0README
zimm_bragg-helix_coil
|-- 0-Old directory - old files
|-- 0README obvious
|-- comall.s R commands: plot zimm-bragg results
|-- comall.ps PS file of output from comall.s
|-- setup.s R commands - source first
`-- zimm.fn.s R commands - read in by setup.s
PHASE TRANSITIONS
Changes in state have been continuing objects of study for the
fields of chemistry, statistical physics, condensed matter physics,
and more recently, biology, and by extension, the social sciences.
The van der Waals equation of state is taught to undergraduates;
the unusual physics of the critical region, to graduate students.
The Ising model is perhaps the central concept for the understanding
of phase transitions. The properties of a phase transition constitute
a definition of complexity: correlation (cooperativity); nonlinearity
(explosive growth); emergence of a new structure (a new phase of differ-
ent order); power law behavior (universality). Biology is rife
with phenomena that can be viewed as phase transitions. A particular
example is the helix-coil transition considered here.
Under the Ising model, a material is viewed as a regular lattice,
with interaction between a lattice site and its neighbors. For the
Ising model of a ferromagnet, each site has a spin, with a favorable
energy of interaction for a neighbor site of the same spin, and an
unfavorable energy for a neighbor of opposite spin. This simple
model, with few variable parameters, reproduces the principal
features of the physics of a ferromagnet: the transition from
disorder to order (magnet, with spins aligned), at a critical
temperature; singularities in some properties (e.g., the heat
capacity); power-law behavior of some properties, related to the
distance over which spins are correlated; and more.
Some phenomena fit a related lattice model, peroclation, for which,
although there is no energy of interaction of a site with a neighbor,
adjacentcy and cooperativity enter through the appearance of a
network linking all regions of the material at the critical point.
The physics of the Ising model was developed largely during the
1950's. In 1959, Zimm and Bragg applied the one-dimensional Ising
model to the unfolding of a protein -- the helix-coil transition.
Several other authors similarly used this model, at about the same
time. The Zimm-Bragg method and nomenclature has stood time's test,
in that experimentally-accessible parameters defined in the theory,
most particularly "s", are sometimes reported in characterizing a
polymeric material.
In 1959, computations were still largely done by hand. Zimm and
Bragg (J. Chem. Phys., 31, 526, 1959) gave relatively few graphs
or tables showing the properties of a helix-coil system for varied
values of the three variable parameters. Now, computer evaluation
for some particular set of parameters is easy.
R functions and an R script for generating plots of the Zimm-Bragg
results for varied parameter values are given here. These can be
used as a teaching tool, or just for fun.
The library "akima" must be part of or have been added to the local
R software for these routines to work.
R FUNCTIONS FOR THE HELIX-COIL TRANSITION
The file "zimm.fn.s" has the two R functions, "dlnqdz" and "newmat"
(and a few others that crept in but are not used).
The file "setup.s" attachs the library "akima", sources the file
"zimm.fn.s", and uses the R functions of this file to create several
data objects "aaa.sigma", where "sigma", the initiation parameter,
has various values, e.g., 0.0001, 0.001, etc. The data objects are lists of
various properties of the transition, for the fixed value of "sigma" and for
ranges of values of "s", the helix-coil equilibrium constant, and "n", the
number of residues in the helix.
The file "comall.s" generates 7 plots, in the file "comall.ps".
All are for values of some property shown in a contour or perspective
plot over the "n"-"s" plane (ranges of values for "n" and "s") and
for a fixed value of "sigma".
The properties shown are:
theta fraction coil state
nu number of coil regions per chain
d theta/d lns sharpness of the transition
l correlation length
Evaluation of the equations of the Zimm-Bragg model is carried out
in the R function "dlnqdz". The results are assembled into the data objects
in the function "newmat". The header for "dlnqdz" is shown following:
"dlnqdz"<-
function(s = seq(0.90000000000000002, 1.1000000000000001, 0.02),
o = cursigma, ds = 1, do = 0, n = 1000)
{
# computation of derivatives of partition function, Q;
#
# 1-d Ising model of helix-coil transition (Zimm and Bragg, 1959);
#
# for mu = 1, where mu is the number of coil segments that must precede
# the helix segment at the start of a helix
#
# input parameters:
# s = single value or sequence of values of s, the factor
# contributed by a helix segment to the statistical weight
# of a chain configuration
# o = value of sigma, the additional factor contributed to the
# statistical weight by the difficulty of initiating
# a helical region
# ds = 1 if the derivative returned is dlnQ/ds; ds = 0 otherwise
# do = 1 if the derivative returned is dlnQ/dsigma; do = 0 otherwise
# n = length of (number of segments in) the chain
#
# value returned:
# vector:
# cbind(s, dlnq, sdlnq, odlnq, l0, l1, theta, dthdlns)
#
# elements:
# s
# dlnq = dlnQ/dz, where dz == ds or do
# sdlnq = s * dlnQ/ds
# = = number of helix segments per chain
# odlnq = sigma * dlnQ/dsigma
# = nu = number of helices per chain
# l0 = larger eigenvalue
# l1 = smaller eigenvalue
# theta = (s/(n-3))*dlnQ/dz
# = fraction helix
# dthdlns = dtheta/dlns
# = sharpness of transition
#
COMMENTS
For a discussion of the Ising model and the significance of the properties
calculated, one might look at, in addition to the Zimm-Bragg paper, a recent
text on statistical physics (e.g. Wang or Kubo). The comments below are
descriptive, not explanatory or analytical.
Some comments on the plots:
Plot 1: Fraction Coil-state = theta, for sigma=.001.
Contour plot.
Plot 2: Fraction Coil-state = theta over n-s plane, for sigma=.001.
Perspective plot.
These are different displays of the same information. The
northeast high plateau is the helix state. Log("s") is
approximately proportional to temperature. Contour lines
like those of plot 1 are superimposed on contour plots for
other properties, discussed following.
(1) The transition is sharp (sensitive to temperature) for
large "n". The transition is diffuse for small "n". The
transition is sharper when the cooperative regions are
larger.
(2) There is a chain length, "n"~10, below which a helix
cannot form, even at very low temperature (large value of
"s"). This is a result of a high cost of initiating a
helix, which for a helix to form must be counterbalanced
by a large number residues going from coil to helical state,
with only a small gain/residue.
Plot 3: Log Coil-regions/chain = nu over n-s plane for sigma=.001,
with theta overlay.
Contour plot.
(1) For short chains (low "n"), log("nu")=0, i.e. there is
1 helical region per chain -- if the chain is short, it is
either all helix or all coil.
(2) Long chains have breaks, with many helical segments per
chain.
(3) For fixed chain length "n", the number of breaks is a
maximum (more and shorter helical regions) in the transition
region. Disorder in the transition region is complementary
to large fluctuations seen in the dynamics near the critical
point.
(4) For fixed "s", in the transition region there is a
maximum average length for a helical region, for "n" > 100
at "sigma"=.001. This is the correlation length.
Plot 4: Sharpness = dtheta/dlns over n-s plane, for sigma=.001.
Perspective plot.
These results underscore the points made in connection with plots 1
and 2. The transition is sharp at long chain length, diffuse for
short chain length, and does not occur for very short chains.
The results also underscore point (4) under plot 3, that
there is a maximum in tbe number of cooperating units, the
correlation length for this system.
Plot 5: Fraction Coil-state = theta over n-s plane, for sigma=.001;
for sigma=.01;
for sigma=1.
3 perspective plots.
(1) The transition is sharper for lower "sigma".
(2) For "sigma"=1, there is no phase transition (nonlinearity);
the dependence of fraction helix ("theta") on log("s") is
that for a simple temperature-dependent equilibrium system
of independent (non-interacting) residues.
Plot 6: Sharpness = dtheta/dlns over n-s plane, for sigma=.001;
for sigma=.01;
for sigma=1.
3 perspective plots.
(1) Sharpness is greater for lower "sigma".
(2) For "sigma"=1, there is no phase transition and the
system behaves like a collection of independent residues.
Plot 7: Log correlation length = over n-s plane, for sigma=.001.
Contour plot.
This plot is complementary to plot 3, log coil regions/chain
= "nu". The correlation length is the fundamental property
for analysis of the phase transition (most particularly,
in the critical region, about the midpoint of the helix-coil
transition).
(1) The correlation length is equal to the chain length, within
the helix region of short to moderate chain length (log("s")
> 0, log("n") = 2 to 3. This is expected, because in this
region there is one helix per chain.
(2) The correlation length is constant at the midpoint of the
transition region, ~ 150, for log("n") > 2.
John Rupley
rupley@u.arizona.edu -or- jar@rupley.com
30 Calle Belleza, Tucson AZ 85716 - (520) 325-4533; fax - (520) 325-4991
Dept. Biochemistry & Molecular Biophysics, Univ. Arizona, Tucson AZ 85721