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1.0 PURPOSE
PENNZYME is a program for fitting trial rate laws to
experimental data from steady state initial velocity experiments.
It determines the set of parameter values for a given rate law
which produces the closest fit to the experimental data, and
provides information as to how good that fit is.
2.0 ABSTRACT
This program finds the set of parameter values in a user
supplied rate law which gives the best fit to the initial velocity
data. The program operates interactively and provides two
algorithms for locating a least-squares fit to the experimental
data. The algorithms have different convergence properties and are
best applied independently or in sequence with independent stopping
criteria. The two algorithms were selected to work together so
their capabilities complement each other.
The program allows the user to interactively alter the initial
parameter values, which parameters are actually being varied in an
optimization, the stopping criteria, the order of application of
the fitting algorithms, and to delete points which the program has
identified as outliers. In addition the user can request that any
of a set of reports pertaining to the input data, and to
intermediate calculations, be typed on his terminal. The user may
also obtain graphs on a suitable graphics terminal or with a
plotter, if the graphics routines described below are included in
the program and used as directed. These operations may be
performed in any order and as many times as desired in a particular
session.
PENNZYME user's document Page 2
BEGINNER'S GUIDE TO PENNZYME
3.0 BEGINNER'S GUIDE TO PENNZYME
The purpose of this section is to give the user who has no
prior experience with computers, enough information to use the
program on an elementary level.
PENNZYME is a program which fits the parameters in trial rate
laws representing possible enzymatic mechanisms to sets of
experimental initial velocity data. It has considerable capability
for inputting initial guesses , following the state of the
computation, etc., which the sophisticated user can employ to
efficiently obtain results interactively. At the same time it is
able to do without all of this and give the unsophisticated user a
result with minimal assistance from him although the computation
may then be less efficient. Nonlinear optimization procedures
generally require good initial guesses for the parameter values.
This program combines two such procedures so that the program
effectively generates its own initial guesses if necessary(in which
case many user inputs are set to default values). A more detailed
description is given in the DETAILED DESCRIPTION section below.
Technical terms are defined in the TECHNICAL GLOSSARY.
To use the program two input files and a FORTRAN function
subroutine representing the trial rate law must be provided. One
of the two input files contains the experimental velocity data.
The other file contains information which depends upon the rate law
and the model to be tested and includes the initial values of the
parameters and the selection indices and convergence criteria for
the fitting algorithms. The experimental velocity data-file
contains only the chemical names and concentrations and the
observed enzymatic velocities. Therefore, the file need not be
changed with each new trial model. The user supplied rate law
subroutine must be compiled and linked to the PENNZYME package
prior to operation of the program. A description of this
subroutine and the linking operation are given below under "RATE
LAW SUBROUTINE". The parameter file and experimental data files
described below must be on FORTRAN logical devices 1 and 20,
respectively. The detailed handling of the rate law file may be
computer-dependent.
The following discussion refers to the sample input files for
the measurements and for the parameters which are shown in Fig 3.1
and 3.2. These files are actually read by the program in this
order, and may come from the same physical device, usually a disk
file. In this and the following discussion, input is described as
being composed of files made up of lines, which have presumably
been typed in at a computer terminal. (In previous versions of
this program these were referred to as cards in a deck.) It should
be noted that columns in the data input file may be separated by
tabs, but that this cannot be done in the parameter file, where ten
spaces should be typed between columns. An optional name for the
velocity may be inserted in the data file on the line preceding the
numerical values, but this will not carry over to any graphic
output.
PENNZYME user's document Page 3
BEGINNER'S GUIDE TO PENNZYME
.c;Experimental Data Input File
3,10
SUBSTR 1
SUBSTR 2
INHIBITR
1.2 5.0 3.5 1.11 .058
1.2 5.0 35.0 .710 .034
1.2 500.0 105.0 2.24 .115
1.2 500.0 350.0 .766 .0426
12.0 50.0 105.0 11.8 .6
12.0 50.0 350.0 5.03 .258
36.0 50.0 3.5 38.5 1.94
36.0 50.0 35.0 36.0 1.64
120.0 150.0 35.0 61.5 3.18
120.0 150.0 105.0 57.9 2.76
FIGURE 3.1
In this example the concentrations of three chemicals are measured
for each of ten velocity observations. The next three lines
contain the names of the three chemicals of our example in the
order in which they will be encountered in the velocity observation
lines that follow. This order must be the same as the order
assumed in the FORTRAN rate law which will be used. The programmer
supplying the rate law is responsible for supplying the
documentation needed to determine this order. In our example the
chemical names are:
SUBSTR 1
SUBSTR 2
INHIBITR
The next ten lines, in our example, each contains the data for one
velocity observation. Each of these lines contains the following
information in the same order:
1. Concentrations of the measured chemicals in the same order as
that in which their names were specified on the name lines.
2. Observed reaction velocity.
3. Experimental error of the velocity measurement.
These numbers may appear with or without decimal points and
exponents, and must be separated by either blanks or commas. The
following rule apply for input:
PENNZYME user's document Page 4
BEGINNER'S GUIDE TO PENNZYME
1. All numbers are space delimited and may contain decimal points
and exponents.
2. Any names which are followed in the record by other input must
be padded with spaces to match the required character count.
3. Records must be placed in the input file in the required order,
and are not numbered or otherwise identified.
Optionally the position normally occupied by the first
experimental point may be occupied by a set delimiter line, in
which case the first velocity data line will follow the last
contiguous delimiter line. If the data cannot fit on one line, it
may be placed on following contiguous records provided that each
line but the last has exactly "n" values, (where "n" is the
"maximum values per record" value included (optionally) in the
first record of the file). The last line must have at most "n"
values but may have as few as one.
Lines in the experimental file may be divided into sets. Set
delimiter lines may beall spaces or begin with non-numeric
characters other than tab, decimal point, or space. (These lines
may also include comments for identification.) The program numbers
the data sets so defined. They may then be manipulated
independtly.
The numbers on the first of these lines is interpreted as
follows:
line image:
1.2 5 3.5 1.113 .0580
Number Interpretation
1.2 Concentration of "SUBSTR 1".
5 Concentration of "SUBSTR 2".
3.5 Concentration of "INHIBTOR".
1.113 Observed reaction velocity.
0.058 Experimental error of this measurement.
The first line of the sample data file specifies the number of
chemicals
which have been measured in the experiment(and which appear in
the rate law subroutine), and the number of steady state velocity
measurements,
in that order.
PENNZYME user's document Page 5
BEGINNER'S GUIDE TO PENNZYME
3.1 Rate Law Subroutine
Prior to operation of the PENNZYME program a user supplied
rate law subroutine must be compiled and linked with the PENNZYME
package. This subroutine must be a FORTRAN "REAL FUNCTION"
subroutine and must be named "RATE". Two argument vectors are
passed to the function subroutine and contain the current
concentration values and the current parameter values. It is
recommended that the argument vector names be the same for all user
written rate law subroutines. A sample rate law subroutine is
provided in figure 5.1 .
REAL FUNCTION RATE(PCONC,PARAM)
C DUMMY VARIABLES:
C PCONC CONCENTRATION ARRAY.
C PARAM PARAMETER ARRAY.
DIMENSION PCONC(1),PARAM(1)
DATA QSMALL/1E-35/
IF(PCONC(1).LE.QSMALL.OR.PCONC(2).LE.QSMALL.OR.PARAM(3)
1 .LE.QSMALL) GO TO 100
C---- ELSE, CALCULATE THE REACTION RATE.
QSUB1 = (PARAM(1)/PCONC(1))
QSUB2 = (PARAM(2)/PCONC(2))
QACTV = (PARAM(4)/PCONC(1))
QINHIB = (PCONC(3)/PARAM(3)) + 1.0
RATE = PARAM(5) / (QSUB1*QINHIB
1 + QSUB2*(QACTV*QINHIB+1) + 1)
GO TO 110
C---- THEN, THE RATE WOULD HAVE BEEN ZERO.
100 RATE = 0.0
110 CONTINUE
RETURN
END
FIGURE 5.1
PENNZYME user's document Page 6
BEGINNER'S GUIDE TO PENNZYME
Notice that the adjustable parameters and the chemical
concentrations are represented by the elements of the vectors
"PARAM" and "PCONC" respectively. The order in which they appear
in these vectors must correspond to the order in which they are
specified in the input file. Comparing the data file of figure 3.1
with the rate law subroutine of figure 3.2 we note the following
correspondences:
Vector element User label
PCONC(1) SUBSTR 1
PCONC(2) SUBSTR 2
PCONC(3) INHIBITR
PARAM(1) KM1
PARAM(2) KM2
PARAM(3) KI
PARAM(4) KA
PARAM(5) VMAX
The first line of the routine declares the program to be
a "REAL FUNCTION" subroutine named "RATE", as required, with two
arguments to be
identified by the names "PCONC" and "PARAM".
The next non-comment statement(i.e. statement without a "C" in
column 1) is
the "REAL" statement which declares that the two arguments are of
variable type
"REAL"(i.e. single precision floating point). The next statement
declares that the
two argument names each refers to a vector of values. The first
argument of the
subroutine always refers to the vector of concentration values at
which the rate
of the reaction is to be calculated. The second argument refers to
the vector of
parameter values which, together with the rate law subroutine,
complete our model
of the enzymatic mechanism. It is these values that the
optimization algorithms must
determine.
The rate law itself is written in algebraic equation form with the
name of the
function subroutine("RATE") on the left-hand side of the equation.
PCONC(K), in our example, would refer to the concentration of the
K'th chemical. PARAM(K) would refer to the value of the K'th rate
law
parameter value. The order of the chemicals and parameters may be
defined
arbitrarily when the rate law subroutine is written, but if n
chemicals
are required they must be referenced as the 1 through n'th elements
of the
concentration vector. The same applies to the use of the parameter
values vector.
PENNZYME user's document Page 7
BEGINNER'S GUIDE TO PENNZYME
The user should note that the concentration and parameter names and
values supplied in the
input data files must appear there in the same order in which they
are
referenced in the argument vectors of the rate law subroutine
(i.e. the index numbers of the array elements correspond to the
order of the input data). It should be remembered that this
routine will be called very many times in the course of the
optimization calculation and should be as efficient as possible.
In our example, additional statements are where included to
check for small values of concentrations and parameters which would
cause
an "underflow" in the calculation of the reaction rate. This check
is optional.
For easier use of PENNZYME the user should have access to a library
of these rate law subroutines, each with documentation explaining
the mechanism it represents and the order in which the
concentrations and parameters appear in their respective arrays.
Cleland has published a large collection of rate laws together with
definitions of their Kinetic constants(Biochim. Biophys. Acta.
67(1963) 104-137). A library of FORTRAN implementations of a
number of these rate laws, in the format described above, is
supplied with PENNZYME. At the present time there are 28 rate laws
in this library. A valuable tool for deriving rate laws is the
KINAL program of Cornish-Bowden, Biochem. J. 165(1977),55-59.
PENNZYME user's document Page 8
BEGINNER'S GUIDE TO PENNZYME
The Parameter File
Initial Estimates Input File
15 THIS IS THE TITLE line.
16 5
17 KM1
18 KM2 30.0
19 KI
20 KA
21 VMAX 90.0
c;FIGURE 3.2
The parameter file starts with a line indicating the number of
adjustable parameters in the rate law model whose optimal values we
wish to determine. In our example there are five of these
parameters. The next five lines- one for each parameter- each
contains the name and estimated value of a parameter. The order of
these lines must be the same as the order required by the rate law
subroutine being used. The parameter name must be in the first
eight columns of the line. The estimated value may start in any
column from the ninth on. The value is represented by a number
with or without a decimal point and exponent. If no estimate is
supplied by the user all columns past the eighth must be blank. In
our example the parameter names and values are:
Name Value
"KM1" 0
"KM2" 30
"KI" 0
"KA" 0
"VMAX" 90
A sample rate law subroutine for this data appears in figure 5.1
(see also "RATE LAW SUBROUTINE" below).
PENNZYME user's document Page 9
BEGINNER'S GUIDE TO PENNZYME
FIGURE 3.2
3.1.1 Detailed Description Of Paramter Input File -
1. Title Record: String of up to 60 characters which will appear
in output report headers. Must be the first record in the
input file and should be centered within the first 60 character
positions of the record.
2. Parameter Count Record: Number of adjustable parameters in the
rate law. Must be the second record of the file and may begin
in any character position of the record.
3. Parameter Data Records: Contain the names and initial values of
the adjustable parameters of the rate law. The first of these
words must occupy the third record position. The rest must
follow in contiguous record positions in the order in which
they appear in the parameter array referenced in the rate law
function subprogram. One must be present for each adjustable
parameter as declared in record 2. Format of the record is as
follows:
1. Parameter Name: Occupies record positions 1 through 8.
2. Parameter Value: Occupies any record positions past 8.
PENNZYME user's document Page 10
program operation
3.2 Program Operation
3.2.1 MENU After processing the input data the program outputs
a message indicating the current menu level and a prompt to
enter the desired option. Entering the option "MENU" will
output the following:
GRAPH ENTER GRAPHICS MODE
DEFN ENTER PROBLEM DEFINITION MODE
CONV ALTER CONVERGENCE CRITERIA
SMPITER SET MAX SIMPLEX ITERATIONS
DEVPAR SET MAX RELATIVE DEV OF PARAMS FOR SIMPLEX
DEVERR SET MAX REL. DEV OF LEAST-SQUARE-ERROR FOR SIMPLEX
STEP SET INITIAL SIMPLEX STEP SIZE
FPITER SET MAX FP ITERATIONS
ERREST SET ESTIMATE OF EXPECTED LEAST SQUARE ERROR FOR FP
RELERR SET MAX RELATIVE ERROR BETWEEN ITERATIONS FOR FP
PARAM ALTER PARAMETER VALUES
DELOUT DELETE OUTLIERS
TRACE SET FORMAT FOR TYPING TRACE OF OPTIM
SMPMODE SET OUTPUT DEVICE FOR SIMPLEX REPORT
SMPPER SET PERIOD BETWEEN TRACE OUTPUTS FOR SIMPLEX
FPMODE SET OUTPUT DEVICE FOR FP REPORT
FPPER SET PERIOD BETWEEN TRACE OUTPUTS FOR FP
RPTMODE SET OUTPUT DEVICE FOR FINAL REPORTS
FREEZE FREE OR PIN PARAMETER VALUES
OPTIM ENTER PROBLEM OPTIMIZATION MODE
SIMPLEX PERFORM A SIMPLEX OPTIMIZATION
FP PERFORM A FLETCHER-POWELL OPTIM.
SMPRPT REPORT RESULTS OF SIMPLEX OPTIM
FPRPT REPORT RESULTS OF FLETCHER-POWELL OPTIM
OUTLIER DETERMINE AND OUTPUT OUTLIER DATA POINTS
REPORT ENTER REPORTS MENU
DATA LIST INPUT DATA
RESID OUTPUT RESIDUALS REPORT
STATS DETERMINE AND OUTPUT STATISTICAL REPORT
PLIST LIST PARAMETER VALUES
VARCOV OUTPUT VARIANCE-COVARIANCE MATRIX
The first word of every line is entered by the user to specify
the desired action. Entering an undefined option will generate
an error message followed by Menu Desired? (If so type "yes".)
Responding with a yes will give a partial list of the options
related to the current level. This is followed by the message:
Do you want menu of other levels? A yes answer will obtain a
list of the options that are at the same level as the last
option entered. A valid option may be entered in response to
either of the two questions and will then be executed.
Examples of individual options follow, in order of
increasing sophistication.
PENNZYME user's document Page 11
program operation
3.3 Program Output
3.3.1 REPORT Submenu -
3.3.1.1 DATA Option (list Input Data) -
A sample report follows:
PENNZYME user's document Page 12
program operation
**********************************************************************
* TEST OF PENNZYME: PENNSYLVANIA ENZYME PROGRAM *
* INPUT DATA *
**********************************************************************
NUMBER OF PARAMETERS NUMBER OF CHEMICALS NUMBER OF EXPERIMENTS
5 3 32
OBSERVED STANDARD
CHEMICAL CONCENTRATIONS VELOCITY DEVIATION
POINT 1 2 3
SET # 1
1 1.2000 5.0000 3.5000 1.0800 0.0930
2 1.2000 5.0000 35.0000 0.7300 0.0540
3 1.2000 5.0000 110.0000 0.3320 0.0270
4 1.2000 5.0000 350.0000 0.1330 0.0100
5 1.2000 50.0000 3.5000 5.8100 0.4100
6 1.2000 50.0000 35.0000 2.4400 0.2400
7 1.2000 50.0000 110.0000 1.5900 0.1200
8 1.2000 50.0000 350.0000 0.6300 0.0450
9 1.2000 150.0000 3.5000 7.7900 0.5500
10 1.2000 150.0000 35.0000 3.7800 0.3200
11 1.2000 150.0000 110.0000 1.9100 0.1600
12 1.2000 150.0000 350.0000 0.8190 0.0600
13 1.2000 500.0000 3.5000 7.4000 0.6300
14 1.2000 500.0000 35.0000 4.3200 0.3600
15 1.2000 500.0000 110.0000 2.1300 0.1800
16 1.2000 500.0000 350.0000 0.7150 0.0680
SET # 2
17 12.0000 5.0000 3.5000 5.5500 0.4300
18 12.0000 5.0000 35.0000 4.1200 0.3200
19 12.0000 5.0000 110.0000 2.7400 0.2000
20 12.0000 5.0000 350.0000 1.2600 0.0930
21 12.0000 50.0000 3.5000 23.9000 2.1000
22 12.0000 50.0000 35.0000 18.5000 1.5000
23 12.0000 50.0000 110.0000 11.3000 0.9300
24 12.0000 50.0000 350.0000 4.9600 0.4100
25 12.0000 150.0000 3.5000 38.8000 3.0000
26 12.0000 150.0000 35.0000 26.3000 2.1000
27 12.0000 150.0000 110.0000 15.8000 1.3000
28 12.0000 150.0000 350.0000 6.6500 0.5500
29 12.0000 500.0000 3.5000 44.5000 3.5000
30 12.0000 500.0000 35.0000 28.9000 2.5000
31 12.0000 500.0000 110.0000 19.3000 1.5000
32 12.0000 500.0000 350.0000 7.9300 0.6300
PENNZYME user's document Page 13
program operation
**********************************************************************
* TEST OF PENNZYME: PENNSYLVANIA ENZYME PROGRAM *
* STARTING VALUES *
**********************************************************************
CURRENT LEAST-SQUARES ERROR: 6.12428E-01
INITIAL PARAMETER VALUES WERE:
KM1 1.00000E+01
KM2 1.00000E+01
KI 1.00000E+01
KA 1.00000E+01
VMAX 1.00000E+01
PENNZYME user's document Page 14
program operation
3.3.2 OPTIM Submenu -
3.3.2.1 SIMPLEX Option (perform A Simplex Optimization) -
The Simplex method is a "robust" method and does not
require a good initial estimate to locate a minimum. Its
convergence properties are less sensitive than gradient search
methods to curvature of the least-squares error surface. In
addition, if there are several local minima the Simplex method
is more likely to find the global minimum rather than a local
one, provided that the initial step size is chosen sufficiently
large.
3.3.2.2 SMPRPT Option (report Results Of Simplex Optimization) -
This report will be produced only if the algorithm has
been used at least once in the session. A sample report
follows:
PENNZYME user's document Page 15
program operation
.rm 82
----------------------------------------------------------------------
FITTING BY SIMPLEX METHOD.
CONVERGENCE ACHIEVED AFTER 218 ITERATIONS.
-ELAPSED CPU TIME: 0 MIN 5.1 SEC
RMS RELATIVE DEVIATION TARGET ACTUAL
ABOUT THE CENTROID OF -
-KINETIC PARAMETERS: 2.50000E-02 1.78445E-02
-LEAST-SQUARES ERROR: 1.00000E-03 9.09281E-04
LEAST-SQUARES ERROR: START 6.12428E-01 END 4.85735E-02
PERCENT REDUCTION: 92.07%
OPTIMAL PARAMETER VALUES WERE:
KM1 1.17673E+01
KM2 4.75639E+01
KI 3.46474E+01
KA 6.76959E+00
VMAX 9.65884E+01
This operation causes a least-squares fit of the velocity data
to be performed by the Simplex method using the current
estimate of the optimal parameter values and the current
Simplex option values. If the option values indicated that a
trace report was
3.3.2.3 FP Option (perform A Fletcher-Powell Optimization) -
The Fletcher-Powell method requires a good initial guess
of the parameter values but converges rapidly when such an
estimate is supplied. This initial estimate is automatically
supplied to the Fletcher-Powell algorithm when the Simplex
method is used first. Since it is a gradient search method it
can detect the case where the Simplex method has converged but
the least-squares error surface still has a calculable gradient
allowing the fit to be improved. Trying to do optimizations
with the Fletcher-Powell method alone can be quite expensive in
computer usage.
PENNZYME user's document Page 16
program operation
The recommended procedure is the last one; locate an
initial estimate by the Simplex method, then verify and refine
this estimate by the Fletcher-Powell method.
3.3.2.4 FPRPT Option (report Results Of Fletcher-Powell Optimization) -
This report will be produced only if the algorithm has
been used at least once during the session. A sample report
follows:
.rm 82
----------------------------------------------------------------------
FITTING BY FLETCHER-POWELL METHOD.
CONVERGENCE ACHIEVED AFTER 8 ITERATIONS.
-ELAPSED CPU TIME: 0 MIN 3.8 SEC
LEAST-SQUARES ERROR: START 4.85735E-02 END 4.84583E-02
PERCENT REDUCTION: 0.24%
OPTIMAL PARAMETER VALUES WERE:
KM1 1.17010E+01
KM2 4.90328E+01
KI 3.46888E+01
KA 6.51458E+00
VMAX 9.61637E+01
Fine Points in Using Fletcher-Powell Optimization
Starting Too Close to the Minimum
A warning message may be output by the Fletcher-Powell
optimization routine: CONVERGENCE ACHIEVED IN TOO FEW
ITERATIONS TO ACHIEVE AN ACCURATE COVARIANCE MATRIX
This means that the Fletcher-Powell optimization started at or
near the minimun and converged in fewer iterations than the
number of parameters. In this situation the H matrix will
either be calculated inaccurately or may never be calculated at
all. A tactic to get around this difficulty is to change at
least two and preferably all the paramter values by 10% and
PENNZYME user's document Page 17
program operation
repeat the Fletcher-Powell optimization. This tactic may also
be useful if the Fletcher-Powell optimization appears to be
having difficulty finding the minimum--which may happen because
it started out there.
Optimizing Error Model
Input to the PENNZYME program includes estimates of the
standard deviation of the velocity measurements. These are not
merely values obtained from a few replicate measurements, but
are the standard deviations expected from hundreds of replicate
measurements. PENNZYME weights each squared residual by the
normalized inverse variance. Clearly, such values are
available only if a particular enzymatic assay, using a
particular instrumentation, had been previously calibrated. If
the standard deviations are set to zero, owing to a lack of
information, all the weights are set to unity and PENNZYME
cannot use weighting to account for the variation in the
reliability of the experimental measurements. The extended
least squares option permits the user to estimate the unknown
variance simultaneously with identifying the optimal values of
the kinetic parameters.
Just as the user must supply a function RATE which
calculates the expected enzymatic velocity at a given set of
ligand concentrations, to use extended least squares he must
also supply a function VARIAN which calculates the expected
variance of a given experimental velocity measurement.
PENNZYME calls VARIAN once for each data point. The function
arguments are:
FUNCTION VARIAN (CONC,PARAM,VCALC) where
CONC = array of ligand concentrations forcurrent data
point
PARAM = array of error model parameters to be optimized
VCALC = expected(i.e. calculated) enzymatic velocity for
current
data point.
VARIAN may be any function of concentrations and/or
calculated velocity, but usually it is a polynomial or a power
law function of the expected velocity. The user should make
some attempt at obtaining a qualitative description of the
error structure of his data.
The error model described by VARIAN will include
adjustable parameters just as RATE does. The parameter input
file (on logical unit 1) must have initial estimates of these
parameters as well as the kinetic parameters. For example, if
the error model is a power law in the expected velocity: VARIAN
= PARAM(1) * VCALC **PARAM(2)
PENNZYME user's document Page 18
program operation
then initial estimates of error model parameters PARAM(1)
and PARAM(2) in addition to those for the kinetic parameters
are necessary. The format for the parameter input file is:
title
kinetic parameters, error parameters
kinetic parameter name <value>
. .
. .
. .
error model parameter name <value>
. .
. .
For the above example the input would resemble:
title
3,2
KM <VALUE>
VMAX <VALUE>
KI <VALUE>
CONST <VALUE>
EXPON <VALUE>
VARIAN is called only if the extended least squares option
is set by including a nonzero number of error model parameters
on the parameters input file.
PENNZYME has a default error model: VARIAN = CONST**2 +
VCALC**EXPON where CONST and EXPON are adjustable parameters.
A user-supplied function overrides this default model.
TO ENTER THE GRAPH MODE, TYPE GRAPH. A DETAILED MANUAL
FOR THE GRAPHICS COMMAND LANGUAGE FOLLOWS THE GLOSSARY BELOW.
The REPORTS level submenu
PENNZYME user's document Page 19
STATISTICS
4.0 STATISTICS
4.0.0.1 STATS Option (determine And Output Statistical Report) -
The Statistics Report contains statistical measures
relevent to the fitting calculation. The form of this report
depends upon which fitting algorithms were used, in what order
they were used, and whether or not the calculations converged.
A discussion of the statistical measures which appear in this
report is given under "STATISTICS" below.
PENNZYME computes the values of several statistics which
serve as measures of the reliability of the optimal parameter
values and as indicators of the presence of systematic errors
(bias) in the kinetic model.
Model Reliability
Residual error
A measure of the model's fit to the
kinetic data.
Pure error
The residual error expected if only
experimental errors of velocity
measurements determine the residuals.
Excess variance
(Residual error)**2 - (Pure error)**2
. A measure of bias. The ratio of these
two squared terms may be used as an
"F-test" to determine if the excess
variance is significant.
Parameter Reliability
Standard deviation
A measure of the curvature of the
error surface in the vicinity of the
minimum. Small values indicate high
curvature and a well defined minimum.
95% confidence limit
The value which when subtracted from
or added to the optimal parameter value
defines the range of values in which there
is a 95% probability of finding the true
PENNZYME user's document Page 20
STATISTICS
parameter value.
T-statistic
The ratio (parameter value / standard
deviation). If this ratio is less than a
certain value, the corresponding parameter
value is not significantly different from
zero, and the parameter is redundant.
PENNZYME performs this test automatically.
**********************************************************************
* TEST OF PENNZYME: PENNSYLVANIA ENZYME PROGRAM *
* STATISTICS REPORT *
**********************************************************************
PURE ERROR: 2.78787E-01
RESIDUAL ERROR: 4.84583E-02
AVERAGE OBSERVED VELOCITY: 9.44122E+00
(RES. ERR.)/(AV. VEL.) = RELATIVE ERROR: 5.13264E-03
(RES. ERR.)**2 - (PURE ERR.)**2 = EXCESS VARIANCE: -7.53738E-02
(RES. ERR.)**2 / (PURE ERR.)**2 = MEASURE OF BIAS: 3.02130E-02
PARAMETER
NAME VALUE STD. DEV. CNFD. LIM. T-STATISTIC < T(.95)
KM1 1.17010E+01 1.866E+00 3.830E+00 6.26950E+00
KM2 4.90328E+01 9.786E+00 2.008E+01 5.01032E+00
KI 3.46888E+01 2.196E+00 4.506E+00 1.57980E+01
KA 6.51458E+00 1.057E+00 2.169E+00 6.16224E+00
VMAX 9.61637E+01 1.202E+01 2.466E+01 8.00187E+00
An arrow("<---") will appear in the column directly beneath the
header " , t(.95)" for any parameters whose t-Statistic exceeds the
95% confidence limit. If the arrow appears for a parameter it is
likely that the parameter is redundant for the data set used.
4.0.0.2 RESID Option (output Residuals Report) -
A sample report follows:
PENNZYME user's document Page 21
STATISTICS
**********************************************************************
* RAPID-EQUILIBRIUM BINARY REACTION WITH HILL COEFFICIENTS *
* RESIDUALS REPORT. *
**********************************************************************
OBS V(CALC)V(OBS) V(CALC) WEIGHT WEIGHTED
-V(OBS) RESIDUAL
SET # 1
1 1.5O1E-O1 1.5O4E-O1 -2.946E-O4 1.73OE+OO -5.O98E-O4
2 1.869E-O1 1.9O5E-O1 -3.6O8E-O3 1.551E+OO -5.597E-O3
3 2.473E-O1 2.451E-O1 2.221E-O3 1.173E+OO 2.604E-O3
4 3.658E-O1 3.69OE-O1 -3.236E-O3 6.120E-O1 -1.981E-O3
5 5.7O8E-O1 6.25OE-O1 -5.423E-O2 2.469E-O1 -1.339E-O2
SET # 2
6 1.672E-O1 1.7O2E-O1 -3.OO8E-O3 1.656E+OO -4.98OE-O3
7 2.O82E-O1 2.O51E-O1 3.O66E-O3 1.422E+OO 4.361E-O3
8 2.755E-O1 2.667E-O1 8.774E-O2 1.OO6E+OO 8.822E-O3
9 4.O74E-O1 3.883E-O1 1.91OE-O2 4.954E-O1 9.462E-O3
1O 6.357E-O1 6.25OE-O1 1.O75E-O2 1.965E-O1 2.112E-O3
SET # 3
11 1.87OE-O1 1.923E-O1 -5.282E-O3 1.55OE+OO -8.19OE-O3
12 2.328E-O1 2.235E-O1 9.35OE-O3 1.264E+OO 1.182E-O2
13 3.O81E-O1 3.OO8E-O1 7.339E-O3 8.375E-O1 6.146E-O3
14 4.557E-O1 4.2554E-O1 3.O21E-O2 3.948E-O1 1.193E-O2
15 7.111E-O1 7.519E-O1 -4.O77E-O2 1.548E-O1 -6.313E-O3
SET # 4
16 2.O94E-O1 2.128E-O1 -3.384E-O3 1.414E+OO -4.787E-O3
17 2.6O7E-O1 2.516E-O1 9.137E-O3 1.O91E+OO 9.967E-O3
18 3.45OE-O1 3.419E-O1 3.143E-O3 6.835E-O1 2.148E-O3
19 5.1O3E-O1 4.819E-O1 2.839E-O2 3.123E-O1 8.866E-O3
2O 7.963E-O1 8.696E-O1 -7.33OE-O2 1.216E-O1 -8.915E-O3
SET # 5
21 2.263E-O1 2.439E.O1 -1.757E-O2 1.306E+OO -2.294E-O2<--
22 2.818E-O1 2.721E-O1 9.7O1E-O3 9.7O6E-O1 9.416E-O3
23 3.729E-O1 3.883E-O1 -1.538E-O2 5.896E-O1 -9.O7OE-O3
24 5.515E-O1 5.714E-O1 -1.989E-O2 2.654E-O1 -5.279E-O3
25 8.6O6E-O1 1.O2OE+OO -1.598E-O1 1.O3OE-O1 -1.646E-O2
CURRENT LEAST-SQUARES ERROR; 1.O625OE-O2
CURRENT PARAMETER VALUES WERE:
KMPEP 2.97865E+O1
KMADP 1.838O7E+O2
VMAX 1.O787OE+OO
HCPEP 1.2O112E+OO
SIG 3.96878E-O2
ALPHA 4.29188E+OO
The markers " <--- " at the right margin flag those POINTS
PENNZYME user's document Page 22
STATISTICS
identified as outliers by the program (see index 8 above). The
message "DELETED" appears in the same position for any points which
were removed from the fitting calculation by the user, using the
DELOUT option which is described below. (When the marker " <--- "
appears in any report involving parameters, it indicates a
parameter which is not significantly different from zero in a
statistics report, and a parameter which is pinned in any other
report involving parameters.)
4.0.0.3 OUTLIER Option (determine And Output Outlier Data Points) -
This operation compares the absolute value of the weighted
residuals to twice the standard error. Any observations
(data-points) whose weighed residuals exceed this limit are
numbered, inserted in an internal "outlier table" and printed on
the user's terminal. There is room in the table for ten outliers.
If more than this are found a warning message and the number of the
velocity observation is printed for each outlier for which there is
no room in the table. A sample report follows:
ENTER DESIRED OPTION > OUTLIER
OUTLIER DATA POINT V(OBS) RESIDUAL
1 6 2.63900E+00 3.37986E-01
2 56 2.42100E+01 2.77115E+00
ENTER DESIRED OPTION >
4.0.0.4 PLIST Option (list Parameter Values) -
This operation produces the following report:
ENTER DESIRED OPTION > PLIST
CURRENT PARAMETER VALUES WERE:
KM1 1.24335E+01
KM2 5.06060E+01
KI 3.54142E+01
KA 6.91388E+00
VMAX 1.01837E+02
.s2
(The marker " <--- " indicates a pinned parameter.)
PENNZYME user's document Page 23
STATISTICS
.s2
ENTER DESIRED OPTION > VARCOV
.b3
This option prints out the variance -covariance matrix.
****************************************************************
* RAPID EQUILIBRIUM BINARY REACTION WITH HILL COEFFICIENTS *
* VAR-COVAR MATRIX *
****************************************************************
KMPEP KMADP VMAX HCPEP
1.2454E+01 1.2915E+00 1.8991E-01 -2.2180E-01 KMPEP
7.2989E+01 2.6871E-01 -4.0922E-03 KMADP
6.6961E-03 -1.0788E-02 VMAX
2.3825E-02 HCPEP
****************************************************************
* RAPID-EQUILIBRIUM BINARY REACTION WITH HILL COEFFICIENTS *
* NORM. VAR-COVAR MATRIX *
****************************************************************
KMPEP KMADP VMAX HCPEP
1.0000E+00 4.2836E-02 6.5762E-01 -4.0720E-01 KMPEP
1.0000E+00 3.8436E-01 -3.1032E-03 KMADP
1.0000E+00 -8.5410E-01 VMAX
1.0000E+00 HCPEP
PENNZYME user's document Page 24
STATISTICS
4.0.1 DEFN (PROBLEM DEFINITION) Submenu -
4.0.1.1 CONV Option (alter Convergence Criteria) -
This operation allows the user to alter the values of
variables which were initialized from data in the "method specific"
input file. These variables are described in the section "INPUT
DATA". The following is a sample of the dialogue which the user
will encounter:
Notice that the user may examine a variable's value without
changing the value by entering "n" in response to the "new value"
prompt. Also note that the variable types are mixed. The user
need not know the type of a particular variable (although it is
hoped that he would) since the type conversion of user supplied
data is done automatically.
4.0.1.1.1 Index 3 - "Delete An Outlier" -
"Outlier deletion" is an operation which allows the user to
remove from the fitting calculation points whose weighted residuals
exceed twice the least-squares error. This is done by removing the
weights of these points and renormalizing the remaining weights.
Points which have been deleted may be returned to the calculation
by the reverse operation "Reinsert a deleted point". Both of these
operations are provided in a control subsection initiated by entry
of index value 3 in response to the options prompt. A sample
dialogue follows:
ENTER DESIRED OPTION >
MANIPULATE OUTLIERS.
PIN
(describe here pinning parameters)
.hl4PARAM option (alter parameter values)
This operation is provided to allow the user to adjust his estimate
of the optimal parameter values prior to or between fitting
attempts. Sample dialogue follows:
.tp40
.b3;.lt
ENTER DESIRED OPTION > PARAM
ALTER PARAMETERS.
PLEASE ENTER PARAMETR INDEX(OR <CR>):2
PENNZYME user's document Page 25
STATISTICS
PARAMETR KM2 IS 10.00000
ENTER NEW VALUE (OR N, FOR NO CHANGE):50
PLEASE ENTER PARAMETR INDEX(OR <CR>):4
PARAMETR KA IS 10.00000
ENTER NEW VALUE (OR N, FOR NO CHANGE):N
PLEASE ENTER PARAMETR INDEX(OR <CR>):97
INDEX 97 IS OUT OF BOUNDS, TYPE <CR> FOR A LIST OF OPTIONS.
PLEASE ENTER PARAMETR INDEX(OR <CR>):5
PARAMETR VMAX IS 10.00000
ENTER NEW VALUE (OR N, FOR NO CHANGE):97
PLEASE ENTER PARAMETR INDEX(OR <CR>):
(ENTER "-1" IF NO MORE CHANGES)
INDEX PARAMETER NAME VALUE
1 KM1 1.00000E+01
2 KM2 5.00000E+01
3 KI 3.00000E+01
4 KA 1.00000E+01
5 VMAX 9.70000E+01
PLEASE ENTER PARAMETR INDEX(OR <CR>):-1
'BYE!
PLEASE ENTER CONTROL INDEX(OR <CR>):
Notice the mistake made by the user in selecting the parameter
index. The number entered ("97") was meant to be the value of
parameter #5 rather than a parameter index.
4.0.1.2 TRACE Option (set Format For Typing Trace Of Optim) - I
f the option values indicated that a trace report was to appear on
the user's terminal it will be typed at this point. See the sample
session dialogue for a typical trace report. Upon return to
interactive mode the parameter values will be the optimal values as
determined by the fitting operation. A sample trace report
follows:
PENNZYME user's document Page 26
STATISTICS
**********************************************************************
* TEST OF PENNZYME: PENNSYLVANIA ENZYME PROGRAM *
* SIMPLEX TRACE REPORT. *
**********************************************************************
ITERATION 25 RMS RELATIVE DEVIATION OF KINETIC PARAMETERS 1.29966E-01
CURRENT LEAST-SQUARES ERROR: 1.32720E-01
CURRENT PARAMETER VALUES WERE:
KM1 2.53438E+00
KM2 7.25742E+00
KI 2.79929E+01
KA 1.79910E+01
VMAX 3.24603E+01
ITERATION 50 RMS RELATIVE DEVIATION OF KINETIC PARAMETERS 3.44595E-02
CURRENT LEAST-SQUARES ERROR: 1.11128E-01
CURRENT PARAMETER VALUES WERE:
KM1 4.09943E+00
KM2 6.96557E+00
KI 3.77969E+01
KA 2.00883E+01
VMAX 3.63544E+01
A least-squares fit is found by the Fletcher-Powell gradient search
method, using the current estimate of the optimal parameter values
as a starting estimate, and using the current values of the F-P
option selection indices. A trace report is produced on the user's
terminal if selected through the F-P option values. A sample trace
report follows:
PENNZYME user's document Page 27
STATISTICS
**********************************************************************
* TEST OF PENNZYME: PENNSYLVANIA ENZYME PROGRAM *
* FLETCHER-POWELL TRACE REPORT. *
**********************************************************************
ITERATION 3
CURRENT LEAST-SQUARES ERROR: 4.45696E-02
CURRENT PARAMETER VALUES WERE:
KM1 1.17266E+01
KM2 4.77989E+01
KI 3.48247E+01
KA 6.77243E+00
VMAX 9.62696E+01
ITERATION 6
CURRENT LEAST-SQUARES ERROR: 4.45120E-02
CURRENT PARAMETER VALUES WERE:
KM1 1.17035E+01
KM2 4.90169E+01
KI 3.46901E+01
KA 6.51486E+00
VMAX 9.61774E+01
PENNZYME user's document Page 28
STATISTICS
4. Program Mode Record: Contains data which indicates the initial
optimization strategy and the report generation mode. This record
must follow the last parameter data record and has the following
internal format:
1. Simplex Selector: Must be first value in record. If value is
greater than or equal to zero then a Simplex mode record is
expected.
2. Fletcher-Powell Selector: Must be second value in record. If
value is greater than or equal to zero then a Fletcher-Powell
mode record is expected, and the Fletcher-Powell mode record
must follow the:
1. Program Mode Record if the Simplex selector value is less
than zero.
2. Simplex Mode Record if the Simplex selector value is
greater than or equal to zero.
3. Report Mode Indicator: Must be third number in record. If its
value is "2" the residuals report will be included in the final
report.
5. Simplex Mode Record: Contains input relevant to use of the Simplex
algorithm. All of the following numbers must be present in the
indicated order. A value of zero for any number will cause the
indicated default values to be used.
1. Maximum iterations of algorithm allowed.
Default: 50 + 40 * number-of-adjustable-parameters
2. Initial Simplex Step Size. Should be large enough to assure
that the minimum will be reached before contraction is
completed. It is expressed as a fraction of the initial
parameter values.
Default: 1.0
3. Convergence Mode.
= 1, Test for convergence will be on the RMS relative deviation
of the parameters of the Simplex from those defining the
centroid.
= 2, Test for convergence will be on the RMS relative deviation
of the least-squares errors of the vertices of the simplex
about that of the centroid.
= 3, Both of the above must be satisfied.
Default: 1
4. Convergence criterion "a". Convergence criterion for test #1 .
Default: 0.05
PENNZYME user's document Page 29
STATISTICS
5. Convergence criterion "b". Convergence criterion for test #2 .
Default: 0.05
6. Trace mode.
= -1, no trace report.
= 0, same as 1 (default).
= 1, trace report on user's console.
= 2, trace report on printer.
= 3, trace report in both locations.
Default: 0
7. Trace period. Number of iterations to be performed between
trace printouts.
Default: 25
6. Fletcher-Powell Mode Record: Contains input relevant to use of the
Fletcher-Powell algorithm.
1. Maximum iterations of algorithm allowed.
Default: 10 * number-of-adjustable-parameters
2. Estimate of minimum sum of squared errors obtainable.
Default: 0
3. Expected relative error of the parameters.
Default: 1 * 10**(-6)
4. Trace mode. Same options as for Simplex.
Default: 0
5. Trace period. Number of iterations to be performed between
trace printouts.
Default: 5
4.0.1.3 Simplex And Fletcher-Powell Mode Record Contents -
Simplex Mode Record Contents:
1. Max. iterations of algorithm allowed,
2. initial Simplex step size,
3. convergence mode,
4. convergence criterion "a",
PENNZYME user's document Page 30
STATISTICS
5. convergence criterion "b",
6. trace mode,
7. number of iterations between trace printouts.
Fletcher-Powell Mode Record Contents:
1. Max. iterations of algorithm allowed,
2. estimate of the minimum sum of squared errors
obtainable,
3. expected relative error of the parameters,
4. trace mode,
5. number of iterations between trace printouts.
PENNZYME user's document Page 31
STATISTICS
4.0.1.4 Description Of Simplex And Fletcher-Powell "Mode Record" Variables: -
Variable Description
Simplex specific:
Max. iterations A good rule of thumb for this algorithm is
50 iterations plus 40 for each parameter
to be fit.
Initial step size This should be large enough to enclose the
minimum error point in parameter space .
It is expressed as a fraction of the
initial parameter estimate.
Convergence mode = 1, test for convergence will be on
the RMS relative deviation of
the parameters of the Simplex from
those of the centroid.
= 2, test for convergence will be on
the RMS relative deviation of the
least-squares errors of vertices
of the Simplex about that of the
centroid.
= 3, both of the above must be satisfied.
Criterion "a" Maximum allowable RMS deviation of the
parameter values of the simplex vertices
and centroid for convergence under
convergence modes 1 and 3.
Criterion "b" Maximum allowable RMS deviation of
least-squares errors of the simplex
vertices and centroid for convergence
under convergence modes 2 and 3.
Trace mode = -1, no trace report.
= 0, same as 1 (default).
= 1, trace report on user's console.
= 2, trace report on printer.
= 3, trace report at both locations.
Trace period Number of iterations to be performed
between trace printouts.
Fletcher-Powell specific:
Max. iterations A good rule of thumb is 10 iterations for
each parameter.
Est. of sum of squares If this is not known use "0" .
PENNZYME user's document Page 32
STATISTICS
Est. of error Not less than 10**(-6).
Trace mode See Simplex specific.
Trace period See Simplex specific.
PENNZYME user's document Page 33
TECHNICAL GLOSSARY
5.0 TECHNICAL GLOSSARY
Centroid The centroid of all the vertices of the
simplex but the one with the largest
least-squares error.
Convergence criteria Conditions which must be fulfilled for an
iterative calculation to be considered to
have converged.
Gradient search method An optimization method which uses the
gradient of a function to locate the
extrema of the function.
Parameter Rate law constant to be calculated by
PENNZYME which will result in the best fit
of the experimental data by the rate law
type.
Parameter space The n-dimensional space defined by the n
different rate law parameter variables.
Record A collection of related items of data
treated as a unit.
Residual Difference between calculated and observed
velocity values.
Run-time The interval during which the program is
executing.
Simplex The n+1 points in parameter space(where n
is the number of parameters) which are
used by the Simplex algorithm to locate
the error function minimum.
Vector A variable type consisting of an ordered
set of values.
Vertices The individual points defining a simplex.
1.0 GRAPHICS COMMANDS
There are three types of commands currently implemented in the
graphics system. They are the Setup, Graph-Type and Data-Selection
commands. All command types use the same general input format to
minimize the number of special cases the user must remember.
The Setup commands define the general physical appearance of
the graph. They are used to specify the size of the graph, axis
numbering and labelling, etc. The data is organized as a table,
each line containing the values of the variables to be graphed.
For PENNZYME the data is organized as lines of data, the first
columns being the various chemical concentrations. The next column
contains the observed velocities at the chemical concentrations;
the last column contains the standard deviations associated with
the observed velocities. The Graph-Type commands specify the
variables (columns) to be graphed. Unless restricted by the
Data-Selection commands, all the lines of data will be used when
graphing.
The Graph-Type commands thus specify the variables to be
graphed and also the type of representation that is to be used.
These include displaying data as discrete or connected points, with
or without error bars and displaying several related curves at once
as in the case of PENNZYME specific graphics.
The Data-Selection commands allow the user to define the range
of data points to be graphed for each individual curve. The data
points can be selected by input line using expressions of the
following type: all points up to line 24, points from line 2
through line 32. They can also be selected by specifying a data
set number. This selects one or several of predefined data subsets
such as those defined in PENNZYME. One can also select data lines
by having any line for which the value of a variabale matches that
specified. For example, include all lines for which Mgconc = 3.O.
The complexity of the commands varies from Setup commands,
usually a simple assign statement (set x-axis length to 8 inches)
to the Graph-type commands where the user must include variable
names for the x, y and error components, separated by operators as
well as any optional switches. In between these two extremes lie
the Data-Selection commands where the user enters an expression,
usually a series of line numbers separated by commas.
In all of the commands, the first word must be the command
specifier. There are currently l5 of these including those
implemented for PENNZYME. All of the commands, variables,
operators and switches can be entered by the user as unambiguous
abbreviations of their full representation. They are all separated
by blank characters except for the variable names which allow
imbedded blanks and therefore are terminated by an operator, a
switch, etc. Command specifiers and switches can optionally be
terminated by an equals sign. This makes for a more sentence-like
command line when using the Setup commands and some switches where
PENNZYME user's document Page 2
GRAPHICS COMMANDS
a value is being assigned ('XLENGTH = 8'). Variable names are
those defined by the user either as specified to the stand-alone
graphics or as specified by a host program. Additional names can
be defined for quantities previously unnamed in the host system but
graphed in the graphics system. This is described later in the
description of the interfacing routine. As an example, in
PENNZYME, quantities were not assigned names by the user since only
chemical names are required. These were the observed and
calculated velocities, the associated standard deviations and the
residuals. They were then assigned the names 'VELOCITY', 'VCALC'
'ERROR' and 'RESIDUAL' in the interfacing routines.
Operators are used to separate variable names and to indicate
the relationship between variables. For example, the '.VS.'
operator indicates that the first variable is to be the ordinate
and the second one the abscissa. The '.ERR.' operator uses the
next variable to generate error bars. Operators must be enclosed
by periods to indicate that they are not part of the variable
names.
A switch is text entered by the user at the end of a command
which modifies the value of a graph parameter. The parameters are
initially set to default values and reset to these values when the
switch is turned off. The effect of a switch is limited to the
command to which it is appended. If it is appended to a setup
command, it will stay active until turned off. When appended to a
graph-type command, it is only active for that command and does not
affect succeeding similar graph-type commands. The general format
of a switch is a slash (/) followed by the switch name, which is of
the same format as the command specifiers. If required, this is
followed by a value to be assigned to the switch. One or more
switches may be used in a command, each being separated by a slash.
Switches are meant to be used to alter a convenient default value.
If the default values are often inappropriate then they should be
modified within the program. Defaults for setup commands, for
example, are set once for the entire program whereas the default
symbols for the graph-type commands are rotated through the eight
special symbols.
1.1 Overview Of Process Of Generating A Graph
The graphics were designed for use with a storage tube
graphics display. This requires that the user enter all the
commands and that they be processed before the graph is initiated:
otherwise, this would require partial erasure of the screen which
is not possible on storage tubes.
In a typical graph, the user would not change the defaults for
the axis lengths and origins. It might be desired, however, to
label the axes, and this would be done using the label commands
followed by a literal string containing the text of the label. All
of the setup commands can be entered at anytime prior to the actual
PENNZYME user's document Page 3
GRAPHICS COMMANDS
graphing since they only take effect after all the other commands
have been entered. At this point the command to define a curve
would be entered. In the simplest of cases this consists of the
DRAW command:
DRAW Y .VS. X
Entering the'PLOT' command will then display a graph using the
entire range of points and connecting them with straight line
segments. The scaling and numbering of the axes would be
calculated to insure the largest graph possible while maintaining
reasonable axis numbering values. The appropriate axis labels
would also be displayed.
The data selection command was designed to allow the user to
define the range of data to be graphed and also to make
specification of multiple curve graphs quicker. The typical
'SELECT' command would be of the type:
SELECT 1-15,16-18
Select rows 1 through 15 and 16 through 18 for display. The
select command must be entered after the graph-type command it is
to modify. If several curves of the same variables were to be
graphed then one could alternate between the graph-type and the
select commands for each curve. For a family of 4 curves this
would require 8 command lines. To reduce this, the graph-type
command can be entered only once and each subsequent 'SELECT'
command will refer to that graph-type command. This reduces the
number of commands to 5 lines, one for the graph-type followed by 4
'SELECT' commands. When this feature is used for displaying
families of points, the symbol defaults are automatically rotated
through the eight available special symbols for each set of points.
1.2 Description Of Graphic Commands And Their Effects
These consist of three basic types: LABEL, ORIGIN, and LENGTH.
Each of these must be prefixed by an 'X' or a 'Y' to specify the
axis being set (e.g., 'XLABEL'). These are input in response to
the prompt [] from the program.
LABEL command
The 'LABEL' command draws up to 50 characters (including
blanks) of text centered below or alongside the designated axis.
If no label command is entered, no label will be drawn. Once it is
PENNZYME user's document Page 4
GRAPHICS COMMANDS
defined, it will appear on every succeeding graph until removed by
a new label command.
[] XLABEL = 'X-AXIS'
[] YLAB = carriage return (Remove YLABEL)
ORIGIN command
The 'ORIGIN' command permits the user to place the lower left
corner of the graph at any distance in inches from the lower left
corner of the physical screen. This is useful for moving the graph
around to ensure best placement of the graph within the physical
limits of the screen. This is required when using different
terminals which have different characteristics. The 'ORIGIN'
command does not affect the values at the origin of the variables
graphed.
[] XORIGIN = O.5
[] YOR = carriage return (Reset to default value)
LENGTH command
The 'LENGTH' command specifies the length of the axis in
inches from the point defined by default or by the 'ORIGIN'
COMMAND. The switch 'NUMBER' may be used to ensure or inhibit ('NO
NUMBER') the output of axis numbering.
[] XLENGTH = 7
[]YLEN = 6/NO NUMB
GRAPH-TYPE commands
These consist of the two basic commands; 'DRAW' and 'POINT'
plus those defined specifically for use with PENNZYME; 'FITPLOT',
'RESIDPLT' and 'CALC'.
DRAW and POINT commands
The 'DRAW' command will generate a graph of y vs. x, using
the data values of the first variabale for y and those of the
second variable for x. The two variables are separated by the
'.VS.' operator. The 'DRAW' command connects each point by a line
segment without representing the data point by a symbol. The
'POINT' command displays the data as unconnected symbols. If error
bars are desired, then a third variable must be specified by the
'ERR' operator.
PENNZYME user's document Page 5
GRAPHICS COMMANDS
[] DRAW YVAR .VS. XVAR
[] DRAW YVAR .VS. XVAR .ERR. STDEV
[] POINT YVAR. VS. XVAR
FITPLOT command
The 'FITPLOT' command is the most often used by the PENNZYME
user. It generates a graph like that specified by the 'POINT'
command and then superimposes a curve of calculated velocities
interpolated polated over the range of the chemical concentration
variable specified. The velocities are calculated using the rate
law function linked to PENNZYME. This graph only makes sense if
the velocity is the ordinate and a chemical concentration the
abscissa. All other chemical concentrations should be fixed over
the range that is being graphed. No checking is performed by the
program for nonsensical graphs. A warning message is issued when
more than one chemical concentration is varied. The user then has
the option of completing the graph or starting over.
[] FITPLOT VEL .VS. CHEMCON
[] FITPLOT VEL .VS. CHEMCON .ERR. STDEV
RESIDPLT Command
The 'RESIDPLT' command is used to display the systematic error
of a model and the systematic and random error of the experimental
data. A graph is generated with the abscissa centered vertically
and the data points displayed as symbols. The 'Residplt' command
should always have the residuals as the ordinate, the abscissa
being any of the chemical concentrations or the observed calculated
velocity. Dashed lines are also graphed at plus or minus the 95%
confidence t-statistic times the residual error. This flags
outlying data points.
[] RESIDPLT RESID .VS. CHEMCON
[] RESIDPLT RES .VS. VELOC (The observed velocity)
[] RESID RES .VS. VCALC (The calculated velocity)
CALC command
The 'CALC' command is similar to the 'FITPLOT' command and is used
when more than one chemical concentration is varying at once.
'CALC' does not interpolate over the chemical concentrations but
rather displays the calculated velocity at the experimental
chemical concentrations and then connects these with line segments.
[] CALC VEL. VS. CHEMCON
[] CALC VEL .VS. CHEMCON. ERR. STDEV
PENNZYME user's document Page 6
GRAPHICS COMMANDS
1.3 Switches For The Graph-Type Commands
SYMBOL switch
Instead of representing data values as discontinuities in a
curve, this switch allows the user to represent the data points as
symbols. Any keyboard character or one of the eight special
symbols can be used. All the points of the set or curve will have
the same symbol. This switch is also useful for modifying the
default symbol value of the 'POINT' command. A keyboard character
will be used if it is enclosed by single quotes. The special
symbols, square, diamond, etc. are selected by using an integer in
the range 1 through 8. These may not be available for all
installations.
[] DRAW VEL .VS. CHEMCON/SYMB = '*' use an asterisk
[] POINT YVAR .VS. XVAR/SYMB = 2 use the second special symbol
SIZE switch
The 'SIZE' switch allows the user to specify the size of the
symbols in inches if these are present. This is useful when
displaying error bars since error bars are only drawn when their
vertical dimension exceeds the vertical dimension of the symbol.
Altering the size of the symbol allows control over the error bars
that are to be shown.
[] FITPL VEL .VS. CHEMCON/SYM = 'X'/SIZE = O.2
SPACE switch
The 'SPACE' switch allows the user to specify the amount of
space left araound a symbol. This prevents a connecting line
segment from traversing the symbol. This does not stop unrelated
line segments from traversing the symbol. The space is given in
inches. This switch does not apply to commands which do not
connect the data values.
[] DRAW YVAR .VS. XVAR/SYM=2/SPACE=0.2
1.4 Data-Selection Commands
There are currently three ways to select data to be graphed.
All three used the command specifier 'SELECT' and vary in the type
of arguments expected.
PENNZYME user's document Page 7
GRAPHICS COMMANDS
The 'SELECT' command followed by the symbol '#' indicates that
data input is to be selected by line number. The lines are
selected by expressions involving line numbers separated by commas,
a range of lines specified by bounds, and a range of lines
specified by an inequality operator.
[] SELECT #1-5 Selects lines 1 through 5
[] SELECT #>6 Selects lines 7 through end of data
[] SELECT #<20 Selects lines 1 through 19
[] SELECT #1,2 Selects lines 1 and 2
[] SELECT #1,2,6-9,11,>17 (The various modes can be mixed
when separated by commas.)
The 'SELECT' command followed by a variable name indicates
that data will be selected if it satisfies a constraint. The only
constraint currently allowed is that the data equal a value
specified. The current implementation will only select the first
contiguous set of data lines satisfying the constraint.
[] SEL CHEMCON .EQ. 52 Select the line(s) where the
variable CHEMCON has the value 52.
The 'SELECT' command followed by the symbol '$' indicates that
data ranges will be selected according to predefined data subsets.
These data sets cannot be defined within the graphics. They must
be defined by a host system which allows the user to do this such
as PENNZYME. Several of the data subset indices can be used in the
same SELECT command as long as they are separated by commas.
[] SEL $2 Select Set number 2
[] SEL $2,5 Select Sets 2 and 5
PLOT command
The 'PLOT' command is the last command entered and is used to
initiate the display of the graphics . This should be entered once
all the desired graph parameters have been set for the current
graph. Output can be sent to a hard copy plotter by following the
'PLOT' command with the word 'PLOTTER'. Example [] PLOT = PLOTTER
Entering a carriage return will erase the graph and allow the user
to define a new graph. The previous graph may be redisplayed
simply by entering the 'PLOT' command again, if no Graph-type or
Data-selection commands have been entered since. The Setup
commands may be used to vary the layout of the graph. Similarly
when the graphics is used as an option within a host program, upon
reentering the graphics the last graph can be regenerated by
entering the 'PLOT' commands. The new graph will then use the new
data values if these have been modified by the host program. With
PENNZYME, one can display a quality of fit graph, exit the graphics
and optimize the kinetic parameters, enter the graphics and display
PENNZYME user's document Page 8
GRAPHICS COMMANDS
the new quality of fit graph using the optimum kinetic parameter
values.
QUIT command
The QUIT command is used to exit from the graphics program.
PENNZYME user's document Page 9
EXAMPLES OF GRAPHICS COMMANDS AND RESULTING GRAPHS
2.0 EXAMPLES OF GRAPHICS COMMANDS AND RESULTING GRAPHS
[] DRAW YVAR .VS. XVAR
[] PLOT
|
| /\
| / \
| / \ /\
| / \ / \
| / \ / \
| / \ / \
| -----/ \----
|/
|--------------------------------------------
[] DRAW YVAR .VS. XVAR/SYM = O
[] PLOT
|
| O
| /\
| / \ O
| / \ / \ Use symbol O
| / \ / \ to denote point
| / \ / \
| /O---O \_O/ \
|/
|---------------------------------------------
[] DRAW YVAR .VS. XVAR
[] POINT UVAR .VS. TVAR
[] PLOT
|
| /\ Two different
| / \ sets of
| / \ x /\ variables
| / \ / \ using different
| / x \ / \ representations
|x / \/ \ x may be graphed
| ------ ----- simultaneously
| x
|-------------------------------------
PENNZYME user's document Page 10
EXAMPLES OF GRAPHICS COMMANDS AND RESULTING GRAPHS
[] DRAW YVAR .VS. XVAR/SYM = O
[] SELECT 1-3,5,6
[] PLOT
|
|
| The select command
| -O------O-- is used to include
| O-----O- / or exclude data
| / \-- / values from the
| / \-O- curve defined by
|O the preceding
| graph-type command
|--------------------------------------
[] DRAW YVAR .VS. XVAR
[] SELECT 1-3,5,6
[] POINT UVAR .VS. TVAR
[] PLOT
| The select command does
| not affect graph-type
| x commands which follow
| it
|
| x
| ------- /
| / \ /
| x / \ --/
| / \/x
| / x
|-----------------------------------
[] DRAW CVAR .VS. DVAR The select command
[] SELECT 1-5 when used repeat-
[] SELECT 6-10 edly generates
[] PLOT distinct curves
based on the values
| /\ set by the preced-
| / \ ing graph-type
| / /\ \ command. This is
| ------/ / \ \ used to display
| / / \ \ families of curves
| / /----/ \ \ if the data is
|/ / \ \------ organized in that
|------------------------------------ fashion.
PENNZYME user's document Page 11
EXAMPLES OF GRAPHICS COMMANDS AND RESULTING GRAPHS
2.1 Graphics Commands Specific To PENNZYME
[] FITPLOT VEL .VS. MGADP Display observed
[] SELECT 1-5 velocities and
[] PLOT superimposed curve
calculated veloci-
| ties at interpola-
| O------O ted concentrations.
| O---/ This only makes
| O / sense if the first
| /- variable is the
| O/ velocity variable.
|/
|-------------------------------------
[] FITPLOT VEL .VS. MGADP. ERR. STDEV Displays error
[] SELECT 1-5 bars. These are
[] PLOT drawn only if they
exceed the size of
| - - the symbol.
| | O Vertical lines are
| O - missing from error
| - | bars because they
| - O -/------ cannot be repre-
| | - / sented by print
| /O/-- symbols. (The
| / | points do not fit
| /O - the line so there
|/ is room for the
|-------------------------------------- error bars.)
[] FITPLOT VEL .VS. MGADP
[] SELECT 1-5
[] SELECT 6-1O
[] PLOT
| O
| O/-----
| -----
| O/
| --
| / x Multiple curves
| O/ /-----
| - x---
| / x---/
| O/ --/
| / /
| / /-x
|O/-/
|//x
|---------------------------------------
PENNZYME user's document Page 12
EXAMPLES OF GRAPHICS COMMANDS AND RESULTING GRAPHS
[] CALC VEL .VS. MGADP
[] SELECT 1-5
[] PLOT
| O ------ Displays calculated
| /---/ velocities
| O/ not interpolated
| -/ over concentrations
| /
| O/
| /
| O/
| /
|--------------------------------
[] RESIDPLT RESID .VS. MGADP
[] PLOT
| O
|-------------------- Uses one symbol to
| O O denote data values
|-------------------- of entire data. This
| only makes sense if
| O O O O the first variable
| O is the Resid variable.
|--------------------
|--------------------
[] RESIDPLT RESID .VS. MGADP
[] SELECT 1-5
[] SELECT 6-1O
[] PLOT
O
|----------------
| x O x Uses a different
| symbol for each
| x data subset
|--------------------
| O O
| x O
| x
|---------------------
|--------------------