A.1.1
Introduction.
Conformational exchange on the µs-ms timescale is measured by NMR using the so-called relaxation dispersion experiments. In this work, the dynamics of DHFR were probed using the CPMG based pulse sequences. The process for analyzing these data generally follows three main steps: collecting and processing the NMR spectra, measuring peak intensities and converting the measured peak intensities to Reff2 , and fitting the Reff2 values to a mathematical function in order to extract kinetic and thermodynamic values. Professor Arthur Palmer’s laboratory has made their in-house program CPMGfit freely available to accomplish the third and final step (Loria, Rance and Palmer, 1999). Unfortunately, CPMGfit cannot fit data to a global exchange model and is limited to a single field strength. As shown by Loria and coworkers, fitting relaxation dispersion data using two or more spectrometer frequencies simultaneously is very import for accurate analysis (Kovrigin et al., 2006). Therefore, the program EXchange RATE analysis (exrate) was developed for analysis of relaxation dispersion experiments
exrate is a general use program for fitting the change in Reff
As described in Chapters 2 and 3, exrate can fit the data to the Carver-Richards equation (Carver and Richards, 1972) or the fast approximation equation (Palmer, Kroenke and Loria, 2001) described in section 1.2. In both cases, the data can be fit using local models (individual kex and pa values) or grouped for global analysis
(shared kexand pavalues). exrate was programmed in FORTRAN 77 and compiled
using the g77 fortran compiler. As a result, exrate can be executed on computers running Unix (Linux), Macintosh or Windows based operating systems. Furthermore, since FORTRAN is a compiled language as opposed to an interpreted language (e.g. Python, Perl, Jave, Matlab, etc. . . ), exrate operates very quickly on the standard home computer.
A.1.2
Materials.
exrate was written FORTRAN 77 and compiled with the GNU g77 compiler: http://www.gnu.org/software/fortran/fortran.html.
All real numbers are in single-precision floating point format. Non-linear minimization of the target function is achieved using a modified Levenberg-Marquardt algorithm implemented in sMINPACK (http://www.netlib.org/sminpack/). Error in the fitted values are calculated using a Monte-Carlo procedure (Press et al., 1992).
A.1.3
Protocol.
The compiled exrate executable must be in the user’s PATH. A control file (Figure A.1) and data table must be located in the working directory (Figure A.2). The control file allows the user to select different fitting options without cumbersome execution flags and the input data table contains all the experimental data and pa- rameters. The first line of the data table includes the number of experimental field strengths, the number of data points for field 1, the Larmor frequency for field 1,
the number of data points for field 2, etc. . . exrate can fit up to three fields simul- taneously. The next lines are the values for τcp(in seconds) followed by the actual data and associated error for each spectrometer frequency. Upon execution of the program, data for each residue will be read into the program. However, residues can be excluded using the “{,}” syntax shown in the sample data table (Figure A.2). This functionality allows the user to modify a data set without completely deleting residues or generating multiple data tables.
Figure A.1: The exrate control file. This file controls all of the fitting options within exrate.
Fitting the data to localkex and pavalues is very straight forward. The user must
choose the appropriate fitting function, whether they would like to estimate error and establish appropriate grid searching parameters for the initial guesses. Error is estimated using a Monte Carlo sampling procedure and the user has control over
Figure A.2: A sample exrate data table. The first line indicates the number of fields, number of data points and Larmor frequencies for each field strength. The next ten lines indicate the values of τcp used in the experiment followed by the data. The data set 2.HN will not be fit due to the brackets.
whether the simulated datasets are calculated from experimental or fitted data points (Press et al., 1992). Furthermore, the user can elect to print the results of each Monte Carlo trail to a file for evaluation. As discussed above, multiple spectrometer field strengths for a single residue can be fitted at once. The user has control over whether a single or individual (one for each spectrometer frequency) R02 values should be fit. Upon executing exrate, the user is asked for the name of the input data table, and
Levenberg-Marquart algorithm that minimizes the target function: X2 = n ∑ i=0 √ (Rexp2 (i)−Rcalc 2 (i))2 σ(i)2 (A.1)
where σ is the experimental error and Rexp2 and Rcalc
2 are the experimental and cal- culated R2 values for each τcp value, i, respectively. Upon completion, exrate prints three files: a formatted output file, a log file and a file titled EXRATE.log. The formatted data file contains the final exchange values for each residue in a tabular format. The log file contains information such as input data, fitting parameters, ini- tial fitted values and fitted values after error estimation. Essentially, the log file is a record of everything the program did and should be evaluated carefully. Finally, the EXRATE.log file records any errors that may have occurred while minimizing the target function. The Levenberg-Marquardt algorithm may fail to find a minimum, especially with noisy data. The EXRATE.log file should be inspected after every run to determine the quality of the fitted values.
The process of global fitting assumes a group of two or more residues are expe- riencing the exact same exchange event such that kex and pa are the same for each
member of the group. The values of ∆ω and R0
2 are atom specific and must be fit on a per-residue basis. To initiate global fitting, the user must modify the rdcontrol.inp file by selecting the “fitting method” and inputing appropriate initial guesses for kex and pa. Since the user should have an accurate estimate of the global exchange pa-
rameters after local fitting, these values are not derived from a grid search method to save computational time. Upon executing exrate, the user is asked for the name of the data table, the tabulated values from local fitting, an output file and log file names. The local data is required in order to obtain estimates of ∆ω and R0
2 to use as initial guesses for minimization. Again, these variables are not grid searched to save time. The data are best-fitted to the relaxation equation minimizing the target
function: X2 = resn ∑ j=0 n ∑ i=0 √ (Rexp2 (i)−Rcalc 2 (i))2 σ(i)2 (A.2)
where j represents the number of residues in the data set. The output files created during the global fitting process are identical to those of the local fitting process. It should be noted that the X2 value reported in the output table represents the per residue X2. This value is important for determining the quality of fit on a per- residue basis to the global model of conformational exchange. In the end, the user is responsible for determining the quality of fit and the appropriateness of fitting the data to a global model for conformational exchange.
The program exrate enables users to fit CPMG based relaxation dispersion data at multiple fields to the general form and fast approximation of the Carver-Richards equation. Furthermore, exrate can fit data to models assuming localized and global exchange events.