calckinparsordAB[l.0 5 * .878906 , 150 , 150 , .128205 , .1 , 150 , .00481541 , 150 , .1 ] {{kfe t - > 1.05732 , kl A - > 5.04067 , k B - » 21.1465} }
line2 2 = {kfet , klA , kB } / .
calckinparsordAB[l.0 5 * .878906 , 150 , 150 , .128205 , .1 , 150 , .00481541 , 150 , .1 ] {{1.05732 , 5.04067 , 21.1465} }
calckinparsordAB[.878906 , 150 , 150 , 1.0 5 * .128205 , .1 , 150 , .00481541 , 150 , .1 ] {{kfe t - > 1. , kl A - > 4.71249 , k B - » 20.0372} }
line2 3 = {kfet , klA , kB } / .
calckinparsordAB[.878906 , 150 , 150 , 1.0 5 *.128205 , .1 , 150 , .00481541 , 150 , .1 ] {{1. , 4.71249 , 20.0372} }
calckinparsordAB[.878906 , 150 , 150 , .128205 , .1 , 150 , 1.0 5 * .00481541 , 150 , .1 ] {{kfe t - > 0.993446 , kl A - * 5.26027 , k B - » 18.8859} }
line2 4 = {kfet , klA , kB } / .
calckinparsordAB[.878906 , 150 , 150 , .128205 , .1 , 150 , 1.0 5 * .00481541 , 150 , .1 ] {{0.993446 , 5.26027 , 18.8859} }
Table 3.2 Values of kinetic parameters obtained using {150,150}, {0.1,150}, and {150,0.1} and testing the effects of 5% errors in the measured velocities
TableFonn[Round[{line21[[1]], l i n e 2 2 [ [ 1 ] ] , l i n e 2 3 [ [ l ] ] , l i n e 2 4 [ [ 1 ] ] } , 0 . 0 1 ] ,
TableHeadlngs -» {{"No errors", "1.05*vl", "1.05*v2", "1.05*v3"}, {"kf[E]t", "KIA", "KB"}}]
No e r r o r s
The errors are a little smaller than for {100,100},{1,100},{100,1} in Table 3.1.
Calculation of kinetic parameters from velocities at {20,20}, {5,20}, and {20,5}
Try a narrower range of substrate concentrations.
calckinparsordAB [. 444444, 20, 20, .333333, 5, 20, .166667, 20, 5]
{{kfet -> 0.999993, klA ^ 5 . 0 0 0 0 3 , kB -» 19.9998}}
l i n e 3 1 = { k f e t , klA, kB} / . calckinparsordAB [ .444444, 20, 20, .333333, 5, 20, .166667, 20, 5]
{{0.999993, 5.00003, 19.9998}}
This is correct, as expected, but now calculate the effects of 5% errors in the measured velocities, one at a time.
calckinparsordAB[ 1.05 * .444444, 20, 20, .333333, 5, 20, .166667, 20, 5]
{{kfet -> 1.16666, klA ^ 5 . 7 1 4 3 2 , kB -> 23.333}}
l i n e 3 2 =
{ k f e t , klA, kB} / . calckinparsordAB[1.05 * .444444, 20, 20, .333333, 5, 20, .166667, 20, 5]
{{1.16666, 5.71432, 23.333}}
calckinparsordAB[.444444, 20, 20, 1 . 0 5 * .333333, 5, 20, .166667, 20, 5]
{{kfet -> 0.999993, klA -* 3.86366, kB -* 20.9521}}
l i n e 3 3 =
{ k f e t , klA, kB} / . calckinparsordAB[.444444, 20, 20, 1 . 0 5 * .333333, 5, 20, .166667, 20, 5]
{{0.999993, 3.86366, 20.9521}}
calckinparsordAB[.444444, 20, 20, .333333, 5, 20, 1 . 0 5 * .166667, 20, 5]
{{kfet -» 0.913038, klA -» 5.52635, kB -» 16.5215}}
line3 4 =
{kfet , klA , kB } / . calckinparsordA B [ .444444 , 20 , 20 , .333333 , 5 , 20 , 1.05 * .166667 , 20 , 5 ] {{0.913038 , 5.52635 , 16.5215} }
Table 3.3 Values of kinetic parameters obtained using {20,20}, {5,20}, and {20,5} and testing the effects of 5% errors in the measured velocities
TableForm[Round[{line31[[1]] , line32[[l]] , line33[[l]] , line34[[1]]} , 0.01] ,
TableHeading s - » {{"N o errors" , "1.05*vl" , "1.05*v2" , "1.05*v3"} , {"k f[E] t", "KIA", "KB"}} ]
These errors are larger than in the preceding two cases, and so it is clear that the widest practical range of substrate concentra-tions should be used. The calculaconcentra-tions in this section emphasize the importance of velocity measurements at low substrate concentrations. Thus the accuracy of the estimation of kinetic parameters is very dependent on th analytical equipment used.
Replicate measurements can be used to improve the accuracy of the estimation of the kinetic paremeters.
■ 3.2.2 Calculation of the effects of changes in the kinetic parameters on the velocities
It is of interest to plot v, dv/dkfet, dv/kIA, and dv/kB versus [A] and [B] in a 3D plot to see how sensitive v is to the values of these 3 kinetic parameters. 3D plots of v, dv/d[A], dv/d[B], and d(dv/d[A])/d[B] were given earlier in Fig. 3.6. 3D plots of v, dv/dkfet, dv/dkIA, and dv/dkB are now calculated for Fig. 3.7.
plotA 3 = Plot3 D [Evaluat e [vordAB ] / . kfe t - > 1 / . kl A - * 5 / . k B - > 20 ,
The derivative of the velocity with respect to kfet is D[vordAB, k f e t ]
The second plot in Fig. 3.7 shows that changing the value of kfet has the largest effect on v at high [A] and high [B].
D[vordAB, klA]
a b kB k f e t ( a b + a kB + kB klA)
D [vordAB , klA ] / . kfe t - » 1 / . kl A - » 5 / . k B - » 2 0 20 a b
(10 0 + 2 0 a + ab) 2
plotC 3 = Plot3 D [Evaluat e [ D [vordAB , klA ] ] / . kfet- > 1 / . kl A - » 5 / . k B - » 20 , {a , .0001 , 15} , {b , .0001 , 60} , PlotRang e - » {0 , -0.04} ,
Viewpoint- » {-2 , -2 , 1} , AxesLabel- » {"[A]" , "[B]" , ""} ,
Lightin g - > {{"Ambient" , GrayLevel[1]}} , PlotLabe l - > "dv/dkIA"] ; The third plot in Fig. 3.7 shows that v is most sensitive to klA at low [A] and low [B].
plotD 3 =
Plot3 D [Evaluat e [ D [vordAB , kB] ] / . kfe t - » 1 / . kl A - > 5 / . k B - > 20 , {a , .0001,15} , {b , .0001,60} , (*PlotRange->{0,-0.04 } ,* ) Viewpoin t - > {-2 , -2 , 1} , AxesLabe l - » {"[A]" , "[B]" , ""} ,
Lightin g - > {{"Ambient" , GrayLevel[1]}} , PlotLabe l - » "dv/dkB"] ; This plot shows that changing the value of kB has the largest effect on v at low [A] and low [B].
figlord 3 = GraphicsArray[{{plotA3 , plotB3} , {plotC3 , plotD3}} ]
dv/dkIA dv/dkB
Fig. 3.7 Sensitivities of the velocity for ordered A + B -» products with kfet = 1.00, K\^ = 5.00, and K& = 20.00 to the values of the kinetic parameters, (a) v versus [A] and [B] . (b) dv/dkfet. (c) dv/dkIA. (d) dv/dkB.
This figure supports the recommendation to use {high ajiigh b}, {low a, high b}, and {low ajow b} to obtain the most accurate values of the kinetic parameters.
3.3 Effects of pH on the kinetics of ordered A + B -> products
The effects of pH on the rapid-equilibrium velocity of an enzyme-catalyzed reaction are a consequence of the pKs of the substrates, enzymatic site, enzyme-substrate complexes, and the consumption of hydrogen ions in the rate-determining reaction.
In investigating these effects, the rapid-equilibrium method is especially useful because a large number of chemical reactions have to be taken into account. The effects of pH on the kinetics of A + B -> products has been discussed in the literature [33,35].
There are two ways to write the expression for the velocity of ordered A + B -> products when the enzymatic site and the enzyme-substrate complexes each have two pKs. The factor method, which is discussed first, utilizes pKs and expresses the velocity as a function of pH, as well as [A] and [B] (Sections 3.3.2 and 3.3.3). The second method utilizes acid dissociation constants and expresses the velocity in terms of [H+], as well as [A] and [B]. These two forms of the rate equation yield the same velocities. The advantage of the second method is that it can be used to write a computer program to estimate the kinetic parameters using the minimum number of velocity measurements. Rate equations derived using the factor method cannot be used to estimate kinetic parameters using Solve because Solve has been developed to solve sets of polynomial equations.
■ 3.3.1 Mechanism for ordered A + B -» products when the enzymatic site and enzyme-substrate complexes each have two pKs
It is assumed that the enzymatic site and enzyme-substrate complexes each have two pKs and that the substrates A and B do not have pKs in the pH range of interest.
E" EA"
HpKle HpKlea HE + A = HEA
II pK2e II pK2ea H2 E+ H2 EA+
KHEA = [HE][A]/[HEA] kcHEA=he*a/hea (4)
EA" EAB"
II II pKleab HEA + B = HEAB
II II pK2eab H2 EA+ H2 EAB+
KHEAB = [HEA][B]/[HEAB] kcHEAB=hea*b/heab (5)
E AB"
HEAB -» products v = £f[HEAB] = kf*et*heab/et = kfet*heab/et (6)
H2 EAB+
The chemical equilibrium constants KHEA and KHEAB are always obeyed, kfet is used for kf [E]t because the relative contributions of kf and [E]t are not discussed here. Notice that this mechanism is written in terms of chemical species, whereas mechanism 1-3 was written in terms of reactants (sums of species).
The equilibrium constants involved in this mechanism are defined as follows:
kHE = h*e/he = l()-PKle kH2E = h*he/h2e = 10"PK2e kHEA = h*ea/hea = l(TPKlea kH2EA = h*hea/h2ea = 10"PK2ea kHEAB = h*eab/heab = lO"PKleab kH2EAB = h*heab/h2eab = lO-PK2eab kcHEA = he*a/hea
kcHEAB = hea*b/heab
The specified concentration of hydrogen ions is represented by h. The total concentration of enzymatic sites [E]t = et is given by et = e + he + h2e + ea + hea + h2ea + eab + heab + h2eab
The rapid-equilibrium rate equation can be derived using the factor method (see Section 2.1.4) or using Solve. The equations for the factor method are discussed first.
The following Mathematica program has been written to input these 6 pKs, 2 chemical equilibrium constants, and kf [E] t = kfet in the calculation of the pH dependencies of Vfexp, KIA , KB, Vf/KB, and Vf/KIAKB. The factor approach is used;
see Section 2.1.4. The derordABzero program also derives the equation for the velocity v of the forward reaction as a function of[A],[B],andpH.
derordABzero[pKle_ , pK2e_ , pKlea_ , pK2ea_ , pKleab_ , pK2eab_ , kfet_ , kHEA_, kHEAB_] : = Module[{eafactor , efactor , eabfactor , vfexp , kia , kb , v } ,
(«Thi s progra m derive s th e p H dependencie s o f th e kineti c parameter s o f th e forwar d enzyme-catalyze d reactio n ordere d A+ B = product s whe n n o
hydroge n ion s ar e consumed . Th e outpu t i s a lis t o f 5 function s o f pH : vfexp , kia , kb , vf/kb , an d vf/kiakb . v i s a functio n o f [A ] = a , [B] = b , an d pH.* ) efacto r = 1 + IOPK2°-PH + i 0PH-PKle;
eaf acto r = 1 + lot* 2"-« * + io*»-** 1« ; eabfacto r = 1 + io**2"**-** + 10pH-PKieab . vfex p = kfe t / eabfactor ;
ki a = kHE A * efacto r / ea f actor ; kb = kHEAB * ea f acto r / eabfactor ;
v = vfex p / ( 1 + (k b / b ) * ( 1 + (ki a / a ) ) ) ;
{vfexp , kia , kb , vfex p / kb , vfexp / (ki a * kb ) , v} ]
When the symbols for the parameters are used as input, the most general forms of the kinetic parameters, two ratios, and v are obtained.
derordABzero[pKle, pK2e, pKlea, pK2ea, pKleab, pK2eab, k f e t , kHEA, kHEAB]
k f e t (1 + 10pH-pKle + 10-PH+eK2e) kHEA (l + l0PHPK l e a + l()-PH*PK2ea) kHEAB
1 + l n PH" PK l e a b + i Q-pH+pK2eab -, -,/spH-pKlea -, Q-pH+pK2ea i -, «pH-pKleab -, Q-pH+pK2eab
k f e t k f e t [1 + i0PH-PK l e a + 1 0-ΡΗ*ΡΚ 2«) kHEAB ( l + 1 0 PH p K l e + l0-PH+PK2e) kHEA kHEAB
k f e t
Ί + i op H"p K l e a b + io"PH*pK2eab)
/ (1.10P"-P"«.10-P»-PK!«) HHEAÌ \
(1 + 1 0PH-PK1«» + 1 0 - P » * PK 2« ) 1-f-! ! kHEAB
1 ' ii.ioPH-P'Uea.io-P^PK2"! a
1 + f l + 10p H"p K l e a b + 1 0 'p H , p K 2 e a b) b
The last function in the list gives the velocity vel.
vel
kfet /
( l + 1 0 PH- PK l e a b + jQ-pH*pK2eab\1 + 10PH"PK l e a + 10-P«*pK2ea\ h + i^ > kHEAB 1 +
( l + X ( ) PH- PK l e a b + l Q - P » * p K 2 e a b \ ^
kfet
( l + 1 0 PH~ PK l e a b + !Q-pH+pK2eab\ 1 +
/ [ι.ιοΡΚ-ρΚ1.,10-ΡΗ.ρΚ2β\ k H E Al \
/1 + 10PH-PM«» + 10 Ρ»·ΡΚ2··1 1 + - ! kHEAB
{ 1.10PH-PKlea + 10-PH*PK2ea) a J ( l + 10p H~PK l e a b+10~PH'p K 2 e*b) t)
The expression for Vfexp is readily obtained by setting [A] and [B] equal to infinity.
vel / . a -» œ / . b-»oo k f e t
i n QpH-pKleab -i«-pH+pK2eab
When the pH, [A], and [B] are specified, the velocity is given by vel / . pB -> 7 / . a -> 100 / . b -» 100
k f e t (l + io7-PKleab + 1o~7+pK2eab) 1 +
I ίΐ.1θ'-Ρκ 1«.10-7·Ρ«2·) MUA \
(1 + 1 07 PK 1" + 1 0 - ' * PK 2" 1 + - ! kHEAB { loo (i.io7-PK l e a.io-7 +PK 2 e a) J
100 ( l + 1 07- PK I , a b* 1 0 -7* PK 2"b)
This rate equation cannot be used in Solve to obtain the values of the kinetic parameters from the minimum number of velocity measurements because it is not a polynomial equation. It can be converted into a polynomial equation, but that is not done here.
The following values for kinetic parameters are chosen arbitrarily for test calculations.
Grid[{{pKle , pK2e , pKlea , pK2ea , pKleab , pK2eab , kfet , kHEA, kHEAB} , {7 , 6 , 8 , 6 , 8 , 7 , 2 , 1 , 2}} ] pKl e pK2 e pKle a pK2e a pKlea b pK2ea b kfe t kHE A kHEA B
7 6 8 6 8 7 2 1 2
The pH dependencies of vfexp, kia, kb, vfexp/kb, vfexp/(kia*kb), and v are obtained as by substituting these kinetic parameters in the expressions for the kinetic parameters and v.
vordABzero = derordABzero[7, 6, 8, 6, 8, 7, 2, 1, 2]
2 1 + 106 p H + l(T7+pH 2 (l + 106-p H + 1 0 - ^p H) 1 1 + 1 07 p H + l(T8 + p H 1 + 106-p H + l(T8 + p H 1 + 107-p H + l(T8 + p H 1 + 1 06 p H + 1(Γ8 + ρ Η
1 2 1 + 10 6-pH + 1 0 -7+pH
( i + i o7-p H + i o -8 + p Hi
2 ( l * 1 06- P " + 10-a*PH) 1 +
I . I OS- PH. I O -7- PH [ I . I O6- PH. I O »·Ρ») ( l + 107-P" + 10-8-PH) b
Notice that the expression for the velocity includes a and b.
The pH dependencies of the kinetic parameters can be plotted as follows:
plotlf r = Plot[Evaluate[vordABzero[[1]]] ,
{pB, 5 , 9} , Ax e s Labe l - » {"pB" , "Vfexp"J , DisplayFunctio n - » Identity ] ;
plot2f r = Plot[Evaluate[vordABzero[[2]] ] ,
{pH, 5 , 9} , AxesLabe l - » {"pH" , "ÊΙΛ"} , DisplayFunctio n - » Identity ] ; plot3f r = Plot[Evaluate[vordABzero[[3]] ] ,
{pH, 5 , 9} , AxesLabe l - > {"pH" , "KB"} , DisplayFunctio n - » Identity ] ; plot4£ r = Plot[Evaluate[vordABzero[[4]] ] ,
{pH, 5 , 9} , AxesLabe l - > ("pH" , "V fexp/KB"] , DisplayFunction- » Identity] ; plot5£ r = Plot[Evaluate[vordABzero[[5]]] , {pH , 5 , 9} ,
AxesLabel - > f "pH" , "V£exp/KIAKB"} , DisplayFunctio n - » Identity ] ;
£ig3jp c = GraphicsGrid[{{plotlfr , plot2fr} , {plot3fr , plot4fr} , {plot5fr}} ]
2.0 1.5 1.0 0.5
pH
pH
pH