NONLIN DOC FILE PURPOSE: NONLIN determines the best straight or curved line to fit a set of data points by a process known as nonlinear least-squares curve-fitting. It then prints the parameter values that define the curve and plots the results using one of several formats. The program readily determines the best line through a group of points, the rate constant of an exponential decay process, the Km and Vmax (or Kd and N) of a saturable process, and solves related problems for up to seven unknown parameters. ANALYSES SUPPORTED: Data may be analyzed according to three basic equations: POLYNOMIALS: Polynomial curves are defined by the equation: Y = B(1) + B(2)*X + B(3)*X^2 + B(4)*X^3 + B(5)*X^4 + B(6)*X^5 + B(7)*X^6 Where B(1), B(2), etc. are the unknown parameters which will be optimized to the data. From two to seven parameters can be fitted to the data. In its simplist form (two parameters), the equation defines a straight line where B(1) is the Y intercept and B(2) is the slope, and the analysis is called linear regression. If more than two parameters are selected, the result is a curve and the analysis is called nonlinear regression. Parameters are added in the order that they appear in the above equation. Most smoothly varying data can be fit to this equation if a sufficient number of parameters is used. EXPONENTIALS: Exponential curves are defined by the equation: Y = B(1)*e^(B(2)*X) + B(3)*e^(B(4)*X) + B(5)*e^(B(6)*X) + B(7) where X is typically time and Y is typically a number, amount or concentration. Again, from 2 to 7 parameters can be fitted. For a two parameter fit, the computer determines B(1) (the initial value of Y for X = 0) and B(2) (the exponential rate constant, which is negative for decay processes and positive for exponential growth). The result is a single exponential curve. When 4 or 6 parameters are fitted, the result is the sum of two or three exponential curves. When an odd number of parameters is selected, the final parameter is treated as a constant (e.g., its value does not change with time, as in the case of the 7th parameter in the above equation). This equation is useful for fitting exponential decay (e.g., radioactivity) and growth (e.g., population) data. HYPERBOLAS: This equation has the form: Y = (B(1)*X)/(B(2)+X) + (B(3)*X)/(B(4)+X) + (B(5)*X)/(B(6)+X) + B(7)*X where X is typically concentration and Y is typically the velocity of an enzyme. When binding of ligand to a receptor or binding protein is studied, X is the "free" and Y is the "bound" concentration respectively. The resulting value of B(1) is the Vmax or concentration of binding sites, and B(2) is the Km of the enzyme or Kd of the receptor or binding protein. Data can be fit to the sum of two or three saturable processes by selecting four or six parameters, respectively. If an odd number of parameters is selected, the last parameter (e.g., B(3), B(5) or B(7)) is treated as a nonsaturable process: B(i)*X. Examples are nonspecific binding or a nonenzymatic reaction rate. DATA ENTRY: After selecting which equation to use, you must enter the data. Up to 100 data pairs can be entered. If a mistake is made, scan and correct the data with the up/down cursor and delete keys. After entering the last data pair, enter "E" in place of the next X value. You will be asked if you wish to accept the data. If not, the computer will review each entry to allow errors to be found and corrected. You will next be asked if you want to save the data. Saved files can be loaded by entering "L" in place of an X value during data entry, and will overwrite previous data. If you forget the name of your file, enter "D" to display the directory. The Cardco numeric keypad is supported on the C-64 version of this program. NUMBER OF PARAMETERS: You will then specify the number of parameters (2 - 7) to be fitted to the data. This number must be less (ideally, much less) than the number of data points. Fits using large numbers of parameters typically take much longer and require higher quality data for meaningful results. In general, it is wise to start with two or three parameters, and then add parameters only if needed. WEIGHTING POWER: This number specifies the "weight" that is given to each Y value in the analysis. A weighting power of two is used when the standard error of the mean (SEM) of each Y value (i.e., the uncertainty in the true value of Y) is proportional to Y, as is commonly true. A value of zero weights all points equally, and is appropriate when the standard error is effectively constant regardless of the value of Y. Use of zero weighting under other conditions tends to overemphasize the importance of large values of Y. When high quality data are available the weighting factor has little effect. X OFFSET: In exponential time curves, it is sometimes useful to subtract a constant from all measured times. The results will reflect this change, although the data are printed and saved unchanged. Enter zero for no offset. INITIAL ESTIMATES: Initial estimates of the expected values for each parameter must be provided. These may be very approximate, but accurate estimates speed the analysis and are essential when many parameters are being fitted. If the experiment has been done previously, enter the earlier results. Extremely inaccurate initial estimates may lead to an error condition during analysis. When this happens, the screen border turns red and the program attempts to recover by reducing the values of all parameters and restarting the analysis. Such results may occasionally be inaccurate and should be validated by reanalysis. Inaccurate extimates may rarely cause to program to home on a local rather than global solution. If a result does not appear reasonable, repeat the analysis with a better starting estimate. ANALYSIS: The analysis is automatic. Current parameter values are displayed along with the current sum of the squares. The data and current best-fit line can be viewed during the analysis by holding down the space bar or F3, F5, F7 or F2, but only if a Simon's Basic cartridge is installed. When further reduction in the sum of the squares is not possible, the results are output to the screen. If the analysis does not terminate within a reasonable time or the sum of the squares is not decreasing, output can be forced by holding down the "F" key. OUTPUT: Values are listed for each parameter along with the standard error (uncertainty) of that value, presented both as an absolute amount and as a percentage. Large uncertainties (over 40%) indicate that too many parameters have been selected for fitting, that the data are being fitted to an inappropriate equation, or that the quality of the data is poor. Also provided are the weighted and unweighted sum of the squares, the standard error of the analysis and the coefficient of determination (the square of the correlation coefficient). These values provide a measure of the quality of the fit, and are useful in determining the best equation and minimum number of parameters necessary to fit the data. PRINTOUT: Results can be printed by pushing the "P" key. A label of up to 60 characters can be added. This should work with most printers on the serial bus. Pressing RETURN sends a line feed and the UP ARROW sends a form feed. If everything is printed on the same line or other problems exist, try changing the secondary address sent to the printer by pressing "S". If your printer is not device #4, other numbers may be selected by pressing "N". PLOTTING THE DATA: Graphic features (only) require that the Simon's Basic cartridge be inserted in the cartridge slot. Push F3 for a linear plot, F5 for a log plot, F7 for a double-reciprocal (Lineweaver-Burke) plot, and F2 for a Scatchard plot of the data. The best fit line will be drawn automatically. If you prefer that all points be joined by line segments, hold down "L" while the data points are being plotted. To plot the data points without any line, hold down "D". Holding down any key will suppress printing the numeric data to the screen. SETTING PLOT LIMITS: Plot scaling is automatic. The X = 0 and Y = 0 axes are shown as dotted lines if they occur within the plot area. To set the plot limits manually, hold down the function key used to select the plot until a screen appears. SCREEN DUMP: Pressing F2 will dump the hires Simon's screen to the printer. This is known to work with the Gemini 10x and SG-10 printers with Cardco interfaces, and may work with other Epson-like printers although this is untested. ENDING THE ANALYSIS: Three options are available when the analysis is complete. 1. Pressing "Q" will quit the program and return you to Basic (this is the *only* method, and also works during the analysis). 2. Pressing F6 allows reanalysis of the same data set (for example, with a different number of parameters). 3. Pressing F8 resets the program for entry of new data. During data entry, if any of the new entries is unchanged, pressing return will reenter it automatically. PRINCIPLES OF OPERATION: This program finds the best values of the parameters by successive approximation. Matrix inversion is used to solve a system of partial differential equations for the parameter values expected to give the lowest values for the sum of the squares. The parameters are then adjusted to these new values and minor adjustments made to further improve the fit (these steps are labelled "iteration" and "subiteration" respectively). This process is repeated until further improvement is not possible. Standard error estimates for each parameter are then found by determining the sensitivity of the sum of the squares to small changes in the value of the parameter. SUMMARY OF COMMANDS: C Disk command D Plot only data (Hold down during plotting) E Edit data set F Force results (Hold down during analysis) G Display graphic screen L Connect points (Hold down during plotting) N Change printer device number P Send results to printer Q Quit S Change secondary address sent to printer T Display prior text screen <^> Send formfeed to printer Redisplay parameter values of last analysis Linear Plot (Hold down to set limits) Log Plot (Hold down to set limits) Double Inverse Plot (Hold down to set limits) Scatchard Plot Screen dump to printer Reanalyze same data set Reset to opening screen Display current best-fit line (hold down during analysis) Send linefeed to printer COPYRIGHT NOTICE: Copyright 1985 by Richard A. Weisiger. All rights reserved. Permission is granted to duplicate this program for personal use and to distribute it through electronic data bases provided both the program and the opening screen are not altered. This program may not be sold. Please notify the author of any errors (Compuserve user # 75015,260 or write University of California, San Francisco, CA 94143).