this article treats the rapid - equilibrium kinetics of the forward and reverse reactions of the enzyme - catalyzed a + b = p + q together and emphasizes the importance of reporting the values of the full set of equilibrium constants .
a computer is used to derive rate equations for three mechanisms and to estimate the kinetic parameters with the minimum number of velocity measurements .
the concept that n kinetic parameters can be calculated from n velocity measurements was introduced by duggleby(1 ) in connection with his discussion of the statistics of the determination of the kinetic parameters . before that , frieden,(2 ) in an article about malate dehydrogenase , showed that it was possible to determine four kinetic parameters with four velocity measurements .
a recent paper(3 ) showed how this can be done for a + b  products , and another paper(4 ) showed how this can be done with five mechanisms for a + b + c  products .
the current article shows that there are advantages in treating the forward and reverse reactions at the same time by making calculations on ordered a + b = ordered p + q , ordered a + b = random p + q , and random a + b = random p + q. three programs are used to estimate the kinetic parameters using the minimum number of velocity measurements .
the most general program of these three programs is the one to use first when the mechanism is studied for the first time and the kinetic parameters are unknown .
this article shows the effects of experimental errors in velocity measurements on the values of the kinetic parameters and on the apparent equilibrium constant calculated using the haldane relation .
various mechanisms for a + b = p + q have been discussed a great deal .
the objective here is to discuss the estimation of the rapid - equilibrium kinetic parameters of the forward and reverse reactions at the same time using mathematica .
mulquiney and kuchel(7 ) have written a program in mathematica to derive steady - state rate equations for enzyme - catalyzed reactions and have given these rate equations for about 30 reactions .
the rapid - equilibrium mechanism is 

 the third reaction is rate - determining , and the other four reactions remain at equilibrium .
the values of the four dissociation constants are assigned arbitrarily so that numerical calculations can be made .
the total concentration of enzymatic sites is given by 

 there are nine reactants , four equilibrium equations , and five components ( a , b , p , q , and e ) .
linear algebra(8 ) shows that the number of reactants n in an equilibrium calculation must be equal to the number of components c plus the number of independent reactions
r:n = c + r. in this case , the number of reactants is given by 9 = 5 + 4 .
it is convenient to solve the five equations that have to be satisfied at equilibrium using solve in mathematica , maple , or matlab to obtain expressions for [ eab ] and [ epq ] at equilibrium .
the calculations described here were made using solve[eqs , vars , elims ] in mathematica .
the following rapid - equilibrium rate equation is obtained 

 this rate equation can be written in various ways , but this is the way mathematica writes it .
this method of deriving a rapid - equilibrium rate equation has three advantages : ( 1 ) the rate equation is obtained in computer - readable form so that it does not need to be typed .
( 3 ) this method can be applied to much more complicated mechanisms,(4 ) without having to make lengthy derivations by hand .
equation 7 in a computer with specified values of the six kinetic parameters can be treated like an experimental system in the sense that substituting a quadruplet of substrate concentrations ( [ a],[b],[p],[q ] ) yields a reaction velocity . when the velocity is equal to zero , the haldane relation for the apparent equilibrium constant is obtained . 
  in 2008 , alberty(3 ) showed that solve in mathematica can be used to obtain the kinetic parameters with the minimum number of velocity measurements .
the use of solve to estimate kinetic parameters by solving n simultaneous polynomial equations has been tested by arbitrarily specifying values of the kinetic parameters and calculating velocities .
the following values of kinetic parameters are used in this section : vfexp = 1 , vrexp = 0.5 , kia = 5 , kb = 20 , kiq = 10 , and kp = 15 . in making these calculations , it is not necessary to assign units to these kinetic parameters .
since there are six kinetic parameters , it is necessary to measure six velocities to estimate the kinetic parameters with the minimum number of velocities . to illustrate this calculation ,
the choices of substrate concentrations have been discussed in two previous articles and are reviewed in the .
the velocities calculated using eq 7 are 1(100,100,0,0 ) = 0.8264 , 2(1,100,0,0 ) = 0.4545 , 3(100,1,0,0 ) = 0.04545 , 4(0,0,100,100 ) = 0.4292 , 5(0,0,100,1 ) = 0.01887 , and 6(0,0,1,100 ) = 0.02857 . a program calckinparsordabordpqfr has been written in mathematica to calculate vfexp , vrexp , kia , kb , kiq , and kp from six experimental velocities for ordered
a + b = ordered p + q. the program s name indicates that it calculates the kinetic parameters for the forward reaction ordered a + b = ordered p + q. this program , given in the , simply provides an orderly way to put six velocities with the corresponding versions of eq 7 into solve .
this yields correct values of the six kinetic parameters , but it is necessary to take experimental errors in the measurements of velocities into account .
this is done by multiplying the six velocities by 1.05 , one at a time .
the values of these parameters can be used to calculate k using the haldane relation .
the use of 5 % experimental errors in velocities may be optimistic ; however , the errors in the table can be doubled or tripled , and replicate measurements can be used .
the first velocity determination { 100,100,0,0 } is targeted on vfexp , and the 5 % error in 1 yields vfexp = 1.06 . the second velocity determination { 1,100,0,0 } is targeted on kia , and the 5 % error in 2 yields kia = 4.45 . the third velocity determination { 100,1,0,0 }
is targeted on kb , and the 5 % error in 3 yields kb = 18.74 . the fourth velocity determination { 0,0,100,100 }
is targeted on vrexp , and the 5 % error in 4 yields vrexp = 0.53 . the fifth velocity determination { 0,0,1,100 } is targeted on kiq , and the 5 % error in 5 yields kiq = 9.07 . the sixth velocity determination { 0,0,100,1 }
is targeted on kp , and the 5 % error in 6 yields kp = 14.04 .
more accurate values of kinetic parameters can be obtained by measuring velocities over a wider range of substrate concentrations .
for example , for ordered a + b = ordered p + q , { 200,200,0,0 } , { 0.1,200,0,0 } , { 200,0.1,0,0 } , { 0,0,200,200 } , { 0,0,0.1,200 } , and { 0,0,200,0.1 } can be used .
the recommendations for choices of substrate concentrations are very general : high [ a ] , [ b ] , [ p ] , and [ q ] means as high as practical , and low [ a ] , [ b ] , [ p ] , and [ q ] means as low as practical , considering the analytical method . when low substrate concentrations are used , rates are low and more difficult to determine , but the enzyme concentration can be increased . when the enzyme concentration is increased 10-fold for a measurement ,
the velocities have to be divided by a factor of 10 before comparing them with velocities at lower enzyme concentrations .
the mechanism is 

 the values of the dissociation constants are assigned arbitrarily so that numerical calculations can be made .
the total concentration of enzymatic sites is given by 

 there are ten reactants , five equilibrium equations , and five components ( a , b , p , q , and e ) , so that n = c + r is 10 = 5 + 5 .
the six equations that have to be satisfied can be solved to obtain expressions for [ eab ] and [ epq ] at equilibrium .
it is not necessary to include the reaction ep + q = epq because when reactions 1214 are at equilibrium this reaction is also at equilibrium .
when this is done in mathematica , the following rapid - equilibrium rate equation is obtained 
  to test the calculation of seven kinetic parameters from seven measured velocities , the following values of the kinetic parameters are chosen : vfexp = 1 , vrexp = 0.5 , kia = 5 , kb = 20 , kip = 25 , kiq = 10 , and kp = 25 .
velocities are calculated at seven quadruplets of substrate concentrations{[a ] , [ b ] , [ p ] , [ q ] } : { 100 , 100 , 0 , 0 } , { 1 , 100 , 0 , 0 } , { 100 , 1 , 0 , 0 } , { 0,0 , 100 , 100 } , { 0,0,1,100 } , { 0 , 0 , 100,1 } , and { 0,0,5,5 } using eq 16 . the last quadruplet is new and was recommended in a previous article.(3 ) the use of eq 16 yields 1(100,100,0,0 ) = 0.8264 , 2(1,100,0,0 ) = 0.4545 , 3(100,1,0,0 ) = 0.04545 , 4(0,0,100,100 ) = 0.4082 , 5(0,0,1,100 ) = 0.04827 , 6(0,0,100,1 ) = 0.05780 , and 7(0,0,5,5 ) = 0.4464 .
equation 16 is used to write a program calckinparsordabrandpqfr to estimate the seven kinetic parameters .
the program name indicates that it calculates the kinetic parameters for the forward reaction ordered a + b = random p + q. this program has the same basic structure as calckinparsordabordpqfr .
this program yields correct values for the seven kinetic parameters , and the effects of 5% errors in the measured velocities , one at a time , are shown in table 2 .
this table gives values for kq calculated using the following thermodynamic cycle 

 when there are no experimental errors , kiqkp = kipkq yields 

 when a reaction with this mechanism is studied experimentally , the value of kq should be reported because it is a thermodynamic equilibrium constant .
another investigator may use a mechanism with kq in the mechanism and report the value of kq . the same haldane relation ( see eq 8) applies , and the effects on k of 5% errors in the measured velocities are shown in table 2 .
the total concentration of enzymatic sites is given by 

 there are eleven reactants , six equilibrium equations , and five components ( a , b , p , q , and e ) , so that n = c + r is 11 = 5 + 6 .
it is convenient to solve the seven equations that have to be satisfied using solve to obtain expressions for [ eab ] and [ epq ] at equilibrium .
when this is done , the following rapid - equilibrium rate equation is obtained 

 this rate equation involves eight kinetic parameters , and so eight velocities have to be measured .
the following velocities were calculated using eq 27 : 1(100,100,0,0 ) = 0.8043 , 2(1,100,0,0 ) = 0.1807 , 3(100,1,0,0 ) = 0.04539 , 4(5,5,0,0 ) = 0.1034 , 5(0,0,100,100 ) = 0.4082 , 6(0,0,1,100 ) = 0.02847 , 7(0,0,100,1 ) = 0.05780 , and 8(0,0,5,5 ) = 0.04464 .
a computer program calckinparsrandabpqfr was written to estimate eight kinetic parameters from eight measured velocities .
the program name indicates that it calculates the kinetic parameters for the forward reaction random a + b = random p + q. kq was calculated using the thermodynamic cycle in eq 17 , and ka was calculated using the following thermodynamic cycle . 

 since kiakb = kakib , it is possible to calculate ka when there are no experimental errors 

 table 3 gives the values of ten kinetic parameters and values of the apparent equilibrium constants calculated using the haldane relation .
table 3 shows the effect of experimental errors on the value of ka . the haldane relation in eq 8
was used to calculate apparent equilibrium constants when there are 5% errors in the velocities , one at a time .
calculated using thermodynamic cycle 17 . calculated using thermodynamic cycle 28 . when a mechanism is being studied for the first time ,
the limiting velocities and michaelis constants are not known . as recommended earlier,(3 ) when the kinetics of a + b = p + q is studied for the first time , there is an advantage in using a more general program first since the mechanism is unknown .
when the general program yields reasonable values , that shows that the more general program applies .
when the general program yields unreasonable values , that shows that the simpler mechanism applies .
for ordered a + b = ordered p + q , six velocities had to be measured ( see table 1 ) , but to use the more general program calckinparsrandabpqfr , two more velocities are needed .
these velocities can be measured at { 5,5,0,0 } and { 0,0,5,5}. these velocities calculated for ordered a + b = ordered p + q are 4(5,5,0,0 ) = 0.1034 and 8(0,0,5,5 ) = 0.05000 . when these velocities are put in calckinparsrandabpqfr ,
all of the kip and kib values are unreasonable , and this shows that using the most general program can identify the mechanism and yield correct values for the kinetic parameters in the actual mechanism .
notice that the effects of 5% errors in measured velocities on kip and kib are highly variable as well as being unreasonable ; this is another sign that the velocity data are not from a random a + b = random p + q mechanism but are from an ordered a + b = ordered p + q mechanism .
seven velocities were calculated earlier using the rate equation for ordered a + b = random p + q. to apply calckinparsrandabpqfr , it is necessary to have eight velocities .
the following velocity was calculated : (5,5,0,0 ) = 0.1111 . including this velocity makes it possible to calculate eight kinetic parameters .
when these velocities are put in calckinparsrandabpqfr , the results in table 5 are obtained .
the values for kib are unreasonable and very sensitive to 5% errors , so e + b = eb is not in the mechanism .
it is of interest to consider the eight velocities for the three mechanisms that are given in table 6 .
there are some small differences between the columns that are not really significant , but ordered a + b = ordered p + q and ordered a + b = random p + q differ mainly at { 0,0,100,1 } , where there is a 3-fold difference in velocities .
the main difference between ordered a + b = random p + q and random a + b = random p + q is at { 1,100,0,0 } , where there is a 2.5-fold difference in velocities .
in general , rapid - equilibrium rate equations are simpler than steady - state rate equations , and so they are the place to start when an enzyme - catalyzed reaction is studied for the first time .
if more kinetic parameters are required to fit the kinetic data , steady - state rate equations can be used and even arbitrary terms can be added .
when rate equations are polynomial , solve can be used to determine the kinetic parameters .
the choice of substrate concentrations for the determination of the rapid - equilibrium kinetic parameters with the minimal number of velocity measurements has been discussed earlier(3 ) by using partial derivatives of the velocity with respect to substrate concentrations .
these choices , which have been described in this article , emphasize using the highest and lowest practical concentration .
the highest concentrations provide information about vfexp and vrexp , and the lowest concentrations provide information about the michaelis constants .
the accuracy of the measurements at low concentrations of substrates is very dependent on the equipment for the measurements of velocities .
when the values of kinetic parameters have been determined with one minimal set of measured velocities , it is recommended that the calculation be repeated with a second set of substrate concentrations to be sure the same values ( within experimental errors ) are obtained .
this article has emphasized the use of a more general mathematica program to determine kinetic parameters when an enzyme - catalyzed reaction is being studied for the first time .
the calculations described here show that a program written to estimate the kinetic parameters for random a + b = random p + q can be used to identify the mechanism and to determine correct kinetic parameters when the mechanism is actually ordered a + b = ordered p + q or ordered a + b = random p + q. the kinetic parameters discussed in this paper are usually functions of ph , and when the velocities are measured at a small number of phs , the pks of the enzymatic site and the enzymesubstrate complexes can be determined.(9 ) some enzyme - catalyzed reactions require metal ions so that the kinetic parameters depend on the free concentration of the metal ion .
 this article deals with the rapid - equilibrium kinetics of the forward and reverse reactions together for the ordered and random enzyme - catalyzed a + b = p + q and emphasizes the importance of reporting the values of the full set of equilibrium constants . 
 equilibrium constants that are not in the rate equation can be calculated for random mechanisms using thermodynamic cycles . 
 this treatment is based on the use of a computer to derive rate equations for three mechanisms and to estimate the kinetic parameters with the minimum number of velocity measurements . 
 the most general of these three programs is the one to use first when the mechanism for a + b = p + q is studied for the first time . 
 this article shows the effects of experimental errors in velocity measurements on the values of the kinetic parameters and on the apparent equilibrium constant calculated using the haldane relation . 
Introduction
Discussion