John L.

Falconer, University of C
Vmax= 0.3767
Km= 544.92 r= Vmax*S/(S + Km)
Solver minimizes sum of squares (SS, yellow cell) by changing V max and Km

Michaelis-Menten Enzyme kinetics

S rexp rcalc (rexp-rcalc)2 (rexp-rmean)2 rexp-rcalc
25 0.0243 0.017 0.0000605 0.0314 0.0078
50 0.0292 0.032 0.0000061 0.0297 -0.0025
100 0.0546 0.058 0.0000145 0.0216 -0.0038
250 0.1388 0.118 0.0004131 0.0039 0.0203
500 0.1726 0.180 0.0000587 0.0008 -0.0077
1000 0.2374 0.244 0.0000415 0.0013 -0.0064
2500 0.3023 0.309 0.0000490 0.0102 -0.0070
5000 0.3395 0.340 0.0000000 0.0190 -0.0002
7500 0.3515 0.351 0.0000001 0.0225 0.0003
10000 0.3652 0.357 0.0000632 0.0268 0.0080
rmean= 0.2015 SS= 0.000707 0.1672 =SSmean
N= 10
P= 2 R2= 0.9958 run# = 5
df= 8 S.E.= 0.0093983

Reference: A.M. Brown, "A step-by-step guide to non-linear regression analysis

of experimental data using a Microsoft Excel spreadsheet", (2001)
Computer Methods and Programs in Biomedicine 65, 191-200

S X matrix Xt*Xinv
25 0.04387 -2.8995E-05 0.4969 2172.06
50 0.08405 -5.3219E-05 2172.1 18938261
100 0.15506 -9.0574E-05
250 0.31450 -0.00014904 S e2 8.833E-05
500 0.47851 -0.00017251 SVmax 0.006625
1000 0.64728 -0.00015783 SKm 40.90
2500 0.82104 -0.00010 alpha 5%
5000 0.90173 -6.1262E-05 tval 2.306
7500 0.93227 -4.3655E-05
10000 0.94832 -3.3879E-05 parameter value +/- 95% confidence intervals
¶r/¶Vmax ¶r/¶Km Vmax 0.377 +/-
Km 545 +/-
confidence interval for Vmax = tval*Svmax
confidence interval for Km = tval*SKm
Svmax= SQRT (x11 element in Xt*Xinv matrix * Se2)
SKm= SQRT (x22 element in Xt*Xinv matrix * Se2)
 John L. Falconer, University of Colorado Boulder

r= Vmax*S/(S + Km) Michaelis-Menten kinetics

0.4
Notation
df degrees of freedom
0.3
Km Michaelis constant
N number of data points

r (micromol/s)
P number of parameters
0.2
R2 Coefficient of determination
rexp-rcalc residual
rmean mean of rexp
0.1
S.E. standard error of rates (standard deviation of the residuals)=SQRT(SS/df)
Se2 =(S.E.)^2
SS sum of squares of (rexp -rcalc) 0.0
SSmean sum of squares of (rexp -rmean) 0 1000 2000 3000
tval inverse of 2-tailed Student t distribution
Vmax maximum rate
X matrix matrix of partial derivatives where each column is the partial derivation of the independent variable (r in this case)
Xt*Xinv the transpose of the X matrix multiplied by the inverse of the X matrix

0.025

0.020
rexp-rcalc (micromol/s)

0.015

0.010

0.005

0.000
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
-0.005

-0.010
% confidence intervals S (micromol)
0.015
94
 Xt*Xinv matrix * Se2)
Xt*Xinv matrix * Se2)
 0.4

0.3

0.2

0.1

0.0
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
S (micromol)

ndent variable (r in this case) w.r.t. one of the paramters (V max and Km in this case)

9000 10000
 Vmax= 0.3765
Km= 543.47

S rexp rcalc (rexp-rcalc)2 (rexp-rmean)2 rexp-rcalc

25 0.0243 0.017 ### 0.0314 0.0077
50 0.0292 0.032 ### 0.0297 -0.0025
100 0.0546 0.059 ### 0.0216 -0.0039
250 0.1388 0.119 ### 0.0039 0.0202
500 0.1726 0.180 ### 0.0008 -0.0078
1000 0.2374 0.244 ### 0.0013 -0.0065
2500 0.3023 0.309 ### 0.0102 -0.0070
5000 0.3395 0.340 ### 0.0190 -0.0001
7500 0.3515 0.351 ### 0.0225 0.0005
10000 0.3652 0.357 ### 0.0268 0.0081
rmean= 0.2015 SS= 0.000707 0.1672 =SSmean
N= 10
P= 2 R2= 0.9958 run# = 5
df= 8 S.E.= 0.009399100471016

S X matrix Xt*Xinv
25 0.04398 -2.912513E-05 0.4961 2163.97
50 0.08425 -5.344606E-05 2164.0 18846139
100 0.15541 -9.092579E-05
250 0.31507 -0.000149494 S e2 8.834E-05
500 0.47917 -0.000172884 SVmax 0.006620
1000 0.64789 -0.000158034 SKm 40.80
2500 0.82143 -0.00010 alpha 5%
5000 0.90196 -6.125696E-05 tval 2.306
7500 0.93243 -4.364388E-05
10000 0.94845 -3.386741E-05 parameter value +/- 95% confidence intervals
¶r/¶Vmax ¶r/¶Km Vmax 0.376
Km 543
confidence interval for Vmax = tval*Svmax
confidence interval for Km = tval*SKm
Svmax= SQRT (x11 element in Xt*Xinv matrix * Se2)
SKm= SQRT (x22 element in Xt*Xinv matrix * Se2)
 r= Vmax*S/(S + Km) Michaelis-Menten kinetics
0.4
Notation
df degrees of freedom
0.3
Km Michaelis constant
N number of data points

r (micromol/s)
P number of parameters
R2 Coefficient of determination
0.2
rexp-rcalc residual
rmean mean of rexp
0.1
S.E. standard error of rates (standard deviation of the residuals)=SQRT(SS/df)
S e2 =(S.E.)^2
SS sum of squares of (rexp -rcalc) 0.0
SSmean sum of squares of (rexp -rmean) 0 1000 2
tval inverse of 2-tailed Student t distribution
Vmax maximum rate
X matrix matrix of partial derivatives where each column is the partial derivation of the independent variable (r in
Xt*Xinv the transpose of the X matrix multiplied by the inverse of the X matrix

0.025

0.020
rexp-rcalc (micromol/s)

0.015

0.010

0.005

0.000
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
-0.005

-0.010
+/- 95% confidence intervals S (micromol)
+/- 0.015
+/- 94
v matrix * Se2)
matrix * Se2)
 0.4

0.3
r (micromol/s)

0.2

0.1

0.0
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
S (micromol)

the independent variable (r in this case) w.r.t. one of the paramters (V max and Km in this case)

0 8000 9000 10000