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Mathematica-Compatible Notebook
This notebook can be used on any computer system with Mathematica 3.0,
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Cell[TextData[StyleBox["Basic Operating Procedure",
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"1) Evaluate the \"Initialize\" notebook for the case you're interested in; \
then close it and open the corresponding \"Output\" notebook. \n2) Add your \
own experimental data for comparison with theoretical data if desired. \n3) \
Set the \"Input Parameters\" to the desired values. \n4) Evaluate the \
\"Output\" notebook. \n5) Repeat 3) and 4) as desired. ",
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"Since there is some redundancy in the various output parameters that are \
plotted and tabulated, you may want to unlock and make inactive or delete \
cells for output you're not interested in. Nearly all the essential \
calculations are performed in one cell immediately under the first \
\"Calculate\[Ellipsis]\" heading, between the input parameters and the first \
plot (generally the Scatchard plot), so this cell must be re-evaluated each \
time the input parameters are changed. Once this is done, the various other \
plots may generally be performed in any order. However, additional \
calculations must be performed before 3-D or 2-D cluster length distributions \
are plotted; these calculations are performed in the cell immediately under \
the \"Calculate\[Ellipsis]\" heading just before the first 3-D or 2-D cluster \
length plot. ",
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"These Mathematica notebooks deal with a simple model for the \
non-sequence-selective binding of small molecules (ligands) such as proteins \
to long linear polymers such as DNA. They calculate and plot data for the \
binding of ligands to an infinite linear lattice for the following cases: \n \
a) symmetric ligands of one type (\"Sym\"); \n b) symmetric ligands \
of two types binding to an isotropic lattice (\"2Lig\" and \"2L-",
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"\"); \n c) asymmetric ligands of one type binding to an isotropic \
lattice (\"Iso\"); \n d) asymmetric ligands of one type binding to an \
anisotropic lattice (\"Ani\"). \nIn the last case (\"Ani\") two binding \
modes are allowed, i.e., different numbers of lattice residues may be covered \
when the ligand binds in different orientations. Case a is equivalent to the \
case analyzed by McGhee & von Hippel in their well-known paper (J. Mol. Biol. \
vol. 86, 469-489 [1974]. Erratum vol. 103, 679 [1976]). Cases b-d are more \
general. ",
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"For each case there is an initialization notebook that contains the \
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the Scatchard plot, the neighbor-effect parameters, and the conditional \
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" (r) except in the case of one of the sets of programs for two ligands \
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"\"), in which the concentration of ligand 1 is the abscissa. In addition, \
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"\"), the output notebooks calculate the 2D (at a given r) and 3D (for the \
whole range of lattice saturation) cluster length distributions. The method \
used is based on the treatment given in Wolfe & Meehan, J. Mol. Biol. vol. \
223, 1063-1087 (1992). It is important to note that except for the case of \
symmetric ligands of one type (\"Sym\"), the programs will not give correct \
output under all circumstances. IT IS THE RESPONSIBILITY OF THE USER TO \
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"In addition, all the programs except one of the sets of programs for the two \
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"To use the notebooks, first open the initialization notebook for the \
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". This needs to be done only once per Mathematica session. Then, close \
the initialization notebook, open the corresponding output notebook, and \
specify the desired \"input variables\" values. Integer values for n are \
expected but are generally not required. The programs for asymmetric ligands \
on an anisotropic lattice (\"Ani\") with two binding modes (n1 not equal to \
n2), and for type 2 or type 3 isotherms for two kinds of symmetric ligands on \
an isotropic lattice (\"2Lig\") may generate errors at saturation for \
non-integral values of n. The variable \"points\" gives the number of points \
at equally spaced degrees of lattice occupancy above r=0 to be used in the \
plot. Because parameter values frequently change rapidly when the lattice is \
nearly saturated, I have added several \"extra\" points just before rSat (the \
maximum value of r at lattice saturation), using magenta text in the cell \
under the first \"Calculate\[Ellipsis]\" heading. ",
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"). If the program is being executed in sections, the Scatchard plot must \
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executing the group of cells in the right-most bracket starting at the \
\"input variables\" cell). The only exception to this rule is in the case of \
the calculation of cluster length distributions for symmetric ligands of one \
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Scatchard plot. ",
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"The Scatchard plot for an asymmetric ligand with a single binding mode (n1 = \
n2) binding to an isotropic (\"Iso\") or anisotropic (\"Ani\") lattice is \
accompanied by a \"reference\" symmetric ligand Scatchard plot having the \
same intercepts and initial slope (in gray). Since the asymmetric ligand \
Scatchard plot (in black) is plotted after the reference plot, the reference \
plot will be invisible if the two plots are identical or nearly so. The \
reference plot can be eliminated, if desired, by setting the Boolean variable \
\"referencePlot\" to \"False\" (in the first line of the first calculation \
cell). ",
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distributions can be plotted in any order. The only restriction is that \
additional calculations are needed for both the 2D and 3D cluster length \
distributions, and these must be performed (by the cell just below the \
respective \"Calculate\[Ellipsis]\" heading) before the data can be plotted. \
The particular degree of saturation for a 2D cluster length distribution is \
specified by \"rCluster2D\", which equals r/rSat. 2D cluster length \
distributions can be calculated repeatedly for different r values after the \
Scatchard plot has been calculated once. The number of equally spaced values \
of r at which the cluster distribution will be calculated for the 3D plot is \
specified by \"points3D\". The amount of time needed for the cluster length \
distribution calculation increases exponentially with the length of cluster \
being considered (you set the maximum length for the calculations with \
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If this program crashes after a small number of calculations, try increasing \
Mathematica's memory allocation. ",
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Most of the cells in the programs are locked, so they would have to \
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make any changes. Also, many of the cells are closed. In order to view \
their contents, one of two procedures must be followed. If the cell is a \
subgroup (hidden behind another cell with a downward pointing arrow to the \
right), it can be viewed using the \"Cell-> Cell Grouping-> Open All \
Subgroups\" command. If it is not a subgroup, unlock it and open it using \
the \"Cell-> Cell Properties-> Cell Open\" command. The output from the \
programs is presented both graphically and numerically. The numeric output \
may in most cases be eliminated by putting a semicolon at the end of the line \
in the cell just before the corresponding graph. \
\>", "Text",
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"All three sets of programs for the single type of ligand case (\"Sym\", \
\"Iso\", and \"Ani\") allow comparison of the calculated theoretical \
Scatchard plot with an experimental Scatchard plot. In order to perform this \
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" must also be set to True. The programs will then plot the experimental \
and theoretical curves together (in gray and black, respectively), using the \
experimental r values for the theoretical data points. Several plots of the \
residuals (experimental - theoretical) will also be displayed, and a \
correlation coefficient (R) will be calculated. (If the correlation is very \
poor, the resulting value of R will be a complex number.) If the initial \
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isotropic lattice are also intended for comparison of experimental data with \
the theoretical binding curve (the \"2Lig\" programs do not allow this). The \
calculations are performed for an abscissa array of ligand 1 concentrations \
(not for an array of r values of ligand 1, as in the other programs), and do \
not reach lattice saturation. The experimental data needs to be given as an \
array of concentrations of ligand 1 (\"",
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"Problems may occasionally arise with execution of these programs, signaled \
by an unexpected discontinuity or plateau in one or more of the plots, a \
message that FindRoot has encountered a single Jacobian, or a message that \
\"Newton's method has failed to converge to the prescribed accuracy\" after \
the specified iteration limit. The discontinuity or plateau may indicate \
that FindRoot has jumped to an incorrect solution of the equations being \
solved for. This type of problem may not be evident in the Scatchard plot if \
it occurs at a point where the curve is close to the x-axis, but it will \
generally be obvious in the plots of one or more of the neighbor-effect \
parameters or q (particularly in the latter case). These problems usually \
arise when the values of one or more of the neighbor-effect parameters (or q) \
are changing rapidly as a function of r, and the step size used for r in the \
iteration is too large (i.e., the value of \"points\" is too small). ",
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"If meaningless or physically impossible values are generated for one or more \
of the neighbor-effect parameters, q, or conditional probabilities, a flag \
will be set and warning messages will accompany the output. In addition, \
data which has been determined to be incorrect will not be displayed. \
HOWEVER, THE ABSENCE OF A WARNING MESSAGE IS NO GUARANTEE THAT THE OUTPUT IS \
CORRECT. Incorrect values may still be in the physically possible range. If \
the results of any calculation are in doubt, a simple and reliable way to \
test the correctness of the output is to change (e.g. double or quadruple) \
the number of points in the plot. If the initial output was incorrect, the \
numbers above the r value where the mistake occurred will change in the \
second calculation. ",
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"The problem of FindRoot jumping to the wrong root can generally be overcome \
by one of three methods: 1) by increasing the number of points in the plot; \
2) by choosing less extreme values of the input variables; 3) by changing the \
form of the equations being solved. The third method should be used when the \
first doesn't work and you don't want to use the second. To use this \
approach you need to note which set of equations are being solved when the \
problem arises (given by the value of \"ne\"). Then go back into the \
initialization notebook and find the equations in question. In many cases \
there will be a standard version of these equations, given in an \
initialization cell, and one or more \"alternate\" versions, in cells which \
are not initialization cells. The version of the equations in the \
initialization cell is the one that normally executes most rapidly; however, \
one of the slower versions may be less susceptible to jumping to the wrong \
root in a particular situation. Select and execute a cell containing one of \
the alternate versions and retry the calculation. This approach cannot be \
used for errors that occur at saturation, because alternate forms of the \
equations used for this calculation are not available. ",
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"An example of a situation where calculating the correct output is difficult \
is the anisotropic lattice case with k1 = k2 = n1 = n2 = 1, w11 = 10, w22 = \
0, w12 = 0.01, w21 = 0.011. The ligands in this case initially form tightly \
bound head-to-head pairs, but must rearrange into a much weaker tail-to-head \
arrangement close to saturation. Mistakes are often made in this calculation \
without setting a flag. The error is obvious when inspecting the plot for q \
(and the plots for some of the b-b conditional probabilities). The second \
alternative set of ne1 equations works much better for this case than the \
primary version. ",
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"A discontinuity that appears between the last and next-to-last points in a \
plot that does not respond to the above-noted measures may represent a \
program bug (most likely in the anisotropic lattice case (\"Ani\") with n1 \
not equal to n2). If you believe you have encountered a bug, please report \
it to the programmer (Alan R. Wolfe, Dept. of Biopharmaceutical Sciences, \
University of California, San Francisco, CA 94143-0446; e-mail address: \
alwolfe@bigfoot.com). Remember that the parameter values will not always be \
correctly calculated at saturation if you use nonintegral values of n. \n",
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*)
(***********************************************************************
End of Mathematica Notebook file.
***********************************************************************)