Introduction

This tutorial describes the use of DynaFit for the analysis of simple equilibrium binding data. The data were provided in the digital format by Jim Stivers (National Institute for Standards and Technology, Rockville, MD), a co-author of the original article. The tutorial uses data from Figure 2 of the cited paper:

"Mechanism of Neomycin and Rev Peptide Binding to the Rev Responsive Element of HIV-1 As Determined by Fluorescence and NMR Spectroscopy"

K. A. Lacourciere, J. T. Stivers, and J. P. Marino (2000) Biochemistry 39, 5630-5641.

Experimental Procedure

The HIV-1 Rev-responsive element (RRE), a 34-mer portion of the viral RNA, was chemically modified to show strong fluorescence. Importantly, the fluorescence intensity of the chemically modified RRE changes upon the binding of a 22-peptide derived from the Rev protein. RRE (30 nM) was incubated at 10deg C with various amounts of Rev-peptide (0-300 nM) and the fluorescence intensity was measured at equilibrium.

Raw Data

The following data were obtained and assembled in a simple text file named rev.txt.

Lacourciere et al. (2000) [Fig. 2]
Rev,nM	F 
------	-------
0     	1
2     	1.0803
4     	1.1005
8     	1.1749
21.33 	1.3921
34.67 	1.5824
48    	1.7166
68    	1.7993
88    	1.9123
108   	1.9317
148   	1.9436
214.67	1.9781
281.33	1.9298

Any text editor or word processor can be used in the preparation of DynaFit data files, as long as the resulting file is saved in the ASCII text format. Note that the raw data file can contain any number of comment lines or header lines, which are simply ignored by the program.

Script File Editing

DynaFit is a program driven largely by text scripts, as opposed to the graphical user interfaces, which control only a small portion of the program's functionality. Detailed instructions for the composition of DynaFit scripts are contained in a

To prepare a script file for the analysis of the HIV / Rev equilibrium binding data, use any word processor or text editor. We begin composing the [task] section of the script file. The problem at hand consists of fitting (as opposed to simulating) equilibrium binding data (as opposed to kinetic or time-course data). Thus the [task] section of the script will contain the following text:

[task]
   data = equilibria
   task = fit

Reaction Mechanism

Next we decide on the kind of reaction mechanism that might be involved in HIV / Rev interaction. For the purpose of the equilibrium binding study, we begin with the simplest possible mechanism, ignoring the possibility of multi-step or multi-site interactions. The simplest binding mechanism is entered in the DynaFit script using arbitrary symbols for the macromolecules involved. In this case we will choose the symbols RRE and Rev for the Rev-responsive element and the Rev-peptide, respectively.

[mechanism]
   RRE + Rev <==> RRE.Rev  :  Kd  dissoc.

The symbol Kd was chosen arbitrarily for the equilibrium dissociation constant; any similar symbol would suffice, such as K1 or Keq. For researchers who are accustomed to working with association equilibrium constants, the DynaFit association abbreviated as assoc could be used to designate the equilibrium constant accordingly.

Concentrations

Concentrations of the reactants that are constant throughout the entire data set are collected in the section of the script file designated as [concentrations]. In the code fragment below, note the use of comments separated by semicolons from the rest of the line.

[concentrations]
   RRE = 30 ; not 100 nM (typo in paper!)

The concentration unit can be arbitrarily chosen by the investigator, but it is important that all kinetic and equilibrium constants, and all molar response coefficients (see below), use the same concentration unit. In this case we have chosen nanomoles per liter as the concentration unit, because it seems natural for this particular problem: all concentrations of reactants are in the nanomolar range.

N.B.: The concentration of RRE used in this tutorial is different from the information published in Figure 2 of the original paper. The journal publication contains a typographical error: the correct concentration is 30 nM, not 100 nM (J. Stivers, personal communication).

Equilibrium Constants

Every statistical analysis involving nonlinear regression (as is the case here) requires from the investigator an initial estimate of the adjustable parameters. The problem analyzed here involves only one equilibrium binding constant, which makes it relatively easy to estimate its initial value.

Looking at the original data, it is easy to notice that the increase in fluorescence intensity levels off at approximately at [Rev] = 200 nM (saturation effect). Importantly, it seems that the system is approximately half-saturated at about [Rev] = 30 nM ("LD-50" value). This means that the dissociation constant for the RRE-Rev complex must be in the vicinity of 15 nM, because the total concentration of RRE in the assay is 30 nM. In general, we can use this very approximate formula to estimate the equilibrium dissociation constants in similar biochemical systems:

The initial estimate of the dissociation constant is entered into the script file simply as

[constants]
   Kd = 50 ?

The question mark after the initial estimate of the equilibrium constant signifies that this parameter is to be optimized by the nonlinear regression procedure. Again, it is important to realize that the numerical value of the dissociation constant must be entered in the same units as are the concentrations (in this case, nanomoles per liter).

The program calculates the predicted instrumental response (in this case, fluorescence) as being directly proportional to the concentrations of some or all reactants. The proportionality constants are the molar response coefficients, which must be entered in the same concentration units as are concentrations and kinetic (or equilibrium) constants.

In this example, we see that relative fluorescence F = 1.00 is produced by 30 nM of RRE. This means that the molar response coefficient of RRE is approximately 0.033 fluorescence units / nanomole / liter, because one nanomole per liter of RRE would cause fluorescence intensity of 0.033 instrument units. Similarly, we can estimate the molar response coefficient of the RRE.Rev complex at approximately 0.066 fluorescence units / nanomole / liter. Indeed, near the saturation point (at 200 nM Pre) where nearly all RRE is converted to the RRE.Rev complex, we see that the fluorescence intensity is approximately F = 2.00. Thus, one nanomole per liter of RRE.Rev complex would cause fluorescence intensity of 0.066 units. These facts are entered into the DynaFit script file as follows:

[responses]
   RRE = 0.033 ?
   RRE.Rev = 0.066 ?

Note that both molar response coefficients are treated here as adjustable model parameters.

The program must be given the location of the primary data files in a special section of the script file called, in this case, [equilibria] (for the analysis of reaction kinetics, the name of the appropriate section is [progress]).

[equilibria]
   variable Rev
   file ./examples/hiv_rev/data/rev.txt

The file names are preceded by a partial path name using the Unix file system notation (using the forward slash '/' as a path name separator). Important Note: All path names are entered relative to the location of the DynaFit executable file.

Output Directory

By default, DynaFit produces the results of computations in the form of text files (plain text, HTML, and LaTeX) in the subdirectory named output located in the same directory as the executable file DynaFit.exe. In this example problem we chose to redirect the output files into a subdirectory of the examples folder. The location of the output directory is given in the [directory] section of the script.

[output]
   directory ./examples/hiv_rev/output/rev

Summary

[task]
   data = equilibria
   task = fit

[mechanism]
   RRE + Rev <==> RRE.Rev  :  Kd  dissoc.

[concentrations]
   RRE = 30 ; not 100 nM (typo in paper!)

[constants]
   Kd = 15 ? 

[responses]
   RRE = 0.033 ?
   RRE.Rev = 0.066 ?

[equilibria]
   variable Rev
   file ./examples/hiv_rev/data/rev.txt

[output]
   directory ./examples/hiv_rev/output/rev

[end]

Script Execution

To try the script file, first select "Load", and then select "Try" from the "File" in the main program window. The program should display the following overlay of the simulated model on the experimental data:

This agreement between the experimental data and the simulated model seems sufficiently good to warrant using the initial estimate of the dissociation constant (15 nM). However, the reader is encouraged to experiment with changing the script file even in minor ways.

To modify the script file, select "Script" from the "Edit" menu and change the value of Kd from 15 nM to 10 nM. Then save the script back to the hard disk and choose "Try" from the "File" menu. This will re-load the modified script and display the overlap of the simulated model on the experimental data, using the modified values.

After the agreement between the experimental data and the simulated model is sufficiently good, select "Run" from the file menu. This will execute the script and perform the nonlinear least-squares regression.

Examining the Results

Plain text format

PARAMETERS & STANDARD ERRORS
    Parameter  Initial    Fitted     Error
          Kd        15     12.61       2.1
       r:RRE     0.033   0.03344   0.00059
   r:RRE.Rev     0.066   0.06785   0.00087

HTML Output Files

To view the results of DynaFit computations in the HTML format (with graphic files embedded in the GIF format) select "Browse Results" from the "View" menu. This will display the results of fit in the default HTML browser window installed on your computer.

Important Installation Note

For the browsing feature to be functional, the program must be installed in a directory that does not contain spaces anywhere in the path name. For example, the installation directory 'C:/Program Files/BioKin/DynaFit' does not allow the HTML browsing of results. Install DynaFit into a directory named (e.g.) 'C:/Programs/BioKin/DynaFit', which does not contain blank spaces.

LaTeX Output Files

TAB-Delimited Output Files

DynaFit exports all graphs as TAB-delimited text files for direct import into third-party graphic packages. In this case the output file named fit_0101.tab is located in the subdirectory ./examples/hiv_rev/output/rev/tab. You may open this file in any text processor to examine its structure or simply copy and paste the contents into your favored graphic software package.

Conclusions

We have used the program DynaFit to determine the dissociation equilibrium constant of the molecular complex formed between HIV-1 Rev-responsive element and the a peptide fragment derived from the Rev protein. The literature results (dissociation constant 13 ± 2 nM) are exactly identical to the values obtained here.

The advantage of using DynaFit for this kind of analysis is that the user can completely avoid using algebraic fitting models. In this particular example problem, the authors analyzed their data using equation (1) below:

For many practicing biochemists, this equation is not too difficult to derive (or verify, after looking it up in the primary literature). However, there are many other experimentalists who are somewhat "mathematically challenged" and probably will prefer to achieve the same results using the symbolic script file:

[mechanism]
   RRE + Rev <==> RRE.Rev  :  Kd  dissoc.

[constants]
   Kd = 50 ?

Of course, the real advantage of DynaFit will surface in the analysis of slightly more complex problems, such as the dual binding of both Rev and glycoside ligands to the Rev-responsible elements, also studied by Lacourciere et al.. It is important to note that those kinds of complex equilibria cannot be described by algebraic equations, as a matter of principle. This is where DynaFit (or similar computational tools) represents the only available method to analyze experimental data. An example will be shown in another tutorial.