LMMpro User`s Manual version 1.xx
Contents
1. 22y db To simplify the notation for number of points n Let Sx x Let Sy y Let Sxy 2 x y Let Sxx 2 x Let nb b From step 5 solve for m b Sx Sxy m Sxx If the intercept s value b is fixed then all parameters are known and solve for m From step 6 solve for b b Sy m Sx n If the slope s value m is fixed then all parameters are known and solve for b If neither the slope nor the intercept are known then combine step 9 and 8 and solve for m 0 m Sxx Sx Sy m Sx n Sxy 0 m Sxx SxSy n m SxSy n Sxy m n Sx Sxx Sxy SxSy n Sxy Sx Sy m n Sx Sxx Now plug this value of m into step 9 to get the value of b LMMpro User s Manual v 1 xx 14 Note that data will be algebraically balanced above and below the regression line This remains true even if the m or b values but not both are fixed by the user This regression analysis is easy enough to do by hand It is even easier today with computers that do it for us Note that his method was discovered by Carl Friedrich Gauss in 1794 He was 17 years old at the time The method was published by Adrien Marie Legendre in 1806 lt Reference CTT 6 gt LMMpro User s Manual v 1 xx 15 NLLS Regressions With computers it is rather easy to optimize a parabolic equation using nonlinear methods The optimization minimizes the squares of the error terms and hence these methods are known as Nonlinear Leas2. Typically you would use one or the other If so then you can hide the start button of the one that you rarely use Note however that at least one of these must be checked e Restore All Options to Defaults As stated click here to reset all options to the way it was when you first downloaded the software This affects the options only It does not affect the registration information lt Reference CTS 4 gt LMMpro User s Manual v 1 xx 36 Run Demo LMMpro has three demo data sets that are intended to serve two purposes 1 use as a teaching tool and 2 use as a practice or demonstration tool Its use as a teaching tool is most important To run a demo simply click on the Run Demo button on the start up window If the button is not there check the selections made on the start up tab under options adjust as needed and restart the program If the program is not registered then it is assumed that your only objective for this software is to use it as a basic teaching tool about the mathematics of various regression analyses about the impact of data error and about the impact of slight errors in the theory When you select Run Demo you will see a small window with options to select which demo to run It may also ask you to select which type of example to run Langmuir or Michaelis Menten but this question is hidden if the start up tab under options has only one run type pre selected for default start up conditions
3. 8 k ES ES ES ES e Rearranging Equation 8 Equation 9 which is known as the Michaelis Menten Equation If Ku is known if eS 1s known and if the reaction is as described above then the reaction rate v can be predicted for any initial substrate concentration S The two parameters V nax and Kyy are obtained by regression analysis of known S and v values It is worth noting that when S K the observed reaction rate v will equal V nax 2 lt Reference CTT 3 gt LMMpro User s Manual v 1 xx 7 History of Linear Regression Methods for Langmuir amp Michaelis Menten Equations The Langmuir and Michaelis Menten Equations are very similar to each other in form They are both parabolic equations Accordingly the methods developed to solve these equations are also similar The objective is to find the best K and T nay values of the Langmuir Equation or the best Ky and Via i values of the Michaelis Menten Equation so that the parabolic equation fits the data with minimal error There are several methods available Without computers it is best to convert the parabolic equation into a linear form which can then be solved by linear regression techniques Langmuir proposed a linearization method in 1918 known as the Langmuir linear regression but it was not known to those using the Michaelis Menten Equation Accordingly Hanes and later Woolf proposed a similar regression technique in 1932 known as t
4. and a better fit is observed for the data with low c or S values In principle modification of the ACF value can be used to highlight sensitivity of the regression results toward deviations in the x axis or y axis data values No sensitivity is observed when the data and theory are perfect In spite of this interesting observation with modified ACF values I recommend that only one ACF value be used with your data and that it be calculated similarly to the one example illustrated above that resulted in ACF 0 924 If alternate ACF values cannot be defended then they should not be used in the final presentation of the results That is I recommend that the ACF value be used only as a unit conversion factor and thus remain true to the experimental propagation of error from the x axis to the y axis values collected and displayed on the original plot lt Reference CTT 10 gt LMMpro User s Manual v 1 xx 25 Correlation Coefficient r When there is no pattern to a set of data we can always determine the average value of the data In this way we d be predicting too high half the time and too low the rest of the time In principle we ll never be too far from the correct answer and the error in our answer will be small Without a pattern the best we can do is Y Y avora ie for any value of x where the probable error is expressed by the standard deviation 0 On the other hand if there is a pattern then we should be able to t
5. A AAA AR oe ee AAA ALEA 20 Axes Conversion Facto ACR E Sateen Away ate ees 23 Correlation Coefficient f 4 ssicesapennreedschadesesseatee ed 25 Correlation Coefficient i Teis e tow naa gee and A AA Gd LORE Ow EES 26 Correlation Coefficient n2 n nanana sides 27 Dors DOW lS sesionar dd lee 28 Tutorial Overview of Software 0 00 30 Demo Mode versus Full Access Mode iii ts eae ew aoe fe SW TRE 31 How to Register thers Oates amp fi rants as Ra 32 How to Check for Software Updates o ib AAA A 33 OP a o de al o TEE 34 R n Demo set ts A a ren PENSE RS 36 Start Optimization Program O P uti A da 38 Data EU A A A A E 39 Regression Exec tion Errors e da A as 41 Graph Commands 0 dra as a AS RRA A a ah 43 LMMpro User s Manual v 1 xx Welcome to LMMpro Tutorial Overview of Theory Select a topic What does LMMpro do Langmuir Equation Overview Michaelis Menten Equation Overview History of Linear Regression Methods Linear Regressions Deriving Linear Regression Methods NLLS Regressions v NLLS Optimization n NLLS Optimization Axes Conversion Factor ACF Correlation Coefficient 1 Correlation Coefficient m2 Correlation Coefficient MA e Do s amp Don ts e Tutorial Overview of Software lt Reference CTT 0 gt LMMpro User s Manual v 1 xx What does LMMpro do This program will help you optimize the equation parameters for the L
6. Note the definition of D above is correct It is not correct in the original publication Schulthess amp Dey 1996 LMMpro User s Manual v 1 xx 22 Also note that n and T pax are multiplied by the axes conversion factor ACF in order to maintain the parity of the units in each component of this equation of the fourth degree 6 For the Michaelis Menten Equation Let A Ky Let B 3K 7 mK Let C 3 K 3mKyy Let D 1 3m Ky Kyg V nar MK V max Let E m KV max Note that n and V nax are multiplied by the axes conversion factor ACF in order to maintain the parity of the units in each component of this equation of the fourth degree 7 Choose the feasible root for the value of r The solution to the equation of the fourth degree yields four roots Only one of these is the feasible answer 8 Once ris known use Equation 2 and 3 to evaluate the values of s and Q Note that the n NLLS regression will optimize the parameters assuming that minimizing the normal error yields the best results This method does not have any known bias in favor of any particular region of the curve Furthermore the ACF factor gives much flexibility to how one wishes to weigh the axes units Also note that the n NLLS regression will result in an optimized curve with the data evenly distributed above and below the curve That is the sum of the errors above the curve will be the same value as the sum of the errors below the
7. The three demos shown are as follows e Demo 1 This is a very simple situation The data were artificially selected so that a perfect fit would be found for K 0 0106 and T nax 838 or Ky 94 34 and V nax 838 The results are not affected by the number of data points available for the regression analysis Lesson If the data are perfect and the theory is perfect then it does not matter how the regression is perfomed All regression methods will reach the same conclusions Much discussion can be pursued here about the nature of mathematics and the multiple paths allowed by mathematics to reach a similar conclusion e Demo 2 This is a slightly more complex situation The data were artificially selected based on the same formulation applied to Demo 1 This time however the data were evenly increased or decreased by 10 The regression results vary and this is especially true when some of the data are checked not used so as to slighltly disturb the evenness of the error distribution Lesson Although the theory may be perfect the extraction of the correct values of the equation parameters can be difficult with real world data sets This is particularly true if the data collected for the regression analysis are few in number Much discussion can be pursued here about the primary cause of regression bias data error How is it recognized How is it minimized e Demo 3 This is a more complex situation The data are real The data error
8. e Define the total number of reactive sites F DS SA S e Combine the equations above in adsorption terms that is using SA terms max LMMpro User s Manual v 1 xx If K is known if T nag is known and if the reaction is as described above then the amount adsorbed I can be predicted for any concentration c of compound A The two parameters max nd K are obtained by regression analysis of known e and P values lt Reference CTT 2 gt LMMpro User s Manual v 1 xx 5 The Michaelis Menten Equation an overview The Michaelis Menten Equation was introduced in 1913 It correlates the initial enzyme reaction rate v with the amount of substrate present S The equation can be derived as follows e Let the enzyme E combine with the substrate to form the enzyme substrate complex ES which then proceeds to form the product P The enzyme E is recycled to be used again where k values are the reaction rate constants e The rate of production of product P is Equation 1 which can be simplified since we seek to describe the initial reaction rate only That is the initial condition is P 0 Note that brackets i denote concentration of species i Accordingly we now have e The maximum rate possible will occur when the maximum amount of enzyme exists in the ES complex form From mass balance principles we know that the total amount of enzyme Er present is the sum of the un
9. is small as noticed by the close reproducibility of the data This time however there is probably a small error in the theory For the most part the theory seems correct and the slight error may go unnoticed Lesson Although the data may have little scatter and the data pattern may be parabolic it does not necessarily mean that everything is perfect We now see significant expressions of regression bias Much discussion can be pursued here about the second cause of regression bias theory error How is it recognized If present then is it that the parameter values have changed and if so how and where Note that the LMMpro User s Manual v 1 xx 37 identification of how and where the parameters changed can be a strong contribution to science Finally if there is some theory error present then should the theory be dropped modified or simply noted as almost correct If it is almost correct then is it ethical to drop the word almost in your reports or publications about the process studied Review Schulthess amp Dey 1996 Soil Sci Soc Am J 60 433 442 for a detailed discussion about this particular data set For the three data sets the axes conversion factor ACF value is fixed at 0 924 If the program is registered this value can be changed if so desired Clearly if the program is registered then you will be able to construct your own data sets and thus challenge your students with more complex analytical problems
10. known or fixed by the user then use Equation 5 above to get the optimized V nax Value 7 If Ku 1s not known guess Ku values and then use Equation 5 to get the corresponding best V pax value Finally use Equation 2 to determine the error that corresponds to the guesses made Repeat the process by incrementally increasing or decreasing the K value guesses until the minimum error condition is found 8 If a is known or fixed by the user then Equation 5 is not needed The best value of Kw however is still determined with Equation 2 and an iteration loop to find the condition with the minimum error Note that the v NLLS regression will optimize the parameters assuming that minimizing the vertical error yields the best results This method does have some bias in favor of any region of the curve where the vertical changes are most pronounced In other words for a parabolic equation such as the Langmuir Equation or the Michaelis Menten Equation the v NLLS regression has some bias toward the lower left region of the graph Also note that the v NLLS regression does not result in an optimized curve with the data evenly distributed above and below the curve That is the sum of the errors above the curve is not the same value as the sum of the errors below the curve Least squares regressions will only balance the data around the curve if the function has a constant term For example in y f x b b is the constant term The La
11. or Ne values for this Do compare the regression results among the various regression methods using only the n or ne values Do not compare the n value with the Ne value even if both are applied to the same graph because they refer to different goodness of fit criteria Do not use r to descirbe the goodness of fit of the original untransformed graphs The r correlation coefficient is for the transformed plots only It describes the goodness of fit of the regression line on the transformed data LMMpro User s Manual v 1 xx 29 Do not compare the r of one regression method with the r of another regression method The r value describes the goodness of fit of the regression line for a specific graph If either the y axis or x axis units change then the corresponding r values of these two graphs cannot be compared Such a comparison is nonsensical Do not compare the e n or ne values of a given regression method They each refer to different criteria for the goodness of fit Do not forget that this computer is just a machine It does have some computational limitations These limitations can be easily displayed when using very few data points in the lower left portion of the graph especially when using just a few points that also do not fall on the typical path of a parabolic equation Computer failure to do the math correctly nearly always results in a regression curve that is not anywhere near the middle of the cluster of dat
12. undo last zoom modification Undo All Zoom will return to first scaling options used and it is not the same as the default scale Set Scale to Default note that default scaling options may include extra white space or Scale Options will let user define the scale parameters Its purpose is to help the user control the graph displayed Its purpose is to help the user control the graph displayed
13. 07 Ajyuoredde sem uorjenba orjoqeied ayy ozrujdo 0 pasn ay poyu ay TEGI Ul 9ZIG SON OY P3A19I9I Y YINOYAV 816 Ul 1mu3ue Aq pajuosaJd 38114 Sem onbruyso UOISSIIZOS JEQUI SIY ydes BurStIo y JO IOUIOS 11311 Joddn y sn d ydess y Jo uor rod jppru oY Ul gyep 94 Furor JoSo 9 10 serq auos sey J OLIN Vep 0 JIAHISUIS OPJ AIOA SOY POYJOU UOISSIIGOS STUL Wy 30 1do9 19 ul y 30 1da9 19 u1 ado s Co LMMpro User s Manual v 1 xx 12 Deriving Linear Regression Methods Given a line y mx b how do we determine what the best line s slope m and intercept b should be for a given set of data x y We could randomly guess the m and b values of the line tabulate the error of each and then identify which guess results in the least error This can be done with a computer but due to the lack of computers in the late 1700 s an alternate method was proposed that is much better The method is called a least squares linear regression method because we solve the problem backwards hence the term regression that is we first minimize the error and we then find out which line is it that gives us that low error value The method does not really care what the minimum error value actually is Instead we note that the derivative of the error function reaches a minimum and its slope is equal to zero when the error value is also at a minimum 1 Let e error X y Yp where ion predicted value w
14. 30193u1 2 A Py adojs Ga nr dogs S T Snsioa A T 10 d 9 1 snsi9a 1 1 10 d xeul A A p I uoTyeUDIUOD ISNS S uonenusouos wnuqmnbg snoonby 9 juejsuos wnuqmbo uonoeoy Y xeul OPPI UOHowr UMUWNXEN A ynuenb uondiospe wunurxepy 1 UOTILA JO IPI JJLI9AQ A poqiospe yunoury J jue 3suoo usjuap stpaeyorpy My IM 1 xvul S A Bu my ET 61 UOgenb3 Uu93UIJA SIS BP UIIA 8161 uonenby amusueT P 6D Ang I9ABIMUTT owes 9y peruosso s suonenba o oqesed omy DSO JO woz aq uy s 1 Y MOJJE 9M JI 189 NON SPpOYJOW SUOISSIIZIS SOY JO SIOYINE DU JO SOLU OY YIM SuoTe UMOYS osje SI padoJ9A9p sem popou yora IBI AL MOJ9Q PAJE Nqe ae OIdIAIAT Aq posn spoyjoul UOIssoIsoI IeoUT ay 1 uorenba orjoqesed e azrumdo 0 Aseo IYW ST H SUOISSIASIY AVIUT T 10 LMMpro User s Manual v 1 xx Jord preyoieos L UL IOUS PJU402140Y DU SIZTUNUTUI Jey UOISSOISOI Y 0 JUSTBAINbd SI UOISSOISOI 99ISJOH 9IPPH UY UOISSOIBII 99ISJOH 9IPRH oy OJUT JIDAUOD pJNOM SIY Udy soxe A X DUI MAUT NOA JI yey NON ydes CUISIIO 3y JO IOUIOS 14311 Joddn y ur eyep oy Suryoes ATASOTO JO serq owos sey I JOJJO PJPP 0 JIATISUOS IWOS SLY POU UOISSOIBOI SIUL J0 d 99I5J0H 91peH UB UI JONI P74021 40Y IY SOZIUITUTUL 18y UOISSIIZII 07 JU9JBATMbAI SI UOISSILTIS PIPUYIJLIS Y WOISSOISOI PIBYIJLIS 9 OJUT JIDAUOD P NOM SIY Udy SOXB X IY WOAUT NOA JI 189 ION ydes Bu
15. Full Access Mode This occurs if the program is already registered e Enter the Serial Number in the space provided e Click the submit button The program will automatically retrieve the Registration Code via the internet Allow a little time for this process to complete You will be prompted when the registration process is complete If all went well you will no longer need the internet to run the program Click the cancel button to end your session with the Program Information window Close all windows and restart LMMpro From now on LMMpro will always start in Full Access Mode instead of Demo Mode Note that program registration is restricted to only one compuer Do not register from a computer that is not intended to hold the final resting place of the LMMpro program Once registered your Serial Number and the retrieved Registration Code will cause the software to work in Full Access Mode from only one computer namely the one used during the registration procedure Once registered the Registration Code cannot be changed You may of course purchase another Serial Number at any time which will allow you to retrieve a completely new Registration Code Once again register the software while using the computer that also happens to be the final resting place of the software program lt Reference CTS 2 gt LMMpro User s Manual v 1 xx 33 How to Check for Software Updates LMMpro provides a quick and easy way to see if
16. LMMpro User s Manual version 1 xx The Langmuir Optimization Program plus The Michaelis Menten Optimization Program Cristian P Schulthess LMMpro User s Manual A tutorial on the use of LMMpro version 1 xx Cristian P Schulthess Schulthess C P 2007 LMMpro user s manual version 1 xx Alfisol LLC Coventry CT Copyright 2007 by Cristian P Schulthess All Rights Reserved including those of translation into other languages No part of this publication may be reproduced stored in a retrieval system or transmitted in any form or by any means electronic or mechanical including photocopy recording or any information storage and retrieval system without permission in writing from the copyright holder Reproduction of this material or portions thereof is allowed for teaching purposes only and also only if the name of the copyright holder and www alfisol com are clearly shown This permission does not apply to the posting of this material or portion thereof in any electronic format even if it is for teaching purposes The use of product or corporate names in this book may be trademarks or registered trademarks and are used only for identification and explanation without intent to infringe Alfisol LLC This material is distributed by Alfisol LLC of Coventry Connecticut USA The objectives of this company are to promote products for soil and environmental scientists to attract the general public toward so
17. On the other hand for most teaching environments and needs these three demos will be sufficient To be sure with registration of the program you d also be able to use the software for your own research needs lt Reference CTS 5 gt LMMpro User s Manual v 1 xx 38 Start Optimization Program O P LMMpro can analyze two types of parabolic equations the Langmuir Equation and the Michaelis Menten Equation If your program is registered then you will be able to run the program using your own data If not then the two buttons that perform this function are temporarily hidden and replaced with the words Demo Mode To run the program simply click on Start Langmuir O P or Start Michaelis Menten O P Note that if you usually use only one of these optimization programs you can choose to hide the button that activates the optimization program of the one you do not use Click on Options button amp Start Up tab to set this up according to your preferences lt Reference CTS 6 gt LMMpro User s Manual v 1 xx Data Entry 39 You reach the Data Entry window by clicking on the Start Langmuir O P or Start Michaelis Menten O P buttons on the startup window You can also reach a modified version of this Data Entry window by clicking the Edit button in the Data Display window The software must be registered if you wish to access the Data Entry window There are numerous things that can be done in the Data Entr
18. Options button Regressions tab LMMpro User s Manual v 1 xx 42 Also note that placing data in the not used category may cause the high not used data to be hidden from view in the log log graphics This is probably because the 0 value of those high not used data points resulted in the log of a negative number which is not allowed If you use the arrows to highlight the data points but a not used data point is not highlighted in the log log regression then it is probably because this kind of error has occurred and the point was dropped from the display This situation only applies to not used data With used data this situation will result in an error message and the closing of the log log regression analysis e NaN in regression results table NaN not a number No number was generated Perhaps the regression was automatically closed Fixing the problem Check that enough data are used in the regressions lt Reference CTS 8 gt LMMpro User s Manual v 1 xx Graph Commands 43 There are numerous commands that can be used on the window that displays all of the results These are tabulated below Effect Tool bar buttons Data Edit Options Help Hide the graphics or table box X on upper right corner of an internal If hidden show the graphics or table box If not hidden show the graphics box in a new larger window Selecting an internal graphics box Mouse rollover data
19. Pooltip shows information about the data and regression On results table click on regression name Row is highlighted Corresponding regression lines are also highlighted on graphs Keyboard arrows lt up gt lt down gt Next row up or down of results table is highlighted Keyboard arrows lt left gt lt right gt Data are highlighted in the order entered Opens various windows These are explained elsewhere Its purpose is to give the user more control over what is shown on the monitor Click on the name of the regression above the graph Its purpose is to allow larger or even full screen display of any graph For tooltip to work it must be allowed in the Options window It can be turned off by a right click anywhere inside a graph and deselecting Show Point Values To modify the information presented by the tooltip go to Options window and Mouse tab Its purpose is to help user identify which line is which Corresponding regression lines are also highlighted on graphs Its purpose is to help user identify which line is which Note that the same datum point is highlighted on all the graphs The original values of each datum point is displayed on the lower bar Its purpose is to help the user determine which point is which LMMpro User s Manual v 1 xx lt Reference CTS 9 gt Area traced becomes area 44 To undo right click on graph and select Un Zoom will
20. a points used in the optimization procedure When a regression analysis results in a predicted curve that does not pass through the center of the group of data points plus the optimized parameter values are enormously large or small then you know that the computational limits have been exceeded If the optimized parameter values are indeed very large or very small but the regression curve predicted remains near the middle of the cluster of data points used then you are probably still well within the computational limits of your computer Do not rely too heavily on your conclusions and do not overstate your conclusions when your conclusions are based on just a few data points that were collected over a narrow range of conditions Increase the range of your data and get as much data as possible If you have a cluster of data in one region of the graph then your confidence of the results in that region of the graph will increase However if that region of the graph happens to correspond to a non parabolic behavior in your particular experiment that is you have a slight theory error that is expressed in that region of the graph then a large cluster of data in that region of the graph will affect the entire regression results Therefore do try to collect data evenly across all regions of the graph The log log and semilog graphs are especially useful for determining if the data are evenly collected across all regions of the graph Do not ignore da
21. a new version of LMMpro has been released that fixes any errors encountered Note that your current Serial Number for this software is valid for all 1 xx versions of this software that is 1 00 to 1 99 No further purchases for a new Serial Number are required for upgrades involving versions 1 00 to 1 99 On the other hand if running in Demo Mode then no Serial Number is needed It is strongly recommended that you check for upgrades from time to time as these upgrades are free and these revised programs are intended to fix any coding errors encountered Also as a favor to your colleagues that also enjoy using LMMpro please let Alfisol LLC know of any errors you may find in the program as soon as possible via email To report a problem click on the Report a Bug button which takes you to the www alfisol com message web page Thank you Perform the following steps to check for software updates Start the LMMpro program Click the program information button Click the check for updates button The program will automatically retrieve the latest version information via the internet Allow a little time for this process to complete e The retrieved information is presented to you If no updates are available then close all windows If an update is available then go to www alfisol com for further instructions for your upgrade Prior to installing a new version you must uninstall the old version Typically this will not erase you
22. ample follows that illustrates a useful procedure for calculating the ACF value Example for estimating the ACF value e Assume an adsorption experiment in a closed container This is a batch reaction type of experiment where the amount adsorbed by the solid is a function of the amount in solution at equilibrium e Assume that the data collected are such that the y axis units are ug m Assume that the data collected are such that the x axis units are mg i e The question asked here is one unit change in y equals how many units change in x e Assume solids concentration 9 24 g Cc e Assume specific surface area of the solid 100 m g e Given the above conditions lug 100m 924g mg 0 924 mg x x x m g L 1000 ug L e For this example ACF 0 924 What does ACF mean It will carry a different meaning with every application In the adsorption example above ACF 0 924 Notice that the amount adsorbed was determined based on the amount remaining in solution This practice is common in adsorption isotherm experiments where the y axis values are not truly independent of the x axis values A loss of 0 924 mg L from the liquid phase is equal to an adsorption amount of 1 ug m An error of 0 924 mg L for the liquid phase concentration is equal to an adsorption error of 1 ug m Arithmetically nothing really changes if you determine the amount adsorbed by direct measurement of the solid phase because the adsorption value is direc
23. angmuir Equation and the Michaelis Menten Equation It uses seven different optimization methods As a teaching tool you will be able to e Explain how the plots are generated e Show that the mathematics of analysis is easy to use if the data and theory are perfect e Show that the mathematics of analysis is sensitive to data error resulting in biased results e Show that the mathematics of analysis is sensitive to regression method choice resulting in biased results e Show that the mathematics of analysis is sensitive to theory error resulting in misleading conclusions As a research tool you will be able to get the best values possible for your data You will also find it easier to evaluate if your data and theory are reasonable see errors noted above Please note that every experiment is unique LMMpro does not identify specific errors in either the data set used the regression method used or the theory applied However LMMpro will usually help you determine if everything is looking good or if something somewhere is slightly off With proper use you might extract some hints about where the problems if any are located About the LMMpro name L is for Langmuir MM is for Michaelis Menten and pro is for program Fortunately pro is also for professional and that is totally acceptable lt Reference CTT 1 gt LMMpro User s Manual v 1 xx 3 The Langmuir Equation an overview The Langmuir Equation w
24. arithmetic average of the data The goodness of fit of a curve to a set of data points is numerically expressed by the correlation coefficient m2 The correlation coefficient n coincides with r in the linear regression setup The eta square term is used here for the correlation coefficient value of the untransformed data The n assumes that a vertical minimum for the error terms corresponds to a better curve fit of the data collected It is defined as Yas the predicted value when x x y arithmetic mean value of y for all values of x and y the experimental value measured when x x A perfect fit by a curve will have n 1 00 whereas a poor fit by a curve will have a low value If n 0 then the predicted curve is no better than a simple average of the data collected With all regressions n lt 1 always This is not easy to do by hand Computers are needed to perform these calculations easily and quickly lt Reference CTT 12 gt LMMpro User s Manual v 1 xx 27 Correlation Coefficient n The goodness of fit of a curve to a given set of data is based on how it compares with the goodness of fit of a simple horizontal line at the arithmetic average of the data The goodness of fit of a curve to a set of data points is numerically expressed by the correlation coefficient n eh The eta star square term is used here for the correlation coefficient value of the untransformed data The n assumes tha
25. as introduced by Irwin Langmuir in 1918 It was originally formulated for predicting the sorption of gases onto a solid phase It is now also used to predict the sorption of aqueous compounds onto a solid phase This is a mechanistic model that assumes that one reaction is involved in the sorption process and that the distribution of the compounds between the two phases gas solid or liquid solid is controlled by an equilibrium constant Langmuir received the Nobel Prize in Chemistry in 1932 for his work The Langmuir Equation assumes that the adsorption reaction involves a single reaction with a constant energy of adsorption Using mass action laws for its derivation these are the only assumptions made Because of its simplicity it is often the first theory tested on any adsorption data set Properly optimizing the parameters involved namely I pax and K will greatly impact the a conclusions made about the data collected It may also impact the decisions made by the researcher as to which objective the current project should pursue next The equation can be derived as follows e Assume that only one reaction is involved expressed as where S is the solid surface with nothing on it A is the compound in the liquid or gaseous phase and SA is the solid surface with compound A on it e Let c A equilibrium concentration of compound A in the liquid or gaseous phase and define the equilibrium constant K as products over reactants
26. complexed form E plus the complexed form ES E E ES Equation 3 Accordingly the maximum rate V nax of production of the product P occurs when ES Ely Substituting this into Equation 2 we now have Equation 4 e Similarly the rate of production of the ES complex is d ES dt k E S k_ ES k ES k E P Equation 5 At the start of the process there is no product present P 0 and this allows us to simplify Equation 5 when elapsed time is near zero d ES dt k E S k ES k ES Equation 6 e It is reasonable to assume that steady state conditions are immediately established Here this means that the rate of production of ES is equal to the rate of decomposition of ES Mathematically this means d ES dt 0 where no change in concentration of LMMpro User s Manual v 1 xx 6 ES is observed as a function of time If we plug this into Equation 6 and rearrange the equation we get Equation 7 and Ku the Michaelis Menten constant e Typically we do not know the concentrations E or ES or at least these values are far too difficult to determine So the final mathematical steps shown here seek to present an elegant expression for the reaction involved using measureable parameters We continue with the ratio of Equations 2 and 4 and substitute with Equations 3 and 7 as needed Vinax SIE Elp E ES E S 1 1 Equation
27. curve This balance is a result of the SUM1 definition outlined above While the SUMI1 definition balances the data around the curve the SUM2 definition tightens the curve to get as close to the data as possible lt Reference CTT 9 gt LMMpro User s Manual v 1 xx 23 Axes Conversion Factor ACF The Axes Conversion Factor ACF describes the relative importance weight of an error in x versus an error in y IF ACF 1 then x axis and y axis values have the same weight A large ACF value gt 1 means that the y axis values are weighed more strongly A small ACF value lt 1 means that the x axis values are weighed more strongly The x axis and y axis values are sometimes not determined completely independent of each other This is easy to illustrate with adsorption experiments involving a compound A in the liquid phase adsorbing to a solid phase Typically the total concentration of compound A in the mixture is known It is then an easy calculation to determine the amount of compound A that was adsorbed by the solid phase T based on the amount of compound A remaining in the liquid phase at equilibrium c That is I is not actually measured but rather it is extrapolated from c Hence any error in the measurement of c will also express a similar error in I Although there is no definitive proof for how the ACF value should be determined we can use this relationship or dependency of the x y data to estimate the ACF value An ex
28. d fixed appears on the results table on the top of the column for this parameter Enter the Axes Conversion This value is required for the n NLLS Factor regression Default 1 lt Reference CTS 7 gt LMMpro User s Manual v 1 xx 41 Regression Execution Errors Regression methods work best when there is plenty of data to work with Common errors include e Regression returns extremely high maximum values for I hax or V max This can occur if all of the data are clustered around the lower left corner of the graph If none of the data are in the mid range or higher portions of the graph then a small error in the data can easily result in an unusually sharp rise and an unusually large maximum Typically this problem will also cause the software to exceed the maximum number of iterations allowed Fixing the problem Get more data using higher concentration c or substrate S values This error will also affect the display of the some of the graphs which can be fixed by manually rescaling the graphs e Regression returns negative parameter values This can occur if all of the data fall near the lower left corner of the graph and if the modified data plotted on the linear regressions results in an inverted slope In addition to an inverted slope there can be a y axis intercept in the linear regression that is on the wrong side of the regression graph Technically the regression is completely mathematically correct and
29. he Hanes Woolf linear regression Note that the Langmuir linear regression and the Hanes Woolf linear regression are identical in form A common method used to solve the Michaelis Menten Equation is the Lineweaver Burk linear regression Since several early environmental scientists did not know about Langmuir s linearization methods this Lineweaver Burk linear regression was also used extensively to solve the Langmuir Equation This method is very sensitive to data error and its use is now strongly discouraged A few years later Eadie Hofstee 1942 1952 proposed another method for solving the Michaelis Menten Equation The Eadie Hofstee linear regression was a significant improvement from the earlier Lineweaver Burk linear regression but it is still somewhat sensitive to data error Once again several early environmental scientists did not know about Langmuir s linearization method and this Eadie Hofstee linear regression was also used extensively to solve the Langmuir Equation Another effort to come up with a better Michaelis Menten optimization method was presented by Scatchard in 1949 It still suffers from some sensitivity to data error Once again several early environmental scientists did not know about Langmuir s linearization method and this Scatchard linear regression was also used extensively to solve the Langmuir Equation Finally the best Michaelis Menten optimization method using linear regression techniques was presented b
30. hen x x and y experimental value when x x Note that we work with the square of the error because it eliminates the problem of working with the absolute value of the error Although the error can be either positive or negative its square is always positive It is perfectly resonable to work with other exponents say 4 or 6 but using the power of 2 is easier to solve This also explains the origin of the name of the method it s a least squares linear regression 2 Let y m x b 3 Combine 1 and 2 and expand e Y y mx by n Y m x 2mbx 2mx y b 2by y 4 Step 3 above yields a plot of e as a function of the line s slope m and the line s intercept b As we approach the best value for m and b e approaches its minimum value If we overshot the m and b values the e value begins to increase again If we stop right at the bottom of the plot right where the best m and b values exist then the e is at its minimum and the tangent of this error function right at that point is zero The tangent at any point in the curve is the first derivative of the curve We seek the point where the first derivative equals zero and where the error as a function of m and b is at a minimum de de 0 and 0 dm db LMMpro User s Manual v 1 xx 13 10 Solve Equation 3 for the minimum as a function of m de 2 dm Solve Equation 3 for the minimum as a function of b de 0 2m2x 22b
31. il and environmental sciences and to help young scientists market their products This manual and other products are available on line at www alfisol com Preface The primary purpose of the LMMpro program is to find the best parameter estimates for the Langmuir Equation and the Michaelis Menten Equation These equations are used extensively in a wide range of research projects involving adsorption reactions and enzyme kinetics I hope you will find the software well suited for your research needs These fundamental equations were introduced by Langmuir in 1918 and by Michaelis and Menten in 1913 They are among the best known equations in the biological sciences and environmental sciences Based on my teaching experiences I have found that students find these equations somewhat easy to understand The difficulty however arises with learning how to work with the mathematics of data manipulation the calculation of linear and nonlinear regressions and the interpretation of the results I hope that this software will help instructors with this topic in their lectures and lab exercises Similarly I hope that students will enjoy the fact that this software removes much of the tedium from their work load which in turn will further their ability to learn how to apply these famous equations The LMMpro software was written in C With over 17 500 lines of code I think you will find it rather user friendly and a very big improvement from whatever else
32. ils the computer hopefully does not also crash and freeze up If the regression does not fail but still results in a very strange answer then the auto scaling in the isotherm graphs may appear strange After a few mishaps you can uncheck the regression method that is causing the problems and you thus avoid the constant nuisance of watching it fail or do strange things Typically problems occur when the number of data points are too few to accurately optimize the regression method A typical error results in negative numbers for the maximums or for the constants or for both When negative numbers are obtained the program will ask you whether or not you still wish to have the regression result displayed Start Up This menu allows you to select what to display on the start up window It is intended to remove distracting buttons from the start up window that is the window that first appears when you start the program e Tutorial and Run Demo You may wish to uncheck these once you are familiar with the program Naturally if you are not a registered user or if you use this program for teaching purposes you would probably wish to keep these buttons visible e Registration Information There is probably no harm in leaving this button visible even after you are completely familiar with the program It is through this button that you can get information about program updates e Start Langmuir O P and Start Michaelis Menten O P
33. ized K and I values For the Michaelis Menten Equation the optimization involves two loops as follows 1 Make an initial guess of the Ky and V nax values Let Q normal distance of each datum point to the Michaelis Menten reaction rate predicted The sign of Q is negative if the point is below or to the right of the curve 2 Let SUM 2 Q absolute value of the sum of the errors 3 Let SUM2 xY Q LMMpro User s Manual v 1 xx 21 Repeat continuously steps 2 and 3 using a different V pax value until a minimum SUM value is achieved Repeat continuously steps 2 3 and 4 using a different K value until a minimum SUM2 value is achieved When you exit both of these loops you will have the optimized Ky and V nax values Calculating the coordinates of the closest point on the curve An exact value of Q defined above can be calculated as follows l Let m n coordinates of datum point Let r s coordinates of closest point on the predicted curve The distance beween two points is defined by Q m m P gt Substitute the theoretical equation for s into the equation above Using differential equations we solve to minimize Q as a function ofr That is dQ dr 0 Simplify the result into the following equation of the fourth degree Art Br Cr Dr E 0 For the Langmuir Equation Let A K Let B 3K mK Let C 3K 3mK 2 2 2 Let D 1 3mK KT pa 0K T pax Let E m nKl hax
34. lower edge of the regression analysis window when the keyboard arrows are used Colors This menu allows you to modify how the data are presented You can also modify the shape and size of the points here If modifications are made inside this tab and if a regression analysis window is open then the program will ask if you wish to redraw the regressions immidiately with the new display choices made If you decline then the choices made will take effect on your next regression activity request Normal regular data or lines Marked marked data Highlighted highlighed data or lines Used the data are not ignored in the regression analysis Not Used the data are ignored in the regression analysis LMMpro User s Manual v 1 xx 35 Regressions This menu allows you to decide which regression methods are to be calculated or not calculated For those methods that require an iteration sequence you are also given the option of controlling how many iterations to perform before it flags you for further instructions namely continue regression for another set of iteration loops or cancel regression It is recommended that you uncheck those regressions that you know to be giving you problems If a given regression fails the program will automatically close down the display of that particular regression method The LMMpro program is coded in such a way that it trys to keep it from crashing or freezing up That is if a regression fa
35. n for this action No records are kept of modifications made LMMpro User s Manual v 1 xx Click on column header all boxes are on checked or off unchecked Checked box data in row are used Unchecked box data in row are ignored Click on column header all boxes are turned off unchecked Checked box use special symbols when displaying this point on the graphs Unchecked box use regular symbols when displaying this point on the graphs Checked box Run regression using fixed value entered Unchecked box Ignore value shown if any Checked box Run regression using values shown 40 If data are ignored then regression assumes that they do not exist Caution The all on or all off column header command will not apply to a cell that is currently highlighted Marking the point does not affect the regression Its purpose is only to give the user more control over how to draw the graphs or to emphasize a particular subset of the data in the graphs Caution The mark all off column header command will not apply to a cell that is currently highlighted There is no mark all on command for this function This forces the regressions to find the best results with this parameter fixed If fixed the word fixed appears on the results table on the top of the column for this parameter This forces the regressions to find the best results with this parameter fixed If fixed the wor
36. ngmuir Equation and the Michaelis Menten Equation are lacking the contant term b lt Reference CTT 8 gt LMMpro User s Manual v 1 xx 20 n NLLS Optimization The n NLLS regression method used by LMMpro was presented by Schulthess amp Dey in 1996 Soil Sci Soc Am J 60 433 442 n NLLS stands for normal nonlinear least squares This regression method optimizes the parameters of the equation without converting the equation into another form or shape The best fit is that equation that yields the smallest error The error of each datum point is defined as the distance between the datum point and its nearest point on the parabolic curve The nearest trajectory of a datum point to the curve is a line that is normal that is perpendicular to the tangent of the nearest point on the curve For the Langmuir Equation the optimization involves two loops as follows 1 Make an initial guess of the K and T pax values Let Q normal distance of each datum point to the Langmuir isotherm predicted The sign of Q is negative if the point is below or to the right of the curve 2 Let SUMI 2 Q absolute value of the sum of the errors 3 Let SUM2 Q 4 Repeat continuously steps 2 and 3 using a different T pax value until a minimum SUM1 value is achieved 5 Repeat continuously steps 2 3 and 4 using a different K value until a minimum SUM2 value is achieved When you exit both of these loops you will have the optim
37. r registration codes Your old registration codes will apply to your new 1 xx version upgrades However just in case it does get erased copy the registration information prior to the uninstall procedure You will find this information by clicking on the program information button lt Reference CTS 3 gt LMMpro User s Manual v 1 xx 34 Options The LMMpro options menu can be reached from the startup window or from the regression analysis window This menu allows you to adjust how the program performs Mouse Under the mouse tab you will find four check boxes They all refer to the tooltip information which lights up when the mouse is passed over a datum point on the regression analysis window e On mouse over Check this box to activate the tooltip option If this box is checked then you can turn it off later by right clicking on the graphs and turning off the Show Point Values If this box is not checked then the tooltip is always off e Original x y data Check this box to see the original data values when the mouse touches a datum point e Modified x y data Check this box to see the modified data Note that the modifed data is different with each regression analysis e y mx b for linear regressions Check this box to see the slope and intercept of the linear regression analysis shown in the graph when you touch any data in the graph Note that some of this information is also displayed in the
38. rack it The simplest pattern is a line y mx b where m slope of the line and b y intercept of the line when x 0 We justify that the data follow a linear pattern by showing that using the line results in less error than using a simple arithmetic average This is known as the goodness of fit of the line and it is numerically expressed by the correlation coefficient 12 where v the predicted value when x x y arithmetic mean value of y for all values of x and y the experimental value measured when x x If the predicted line has less error than the average value then its distance from the measured value will usually be smaller than the distance between the average value and the measured value Summed across all the points it will definitely yield a smaller number the numerator will be smaller than the denominator and the value of r will be close to 1 00 A perfect fit by a line will have r 1 00 whereas a poor fit by a line will have a low value If 1 0 then the predicted line is no better than a simple average of the data collected With all linear predictions lt l always This is easy enough to do by hand It is even easier today with computers that do it for us lt Reference CTT 11 gt LMMpro User s Manual v 1 xx 26 Correlation Coefficient 7 The goodness of fit of a curve to a given set of data is based on how it compares with the goodness of fit of a simple horizontal line at the
39. rro 3y JO IOUIOS YO 19M0 OY Ul gp 94 SUI JOSOTO 107 serq IVIOS sey 1 JOJJO PJPP 0 JIATISUOS IWOS SLY POU UOTSSOIBOI SIU Wy Ex 1doosoqutl x Nyt edojs PU Ty 1d90193u1 odojs 6P61 PABUIIEIS XPM 3d90193u1 A I odoys Z7S61 ZP61 99 SJOH o1Ipe gy 11 LMMpro User s Manual v 1 xx lt S LLO VUAJN gt PIATOAUT SSIDOIA oy JO 9 JLU ony y JO UOTSsoIdxo peonewoyew dJo duIOSUT ATIYSI S e 0 onp 9q peo sur Aew 7 Popo Lep 9Y UI JOM UR JO UOISSIIAXO UL IIBSSIIIM JOU SI SUONIPaid JO9Y UdUDPJ STPORYSTPY JO SuoTOIposd ATOSY IMUUELT oY Wo BILP OY Ul UONEIADP BUS e SI JPY 10 119 A1OIY 0 JANISUIS OSTE 9 18 SPOYJIVL UOISSI 1GIA AVIUITUOU PUL IBQUI PE 380 ION O T 03 yenbo jos SI 31 pue POX ST ado S 9y ey SION JOM UOISSIIGOS TEUN IS9 81US IY YIM DUO DY SI INJLA Y SOQ IYL SN LA Q 189 9y IDU PUR INJLA wnwrrxew 159q s uonenba y spu yey doo 3AneIoy UB BIA PIAJOS 9q ISNUU 3 puey Aq 9A OS 0 I NIIJFJIP 00 SI JY POYJOU UOISsoISOI Iva ATUO IY SI SH L JOLJO Vep 0 JIANISUIS OIJ AIOA SOY poyu UOISSIIGOS STUL WADYIOST UONdIospe IMUUBLT DUI DATOS 07 pasn usyM PON UOISSIIZIY Jesu AIMUISURT 94 SB POYJOU UOISSIIGOI STY 0 SIJOI OIT 2J8MJJ0S ST SOWIBU NIYI SILVI UdIZO poyu UOISSOIDOI SIY pue S6 1 auep eH Aq UOISsoIsOI JJOO M SIUPH I SP 0 PoLIOJoI pue Z 61 SIULH Aq poyuosoid 197 SEM 3 S194470 Aq passtur eJ
40. software The only requirement is that it be reinstalled into the same computer where the original registration process took place Also note that you must remember the Serial Number you were given when you first purchased the software through Alfisol LLC To return to Full Access Mode you will also need to re register the software every time you choose to re install it Re registration is free and automatic with the click of a button no purchase is necessary If you do not remember the Serial Number email technical support at www alfisol com for assistance Note that the registration information will often survive the uninstall and re installation procedure If this occurs then the needed codes will be found and your program will remain in Full Access Mode lt Reference CTS 1 gt LMMpro User s Manual v 1 xx 32 How to Register the Software You must acquire a Serial Number in order to register the software If you do not have a Serial Number go to www alfisol com and get one by first purchasing it Continue with the steps below only after you have acquired a Serial Number You must be connected to the internet to complete the registration process Registration of the software is a one time event Perform the following steps to register the software e Start the LMMpro program e Click the program information button e Click the register now button Note that this button is deactivated if the program is already running in
41. st error The error of each datum point is defined as the vertical difference between the predicted value and the actual value The predicted value is the value calculated for a given x and the actual value is the value measured for a given x For the Langmuir Equation the optimization is as follows 1 Let 2 E Y error Y measured predicted 2 Substitute the Langmuir Equation for predicted values LMMpro User s Manual v 1 xx 18 5 Solving Equation 4 for yields max 6 If K is known or fixed by the user then use Equation 5 above to get the optimized r value max 7 If K is not known guess K values and then use Equation 5 to get the corresponding best I nax value Finally use Equation 2 to determine the error that corresponds to the guesses made Repeat the process by incrementally increasing or decreasing the K value guesses until the minimum error condition is found 8 TfT hax is known or fixed by the user then Equation 5 is not needed The best value of K however is still determined with Equation 2 and an iteration loop to find the condition with the minimum error For the Michaelis Menten Equation the optimization is as follows l Let e Y error Y measured predicted 2 Substitute the Michaelis Menten Equation for predicted values LMMpro User s Manual v 1 xx 19 4 To optimize V set the first derivative equal to zero and solve max 6 If Ky is
42. t Squares NLLS methods There are two NLLS methods used by LMMpro one that minimizes the vertical error v NLLS the other minimizes the normal error n NLLS The equation parameters are incrementally changed until a minimum in the sum of the squares of the errors is attained Using v NLLS the error is defined as the square of the vertical distance between each datum point and the predicted curve Using n NLLS the error is defined as the square of the normal distance between each datum point and the predicted curve The normal distance is defined as the perpendicular distance from the nearest place on the curve When using n NLLS the slope of the error line between the datum point and the predicted curve will change depending on the location of the datum point It will be nearly horizontal in the lower left corner of the graph It will be nearly vertical in the upper right corner of the graph Accordingly with n NLLS you must decide upon the relative importance of a horizontal error versus a vertical error A diagonal error line will have both vertical error and horizontal error components The ACF value is used to determine the relative impact of these types of error The Axes Conversion Factor ACF describes the relative importance weight of an error in x versus an error in y IF ACF 1 then x axis and y axis values have the same weight A large ACF value gt 1 means that the y axis values are weighed more strongly A small ACF
43. t a normal minimum for the error terms corresponds to a better curve fit of the data collected It is defined as O E Yi y where Q the normal distance of the data point to the nearest point on the predicted curve y arithmetic mean value of y for all values of x and y the experimental value measured when x x A perfect fit by a curve will have Te 1 00 whereas a poor fit by a curve will have a low value If Ne 0 then the predicted curve is no better than a simple average of the data collected With all regressions ne lt 1 always This is too difficult to do by hand Computers are needed to perform these calculations easily and quickly lt Reference CTT 13 gt LMMpro User s Manual v 1 xx 28 Do s amp Don t s Do try to optimize your data using all of the regression methods available Do try to minimize data error and to use as many data points as possible Try to have an even spread of the data on the plots so as to have data present in low medium and high concentration ranges Do be sure to use all of your original data for your final graphs reports and publications Keep an eye on the N xx of xx in the lower left corner of the graph It is there to remind you if any data have been dropped for the current regression results being displayed N number of data points being used out of the total number of data points available Do present your conclusions about the goodness of fi
44. t of the data by showing the original untransformed data and graph with the optimized regression curve drawn Showing the transformed data and graphs is to be strongly discouraged because it can be potentially very misleading If a linear regression was used to optimize the equation parameters it does not mean that the linear regression and its transformed data are more important than the original graph and its untransformed data The linear regression is just a mathematical tool and the goodness of fit of the linear regression on the transformed data is typically not the objective The objective is to work with the untransformed data and the original graph and to see how well the optimized parabolic equation fits the data in this plot In other words noting how the parameters were optimized is only important as a record of the methods used It s an important footnote that must be told However the actual values of the optimized parameters plus information about how well they predict the untransformed data based on a criteria that applies to the untransformed data are typically the only two things that really matter Do try to locate areas that do not adhere well to the theoretical predictions These areas may be easier to locate if you run several regression analyses each with different data points selected for use in the regression runs Do report the goodness of fit of your optimized equation on the untransformed data set Use either the n
45. ta It is very risky and sometimes unethical to do so This LMMpro allows you to ignore data with just a click of the mouse in the Data window but this practice is only to help you play with the data and thus evaluate various what if situations These various what if situations when used correctly can help you locate areas on the graph that are displaying some kind of error such as data error or theory error You can then proceed with a plan to determine if problems exist with the data such as precision and or accuracy or with the theory In general it will be difficult to detect any theory error if there is also some concern with the precision or accuracy of the experimental data values lt Reference CTT 14 gt LMMpro User s Manual v 1 xx Welcome to LMMpro This program will help you optimize the equation parameters for the Langmuir Equation and the Michaelis Menten Equation Tutorial Overview of Software Select a topic Demo Mode vs Full Access Mode Software Registration Check for Updates Options Run Demo Start Optimization Program Data Entry Regression Execution Errors Graph Commands e Tutorial Overview of Theory lt Reference CTS 0 gt 30 LMMpro User s Manual v 1 xx 31 Demo Mode versus Full Access Mode Without registration of the software the program will only run in Demo Mode This mode will allow you to run all of the features offered by the program but you ma
46. the best parameter values have been found Nevertheless negative parameter values are not logical answers Fixing the problem Get more data using higher concentrations c or substrate S values When this problem occurs LMMpro will automatically informing you that the results are nonsensical and ask you if you wish the program to continue or to cancel the particular regression causing the problem This error will also affect the display of the some of the graphs which can be fixed by manually rescaling the graphs or by instructing LMMpro to cancel the particular regression s causing the problem e Value of data is greater than estimated maximum value 0 gt 1 This problem occurs if the scatter in the data is such that a datum point exists that happens to be higher then the maximum value estimated by the regression method This error applies only to the log log regression method where the estimated 0 value must be less than 1 Fixing the problem LMMpro will automatically inform you of this error and it will also cancel the log log regression when this occurs For subsequent regression runs for log log uncheck the usage box in the Edit Data window for this high datum point However this will also affect the results obtained with all the other regression methods Accordingly this modified data set is probably only useful for a study of the log log regression results An alternate solution is to deactivate the log log regressions see
47. tly related to the liquid phase equilibrium value as a result of the principle of mass balance That is the total sum of the nonadsorbed liquid portion plus the adsorbed solid portion must be constant at all times LMMpro User s Manual v 1 xx 24 We can apply this example to enzyme kinetics In enzyme kinetics studies the rate of reaction v can be evaluated based on the rate of loss of substrate S Accordingly an error in the measurement of S will result in an error in the measurement of v The ACF value measures this relationship Arithmetically nothing really changes if you determine the reaction rate by direct measurement of the product formed because the product concentration is directly related to the reactant concentration as a result of the principle of mass balance That is the total sum of the product formed plus the reactant remaining must be constant at all times Note however that there is presently no definitive definition for ACF Accordingly the ACF parameter can be used as a relative weight conversion factor of the axes instead of a unit conversion factor Changing the value of ACF gives the user more flexibility on how to interpret the regression results The ACF value is not a unique value A large ACF value gt 1 means that the y axis values are weighed more strongly and a better fit is observed for the data with high c or S values A small ACF value lt 1 means that the x axis values are weighed more strongly
48. value lt 1 means that the x axis values are weighed more strongly LMMpro User s Manual v 1 xx 16 NLLS Regression Bias The v NLLS method is biased toward optimizing the fit with the data in the lower left side of the graph This is because this area of the graph will generate large errors if the predicted curve is not properly tracking these data Notice that even a small change in the predicted curve will slightly shift the location of the sharp rise in the parabolic equation which in turn results in a large vertical error with each datum point in that area Using the n NLLS method a small change in the predicted curve will not result in any significant increase in its distance to any of the data points Accordingly the n NLLS method is not biased toward optimizing the fit with the data in any particular area of the graph The n NLLS regression method used by LMMpro was presented by Schulthess amp Dey in 1996 Soil Sci Soc Am J 60 433 442 lt Reference CTT 7 gt LMMpro User s Manual v 1 xx 17 v NLLS Optimization The v NLLS regression method used by LMMpro is based on an adaptation of the optimization method discussed by Persoff amp Thomas in 1988 Soil Sci Soc Am J 52 886 889 v NLLS stands for vertical nonlinear least squares This regression method optimizes the parameters of the equation without converting the equation into another form or shape The best fit is that equation that yields the smalle
49. y Hanes 1932 and referred to as the Hanes Woolf regression by Haldane 1957 As noted earlier this technique is essentially identical to the linear regression technique presented by Langmuir in 1918 It is the least sensitive to data error Once again several early environmental scientists did not know about Langmuir s linearization methods and this Hanes Woolf linear regression was also used extensively to solve the Langmuir Equation That is the technique was incorrectly named the Hanes Woolf linear regression rather than the Langmuir linear regression when used to solve the Langmuir Equation LMMpro User s Manual v 1 xx There is one more linear regression that is worth noting here The data can be presented in a log log form resulting in a linear pattern with a slope 1 0 It is optimized via an iteration process and hence this method is not used That is this method is not easy to use without computers lt Reference CTT 4 gt LMMpro User s Manual v 1 xx yde 13 PUISIIO 3y JO IOUIOS 149 19MO 94 Ul gp 94 Suryoery AT9SO O JOJ serq JUOIS AIOA amp sey Y JOLJO BVP 0 SATIISUDS A 9UI9JJX9 SI poyU UOTSSOIBOI STYL e ep eur3rio a s 94 JO 10 y sn d uonenbg uaJuaJA S oe yor oy 10 uonenbg 1mu3Sue7 oy jowueu 919y passodxo suorjenbo or oqe red OM OY JO JOYITA Aq poyeroues Ydess 0 SI9J9I MOTOG SJUWILIOI DY UL YdADAS DU18140 VIIO DY DON SJU9UNUIO 7 MA 1d9019 u XT 1 3d
50. y only use the data sets provided Running the program in this mode is free of charge and the three demo data sets encoded in the program are intended to satisfy most teaching objectives about these equations as well as about some of the limitations of the various regression methods commonly used It is also recommended that you run this program in this mode prior to registering it so that you can be certain that it will function properly on your computer With registration of the software the program will run in Full Access Mode which simply means that you can now also import your own data sets create your own data sets modify the data sets and save the data sets Registration of the software allows you to construct more complex data sets for your teaching objectives The registration of the software will also allow you to apply these regression tools to your own research data sets and research projects Note that the registration of the software applies to only one computer per Serial Number entered If the software is copied over to another computer it will only run in Demo Mode in the other computer The Serial Number and Registration Code obtained with the first computer will be recognized as an incorrect number code combination in the second computer which means that in the second computer it will run in Demo Mode only You may re install the program into the same computer as often as you like It will not affect the registration of the
51. y window These are tabulated below Efect Data entry in data Numbers are filled in Data units are recorded Comments about the data comments box set are recorded Click on row number Click on x axis all lt Import Data gt Numerically rearranges the data in ascending order LMM data file is created amp saved LMM data file is imported Program executes the regression analysis Close this Window Comments Be sure to have more than one row of data entered These units are not used in the regressions Its purpose is only for clarity to the user about the data values entered The comments entered are not used in the regressions Its purpose is only for clarity to the user about the data entered This allows you to delete the row Press lt delete gt key to delete the data shown in row Caution There is no undo option for this action Program uses the y axis data as a tie breaker Data order does not affect the regressions Its purpose is only for clarity to the user about the data entered If filename already exists prompt for save as or overwrite appears Data must exist for this action to work Data Entry window will close Data Display window will open after data import is completed Window will close Regression Results window will open Review the Options window to control how the regressions are performed and presented Caution There is no undo optio
52. you ve been using for these equations up to now In kind consideration for your fellow colleagues that are also using LMMpro please try to promptly tell us if you find any bugs in the program The software comes with an easy link for report a bug and another easy link for check for update I thank you in advance for your comments Version 1 00 was released on 12 October 2007 Please visit www alfisol com for additional information about LMMpro In closing I thank Samir A Mohamed for his many long hours of work on the code He taught me a great deal about C and helped me with the program s code I would not have completed this project in a timely manner without his help Cristian P Schulthess October 2007 iii Table of Contents Tutorial Overview of Theory nuanua naaar 1 Whatdoes LMMpro do serseri ae A AAA Aaa we de 2 About the LMMpro name 20 ia aie wae he Seles we ee a eee Ra bola 2 The Langmuir Equation an overview y a A ASR ae we 3 The Michaelis Menten Equation an OvervieW 0 cece eee teens 5 History of Linear Regression Methods for Langmuir amp Michaelis Menten Equations 7 Linear Regressions yan sta ea a ea ee a etait eed e r es ae oia 9 Deriving Linear Regression Methods 0 0 cece cece eee eee n ene ees 12 NEES Represa etre ARA oe ee ek Bone eee ae ee Se ees 15 VNEES Opinii ZatiOn pasted sees pea ea eta td Sea a Se We dd oe ee 17 H NLES Opumizaon di weak sey
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