SGF method turns out to be the asymptotic approximation to the method described here for very long systems. Furthermore, by manipulating the GF, we can usually obtain directly an explicit expression of the PFs for systems with arbitrary sizes, without the combinatorial arguments used by other methods. In section 2, the basic method is described. The basic method is simplified in section 3 for the models where the interactions between neighboring units can be grouped into a hierarchy. In this case the method can be applied recursively, reducing the sizes of the matrixes involved considerably. In section 4 a few examples are provided to illustrate the use of the basic method and its simplification. II. Basic Method Let’s denote fn(x) as the PF of a polymer with n units. For the sequence of functions {fn(x), n ) 0, 1, ...}, the ordinary GF is defined as the sum in terms of a parameter z, ∞
G(x, z) ) f0(x) + f1(x)z + f2(x)z2 + ‚‚‚ )
fn(x)zn ∑ n)0
(1)
Our purpose is to derive a general formula for G(x, z). From the definition of eq 1 it is clear that G(x, z) represents the PFs for systems of all sizes. From G(x, z) we can obtain PF for a system with a given size n as the coefficient of zn. The variable x, which represents all variables other than z, will be omitted in the following. Without losing generality, suppose that each unit of the system can have three states, a, b, and c. Each unit of the system can take one of these three states, and the interactions between the units can also be specified by the model under investiga-
10.1021/jp011758n CCC: $20.00 © 2001 American Chemical Society Published on Web 09/19/2001
10112 J. Phys. Chem. B, Vol. 105, No. 41, 2001
Kong
tion. For example, a system with 12 units can take a configuration as
aabbbcbbaccc
[
-gawab -gawac 1 -gbwbc -g w M) 1 b ba -gcwca -gcwcb 1
(2)
Any configuration of the system, like the one shown above, can always be separated into distinct block(s), with the unit(s) in each block in the same state,
aa|bbb|c|bb|a|ccc
(3)
The example given above has a block pattern with a length of 6
ABCBAC
ga(z)wabgb(z)wbcgc(z)wcbgb(z)wbaga(z)wacgc(z)
(5)
where ga(z), gb(z), and gc(z) are the GFs for a block of unit(s) in the state a, b, and c, respectively. In the following we use G(z) to denote a GF that includes the empty system (n ) 0), and g(z) to denote the corresponding GF for systems that are at least one unit long. The definition of g(z) is modified slightly from G(z) in eq 1 as ∞
fin(x)zn, i ) a, b, c ∑ n)1
(6)
The wab, etc. in eq 5 specify the interactions between units in different states. [The interactions between units in the same state are included in gi(z).] To get a GF for all the configurations, all block patterns such as eq 4 should be considered. This can readily be achieved through a matrix
[ ]
0 A A N) B 0 B C C 0
(7)
[1,1,1]Np[A, B, C]t
(8)
The matrix product
contains all the patterns with a length of p + 1. All the block patterns with lengths greater than 1 are given by
[1,1,1](N0 + N1 + N2 + ‚‚‚)[A, B, C]t ) [1,1,1]M-1[A, B, C]t (9)
[
]
(11)
Gabc(z) ) 1 + [1,1,1]M-1[ga, gb, gc]t ) 1 + gabc(z)
(12)
Generally, if each unit has p states, we have
G(z) ) 1 + [1, 1, ..., 1]M-1[g1, g2, ..., gp]t where
[
-g1w12 -g1w13 1 -g2w21 -g2w23 1 M) ·· ·· ·· · · · -gpwp1 -gpwp2 -gpwp3
... -g1w1p ... -g2w2p ·· ·· · · ... 1
(13)
]
(14)
III. Simplification of the Method Equations 13 and 14 provide a general method to get the GF of PFs of any model, as long as the interactions between neighboring units are restrained to the nearest neighbors. From eq 14 we see that the size of the matrix used is equal to the number of states for each unit. In this section we point out that the size of the matrix can be reduced further if the interactions between different units have certain special properties. In these cases, eqs 13 and 14 can be applied recursively to subsets of states of the systems, which makes it possible to use much smaller matrixes in each recursive step. As shown below, these interaction properties are often satisfied in physical chemistry and biophysics models. Suppose that, in the example above, both states b and c have the same interactions with state a, i.e., wab ) wac and wba ) wca.. We can first calculate gbc(z) by the method above, and then use gbc(z) and ga(z) to get the final GF for the whole system. In general, if the states of each unit can be separated into distinct groups R, β, γ, ..., with the states inside each group having the same interactions with the states in the other groups, the GFs of each group, gR(z), gβ(z), gγ(z), ..., can be obtained separately by using eq 13. The GF for the whole system can then be obtained by using eq 13 again. Specifically, if there are only two groups, R and β, and there is no interaction between these two groups, the GF for the system is given by
[
1 -gR GRβ(z) ) 1 + [1,1] -g 1 β
where
1 -A -A 1 -B M ) I - N ) -B -C -C 1
]
Taking the PF of an empty system (n ) 0) as identity, the GF of the systems of any size is given by
(4)
All the configurations that can be partitioned into a pattern like eq 4 (these configurations come from systems with all possible number of units) are enumerated by the following GF
gi(z) ) fi1(x)z + fi2(x)z2 + ‚‚‚ )
between units in different states, we obtain the matrix
)
]
-1
[gR,gβ]t
(15)
GRGβ GR + Gβ - GRGβ
(10)
Substituting ga, gb, and gc for A, B and C in eqs 9 and 10, and putting in the appropriate factors specifying the interactions
In the lattice gas models in statistical mechanics and ligand binding problems in molecular biology, one of the states is usually the empty state, whose statistical weight is usually assigned as identity, and this empty state does not interact
Evaluating Partition Functions of Linear Polymers
J. Phys. Chem. B, Vol. 105, No. 41, 2001 10113
with its neighbors. The GF of a block unit(s) in empty state is simply given by
1 GR(z) ) 1 + z + z + ‚‚‚ ) 1-z 2
Gβ 1 - zGβ
(17)
Hence, if there are only two states for each unit, one of which is the empty state, we even do not need the matrix formula of eq 13; eq 17 is sufficient to obtain the GF for the PFs of the system. IV. Applications Several applications of the general method or its variations will be presented to illustrate the simplicity of the method. A. One-Dimensional Model with Cooperativity. In this model the ligand covers one binding site, and there are interactions between ligands when they bind to adjacent sites. Two parameters specify the model: σ, the unit-less cooperativity parameter, which specifies the nearest-neighbor interactions of bound ligands; and ω ) Kx, the product of the intrinsic binding constant K and the ligand activity x. This is the latticegas version of the one-dimensional Ising model. To solve the model with the method presented here, we notice that there are only two states for each binding site: one is the empty state and the other is the bound state, and the empty state does not interact with the bound state. Equation 17 can be used here, with ∞
Gβ(z) ) 1 +
σn-1ωnzn ) ∑ n)1
1 - σωz + ωz 1 - σωz
(18)
By using eq 17, we obtained directly
G(z) )
1 - (σ - 1)ωz 1 - (1 + σω)z + (σ - 1)ωz2
(19)
If σ ) 1 (no cooperativity),
G(z) )
∞
1 1 - (1 + ω)z
)
(1 + ω)nzn ∑ n)0
we get the trivial solution of fn ) (1 + ω)n. In general, G(z) can be expressed as G(z) ) c1/(1 - a1z) + c2/(1 - a2z) ) c1∑n a1nzn + c2∑n a2nzn, where a1 and a2 are the two roots of z2 - (1 + σω)z + (σ - 1)ω ) 0, and c1 ) (a1 - (σ - 1)ω)/(a1 - a2), c2 ) (a2 - (σ - 1)ω)/(a2 - a1). The closed-form PF is given by fn ) c1a1n + c2a2n. B. Multivalent Binding. In a variety biological binding problems, the ligand usually covers more than one elementary binding site. The problem is usually modeled by three thermodynamic parameters: in addition to ω and σ mentioned above, the third parameter m is used to specify the number of binding sites covered by a bound ligand. Because now each unit covers m binding sites, the z in GF of a block of units in bound state should be changed to zm, ∞
Gβ(z) ) 1 +
∑σ
n)1
n-1
n mn
ωz
)
1 - σωzm + ωzm 1 - σωzm
G(z) )
(16)
If we substitute this as GR in eq 15, we obtain
G(z) )
With use of eq 17 the GF of the model can be obtained as
(20)
1 - (σ - 1)ωzm 1 - z - σωzm + (σ - 1)ωzm+1
(21)
which is the same as eq 23 of ref 6 obtained by using the recurrence relation obeyed by the PFs. C. Helix-Coil Transition of Peptide Chains. The helix-coil transitions in biopolymers have been studied extensively in the past (for review, see ref 1). This is also the first example given by Lifson when introducing the SGF method.3 As did Lifson, we also use the Lifson and Roig model.5 In this model each unit can take one of two states, either the random (coil) state or the helical state. A unit in the coil state contributes a factor u to the statistical weight in PF. A unit in helical state contributes a factor V if its neighbor is in a coil state, and a factor w if it is in the middle of a helical sequence. Because we only have two states for each unit, and the neighbor units do not have explicit interactions, we can use eq 15 directly. The GF for a block of ∞ unzn ) 1/(1 sequence in coil state is given by GR(z) ) ∑n)0 uz), and the GF for a block of sequences in helical state is given by ∞
Gβ(z) ) 1 + Vz + V2z2
∑
wnzn ) 1 + Vz +
n)0
V2z2 1 - wz
(22)
Using eq 15, we can directly obtain the GF for the system as
G(z) )
1 - (w - V)z - V(w - V)z2 1 - (u + w)z + u(w - V)z2 + uV(w - V)z3
(23)
The side-chain interactions can also be easily incorporated by introducing another factor s in the GF for the helical sequences (eq 22).3 If u is set to 1, which is usually the case, an even simpler method to obtain the GF of eq 23 is to use eq 17 directly. D. Binding on a Ladder Model. The method can be used to study all the linear models that are built up by replicating a basic unit. The basic unit can be a single binding site, as illustrated above, a linear chain, or a plane.7 Here we use the ladder model8 as an example. The ladder model extends the traditional one-dimensional linear model by introducing another cooperativity parameter τ to specify the interactions between the ligands bound in the two legs of the ladder lattice. Each rung of the ladder is considered as a basic unit here, and each unit can have four states: {00}, {10}, {01}, and {11}, with {00} as the empty state. The GFs for a block of units in one of the three bound states are given as ∞
g{10}(z) ) g{01}(z) ) ∞
g{11}(z) )
∑
n)1
ωz
σn-1ωnzn ) ∑ 1 - σωz n)1
τnσn-1ω2nzn )
τω2z 1 - τσω2z
(24)
If we assign the three bound states {10}, {01}, and {11} to states a, b, and c in eq 12, with wab ) wba ) 1 and wac ) wca ) wbc ) wcb ) σ, we obtain the Gβ(z) for the bound states. Using eq 17 to take account of the empty state, we can get eq 35 of ref 8
10114 J. Phys. Chem. B, Vol. 105, No. 41, 2001
G(z) )
Kong
1 - ω(σ - 1)(ωτσ + 1 + ωτ)z + τ(σ - 1)3ω3z2 1 - (σω + σ2τω2 + ω + 1)z + ω(σ - 1)(σ2τω2 + ωτσ + 1 + ωτ)z2 - τ(σ - 1)3ω3z3
V. Discussion The method discussed here is similar to Lifson’s SGF method.3 As in the SGF method, we consider a block of unit(s) in the same state and decompose the combinatorial problems of calculating PFs into a simpler problem of enumerating these blocks and their interactions; If we substitute z in eq 6 with x-1, gi(z) will have the same definition of Lifson’s sequencegenerating functions. SGF method ends up with solving the secular equation
|I - N| ) |M| ) 0
(26)
the largest root of which determines the PF when the length of the system is large. This turns out to be the asymptotic expression of eq 13. Because M-1 ) M(*)/| M|, where M(*) is a matrix made up of the cofactors of the elements of the matrix M, we see that the denominator of G(z) is equal to |M|. The asymptotic behavior of the GF is determined by the smallest zero of the denominator of the GF, which is equal to the largest root of eq 26 if z is substituted by x-1. Unlike the SGF method, which only applies to long systems, the method introduced here applies to systems of any size. The GF includes all the PFs in one expression. With the modern mathematical packages such as Maple and Mathematica, a few lines of commands will give all the results for the examples presented earlier. The expansion of the GF to obtain the individual PF (the coefficient of zn) can also be calculated very quickly for small systems. These PFs of short systems are not only of interest themselves; they can also provide initial conditions for the recurrent method, where the order of the recurrence relation obeyed by the PFs is reduced by symmetry.7 In addition, by manipulating the GF, explicit expression of PF for a system of arbitrary size can be obtained, usually in a form
(25)
of sum of binomial terms, or even in a closed-form. This is achieved without using the combinatorial arguments that are often used by other methods. The method presented here applies to linear systems. Linear systems actually have simpler PFs than circular systems, although the symmetry is broken in linear systems because of the edge effects from the ends.7 The order of recurrence of the PFs of a linear system is lower than that of the corresponding circular system with periodical boundary conditions. Other commonly used methods for evaluating PFs, such as the transfer matrix method, are usually more complicated when applied to linear systems. The linear systems are often approximated by circular ones to use these methods. The method described here eliminates these unnecessary approximations. Because of the combinatorial nature of the PFs, the methods described here can readily find applications in fields other than statistical mechanics and biophysics, where the same combinatorial approaches are heavily used. Some examples are the runs and run-related statistics, the study of word and pattern, and context-free language. In general, the method applies to any system that can be divided into blocks containing only one of the basic states. References and Notes (1) Poland, D.; Scheraga, H. A. Theory of Helix-Coil Transitions in Biopolymers; Academic: New York, 1970. (2) Hill, T. L. CooperatiVity Theory in Biochemistry; Springer: New York, 1985. (3) Lifson, S. J. Chem. Phys. 1964, 40, 3705. (4) Poland, D. J. Chem. Phys. 1974, 60, 808. (5) Lifson, S.; Roig, A. J. Chem. Phys. 1961, 34, 1963. (6) Di Cera, E.; Kong, Y. Biophys. Chem. 1996, 61, 107. (7) Kong, Y. J. Chem. Phys. 1999, 111, 4790. (8) Kong, Y. Biophys. Chem. 1999, 81, 7.
