4.2 Use of Solve to estimate the pH-dependent kinetic parameters from the minimum number of velocity measurements for random A + B -» products
■ 4.2.1 Estimation of kinetic parameters using {[A],[B]} = {100,100}, {1,100}, {100,1}, and {5,5}
As Duggleby pointed out in 1979 [9], when a rate equation has N kinetic parameters, it is only necessary to determine N velocities. The application of this method to ordered A + B -> products and random A + B -» products has been discussed [32].
As a test, arbitrary values of the 4 kinetic parameters in mechanism I are used to calculate velocities at {[A],[B]} equal to {100,100}, {1,100}, {100,1}, and {5,5}. These velocities are then treated as experimental data and a computer program calckin-parsrandABI has been written to calculate the 4 kinetic parameters from the 4 velocities. This program yields exact values of the kinetic parameters when there are no experimental errors, but it is necessary to take experimental errors in measuring the veloci-ties into account. This is accomplished by introducing 5% errors in the 4 velociveloci-ties, one at a time. This process is repeated with {[A],[B]} equal to {150,150}, {1,150}, {150,1}, and {1,1} to show how to obtain more accurate values of the 4 parameters. The process is repeated again with {70,70}, {5,70}, {70,5}, and {7,7} to show that less accurate values of the 4 parameters are obtained when a narrower range of substrate concentrations is used.
This program yields the values of vfexp, klA, kB, and kIB from four measured velocities at four pairs of substrate
concentrations for mechanism I. This program is based on vrandABl.
vrandABl
a b kI B vfex p
b k B kl A + a b kI B + a k B kI B + k B kl A kI B
calckinparsrandAB l [vl_ , al_ , bl_ , v2_ , a2_ , b2_ , v3_ , a3_ , b3_ , v4_ , a4_ , b4_ ] : = Module[{} , («Thi s progra m calculate s vfexp , klA ,
kI B an d k B fro m fou r experimenta l velocitie s fo r A + B - >
product s a t fou r pair s o f substrat e concentration s o n th e assumptio n tha t th e mechanis m i s random . Th e firs t velocit y i s a t hig h [A ] an d hig h [B] , th e secon d velocit y i s a t lo w [A ] an d hig h [B] , th e thir d velocit y i s a t
hig h [A ] an d lo w [B] , an d th e fourt h velocit y i s a t lo w [A ] an d lo w [B].* )
Solv e [ {v l ---- a l * b l * kI B * vfex p / (b l * k B * kl A + a l * b l * kI B + a l * k B * kI B + k B * kl A * kIB ) , v2 == a 2 * b 2 * kI B * vfex p / (b 2 * k B * kl A + a 2 * b 2 * kI B + a 2 * k B * kI B + k B * kl A * kIB ) , v3 ~ a 3 * b 3 * kI B * vfex p / (b 3 * k B * kl A + a 3 * b 3 * kI B + a 3 * k B * kI B + k B * kl A * kIB ) , v4 = = a 4 * b 4 * kI B * vfex p / (b 4 * k B * kl A + a 4 * b 4 * kI B + a 4 * k B * kI B + k B * kl A * kIB ) } , {vfexp , klA , kIB , kB}] ]
The velocities at specified temperature, pH, and ionic strength are readily calculated at {[A],[B]} = {100,100},{l,100},{100,l},and{5,5}.
vrandABl / . kl A - » 5 / . kI B - > 4 0 / . k B - » 2 0 / . vfex p - > 1 / . a - > 10 0 / . b - > 10 0 / / N 0.80971 7
vrandABl / . kl A - » 5 / . kI B - » 4 0 / . k B - » 2 0 / . vfex p - > l / . a - » l / . b - > 10 0 / / N 0.21276 6
vrandABl / . kl A - » 5 / . kI B - > 4 0 / . k B - > 2 0 / . vfex p - » 1 / . a - » 10 0 / . b - » 1 / / N 0.04540 3
vrandABl / . klA ->5 / . kXB -» 40 / . kB-»20 / . vfexp - > l / . a - » 5 / . b - » 5 / / N 0.105263
These are now considered to be experimental velocities. The 4 velocities are expressed by the following four rate equations:
vrandABl / . a -> 100 / . b -» 100 10 000 kIB vfexp
100 kB klA + 10 000 kIB + 100 kB kIB + kB klA kIB vrandABl / . a - > l / . b - > 100
100 kIB vfexp
100 kB klA + 100 kIB + kB kIB + kB klA kIB vrandABl / . a -> 100 / . b -> 1
100 kIB vfexp
kB klA + 100 kIB + 100 kB kIB + kB klA kIB vrandABl / . a - > 5 / . b - > 5
25 kIB vfexp
5 kB klA + 25 kIB + 5 kB kIB + kB klA kIB
The program calckinparsrandABl solves these 4 simultaneous equations for vfexp, klA, kB, and kIB. The velocities are rounded to five digits.
The 4 kinetic parameters for random A + B -> products are given by
calckinparsrandABl[.80972 , 100 , 100 , .21277 , 1 , 100 , .045403 , 100 , 1 , .10526 , 5 , 5 ] {{vfex p - » 1. , kl A ^5.00046 , k B - * 20. , kI B - > 40.0062} }
linei 1 is needed to obtain the calculated values of the 4 parameters in the form to make a table.
linel l = {vfexp , klA , kB , kIB } / .
calckinparsrandABl[.80972 , 100 , 100 , .21277 , 1 , 100 , .045403 , 100 , 1 , .10526 , 5 , 5 ] {{1. , 5.00046 , 20. , 40.0062} }
These values are correct, but experimental errors in velocities have to be taken into account. 5% errors in the velocities are introduced, one at a time.
calckinparsrandABl[1.0 5 * .80972 , 100 , 100 , .21277 , 1 , 100 , .045403 , 100 , 1 , .10526 , 5 , 5 ] {{vfex p - > 1.06372 , kl A - > 5.03821 , k B - » 21.3267 , kI B - > 39.6252} }
linel 2 = {vfexp , klA , kB , kIB } / .
calckinparsrandABl[1.0 5 * .80972 , 100 , 100 , .21277 , 1 , 100 , .045403 , 100 , 1 , .10526 , 5 , 5 ] {{1.06372 , 5.03821 , 21.3267 , 39.6252} }
calckinparsrandAB l [.80972 , 100 , 100 , 1.05 * .21277 , 1 , 100 , .045403 , 100 , 1 , .10526 , 5 , 5 ] {{vfex p ^0.997629 , kl A - > 5.06296 , k B - * 19.9406 , kI B ^44.7411} }
linel 3 = {vfexp , klA , kB , kIB } / .
calckinparsrandAB l [.809 7 2 , 100 , 100 , 1.05*.21277 , 1 , 100 , .045403 , 100 , 1 , .10526 , 5 , 5 ] {{0.997629 , 5.06296 , 19.9406 , 44.7411} }
calckinparsrandAB l [.80972 , 100 , 100 , .21277 , 1 , 100 , 1.05 * .045403 , 100 , 1 , .10526 , 5 , 5 ] {{vfex p - » 0.988975 , kl A - > 5.59099 , kB- > 18.6765 , kI B ^43.2002} }
linel 4 = {vfexp , klA , kB , kIB } / .
calckinparsrandAB l [.80972 , 100 , 100 , .21277 , 1 , 100 , 1.05 * .045403 , 100 , 1 , .10526 , 5 , 5 ] {{0.988975 , 5.59099 , 18.6765 , 43.2002} }
calckinparsrandAB l [.809 7 2 , 100 , 100 , .21277 , 1 , 100 , .045403 , 100 , 1 , 1.05 * .10526 , 5 , 5 ] {{vfex p - > 1.00126 , kl A - > 4.34663 , k B ^ 20.1505 , kI B ^33.3227} }
linel 5 = {vfexp , klA , kB , kIB } / .
calckinparsrandAB l [.80972 , 100 , 100 , .21277 , 1 , 100 , .045403 , 100 , 1 , 1.05 * .10526 , 5 , 5 ] {{1.00126 , 4.34663 , 20.1505 , 33.3227} }
Table 4.1 Values of kinetic parameters obtained using {100,100}, {1,100}, {100,1}, and {5,5} and testing the effects of 5%
errors in the measured velocities, one at a time.
T a b l e F o r m [ R o u n d [ { l i n e l l [ [ l ] ] , l i n e l 2 [ [ l ] ] , l i n e l 3 [ [ l ] ] , l i n e l 4 [ [ l ] ] , l i n e l 5 [ [ 1 ] ] } , 0 . 0 1 ] , TableHeadings ->
{{"No e r r o r s " , "1.05*vl", "1.05*v2", "1.05*v3", "1.05*v4"}, {"vfet", "klA", "kB", "kIB"}}]
No error s
These values of dissociation constants can be used to calculate values of kA using kA = klA kB/klB.
Fortunately, the columns in Table 4.1 can be copied and used to calculate the corresponding values of kA.
5 . ' 2 0 . ' 4 0 . 0 1 '
These values can be inserted at the end of each line in Table 1. linei 1 becomes augmented linei la.
l i n e i l a =
{{1.000003292201511', 5.000455337744712', 19.999961325866373', 40.006166902473936', 2.50}}
{ { 1 . , 5.00046, 2 0 . , 40.0062, 2.5}}
linel2a =
{{1.0637169182861397' , 5.03820821056791' , 21.326738245762204' , 39.62519882525381' , 2.71} } {{1.06372 , 5.03821 , 21.3267 , 39.6252 , 2.71} }
linel3 a =
{{0.9976292844615865' , 5.062958705312499' , 19.940611419572804' , 44.74113269985607' , 2.26} } {{0.997629 , 5.06296 , 19.9406 , 44.7411 , 2.26} }
linel4 a =
{{0.9889746023395195' , 5.590991142356963' , 18.676523325976987' , 43.20020929800704' , 2.42} } {{0.988975 , 5.59099 , 18.6765 , 43.2002 , 2.42} }
linel5 a =
{{1.0012580433022464' , 4.346631357956853' , 20.150530913722786' , 33.32266922630096' , 2.63} } {{1.00126 , 4.34663 , 20.1505 , 33.3227 , 2.63} }
Table 4.2 Values of kinetic parameters including kA obtained using {100,100}, {1,100}, {100,1}, and {5,5} and testing the effects of 5% errors in the measured velocities, one at a time.
TableForm[Round[{linella[[1] ] , linel2a[[1] ] , linel3a[[1]] , linel4a[[1]] , linel5a[[1]]} , 0.01] , TableHeadings- » {{"N o errors" , "1.05*vl" , "1.05*v2" , "1.05*v3" , "1.05*v4"} ,
{"vfet" , "klA" , "kB" , "kIB" , "kA"}} ]
Thus measuring 4 velocities yields 5 kinetic parameters, four directly and a fifth using kA = klA kB/klB.
■ 4.2.2 Estimation of kinetic parameters using {[A],[B]} = {150,150}, {1,150}, {150,1}, and {1,1}
Increasing the range of substrate concentrations improves the estimation of the kinetic parameters.
vrandABl / . klA -> 5 / . kIB -» 40 / . kB -» 20 / . vfexp - » 1 / . a -> 150 / . b -» 150 / / N 0.866218
vrandABl / . klA -> 5 / . kIB -> 40 / . kB -» 20 / . vfexp -» 1 / . a -» 1 / . b -» 150 / / N 0.232558
vrandABl / . klA -» 5 / . kIB -» 40 / . kB -» 20 / . vfexp -» 1 / . a -» 150 / . b -» 1 / / N 0.0461184
vrandABl / . kl A - » 5 / . kI B - > 4 0 / . k B - » 2 0 / . vfex p -»l/.a-»l/.b-»l// N 0.0080971 7
The enzyme concentration can be increased to obtain the fourth velocity.
The four kinetic parameters are calculated as follows:
calckinparsrandABl[.86622 , 150 , 150 , .23256 , 1 , 150 , .046118 , 150 , 1 , .0080972 , 1 , 1 ] {{vfex p - > 1. , kl A -^4.99992 , k B - > 20.0003 , kI B - > 40.0002} }
line2 1 = {vfexp , klA , kB , kIB } / .
calckinparsrandAB l [.86622 , 150 , 150 , .23256 , 1 , 150 , .046118 , 150 , 1 , .0080972 , 1 , 1 ] {{1. , 4.99992 , 20.0003 , 40.0002} }
5% errors in the velocities are introduced, one at a time.
calckinparsrandABl[1.0 5 * .86622 , 150 , 150 , .23256 , 1 , 150 , .046118 , 150 , 1 , .0080972 , 1 , 1 ] {{vfex p - > 1.05901 , kl A ^4.98325 , k B - > 21.2393 , kI B - > 39.1064} }
line2 2 = {vfexp , klA , kB , kIB } / .
calckinparsrandAB l [1.0 5 * .86622 , 150 , 150 , .23256 , 1 , 150 , .046118 , 150 , 1 , .0080972 , 1 , 1 ] {{1.05901 , 4.98325 , 21.2393 , 39.1064} }
calckinparsrandABl[.86622 , 150 , 150 , 1.05*.23256 , 1 , 150 , .046118 , 150 , 1 , .0080972 , 1 , 1 ] {{vfex p - » 0.998623 , kl A - > 5 .01064 , k B - > 19.9713 , kI B - > 43.7116} }
line2 3 = {vfexp , klA , kB , kIB } / .
calckinparsrandAB l [.86622 , 150 , 150 , 1.05*.23256 , 1 , 150 , .046118 , 150 , 1 , .0080972 , 1 , 1 ] {{0.998623 , 5.01064 , 19.9713 , 43.7116} }
calckinparsrandAB l [.86622 , 150 , 150 , .23256 , 1 , 150 , 1.05 * .046118 , 150 , 1 , .0080972 , 1 , 1 ] {{vfex p - » 0.993076 , kl A - > 5.33118 , k B - > 18.8225 , kI B - > 40.5319} }
line2 4 = {vfexp , klA , kB , kIB } / .
Table 4.3 Calculations of effects of 5% errors in velocities on kinetic parameters for random A + B -» products using {150,150},{1,150},{150,1},{1,1}
This shows that when the substrate concentrations cover a wider range (with respect to the unknown Michaelis constants), more accurate values of kinetic parameters are obtained. Thus the objective of the velocity measurements is to cover as wide a range of substrate concentrations as practical.
These values of dissociation constants can be used to calculate values of kA using kA = klA kB/klB.
5 *20 / 40 / / N 2.5
4.983 « 2 1 . 2 4 / 39.11 2.70619
The columns in Table 4.3 can be copied and used to calculate the corresponding values of kA.
5 . ' 2 0 .- 4 0 .'
{{1.0000043218428698' , 4.999920674427261' , 20.000269600576466' , 40.000160958433504' , 2.50} } {{1. , 4.99992 , 20.0003 , 40.0002 , 2.5} }
line22a =
{{1.059005771687429 , 4.983253349602989 , 21.239310599196592" , 39.10636087929682 , 2.70} } {{1.05901 , 4.98325 , 21.2393 , 39.1064 , 2.7} }
line23 a =
{{0.9986227682097853" , 5.010643073171724" , 19.971256727203116" , 43.71158321572731" , 2.29} } {{0.998623 , 5.01064 , 19.9713 , 43.7116 , 2.29} }
line24a =
{{0.9930762333264954" , 5.331182250633359" , 18.822497775120926" , 40.53185428377968" , 2.47} } {{0.993076 , 5.33118 , 18.8225 , 40.5319 , 2.47} }
line25 a =
{{1.000269289041362" , 4.69259410333882" , 20.045313901081048" , 37.02757207803543" , 2.54} } {{1.00027 , 4.69259 , 20.0453 , 37.0276 , 2.54} }
Table 4.4 Calculations of effects of 5% errors in velocities on kinetic parameters including kA for random A + B -> products using {150,150},{ 1,150},{ 150,1},{1,1}
TableForm[Round[{line21a[[1] ] , line22a[[1]] , line23a[[1]] , line24a[[1]] , line25a[[1]]} , 0.01] , TableHeadings- > {{"N o errors" , "1.05*vl" , "1.05*v2" , "1.05*v3" , "1.05*v4"} ,
{"vfexp" , "klA" , "kB" , "kIB" , "kA"}} ]
This shows that more accurate values are obtained with a broader range of substrate concentrations.
■ 4.2.3 Estimation of kinetic parameters using {[A],[B]} = {70,70}, {5,70}, {70,5}, and {7,7}
The velocities at specified temperature, pH, and ionic strength are readily calculated at {70,70},{5,70},{70,5},{7,7}.
vrandABl / . klA -» 5 / . kIB -» 40 / . kB -» 20 / . vfexp - > l / . a - > 7 0 / . b - > 7 0 / / N
These velocies are rounded to five digits to calculate the values of kinetic parameters.
calckinparsrandAB l [.74525 , 70 , 70 , .48276 , 5 , 70 , .18792 , 70 , 5 , .15987 , 7 , 7 ] {{vfex p - > 1. , kl A - > 4 . 99999 , k B ^ 20. , kI B - » 39.9998} }
line3 1 = {vfexp , klA , kB , kIB } / .
calckinparsrandABl[.74525 , 70 , 70 , .48276 , 5 , 70 , .18792 , 70 , 5 , .15987 , 7 , 7 ] {{1. , 4.99999 , 20. , 39.9998} }