Optimisation of assay conditions for tRNAs using a central composite design

The influence of three factors, namely pH, enzyme concentration and amino acid concentration is to be studied on the esterification of tRNA arginyl-tRNA synthetase by counting the radioactivity of the final product, using 14C labelled arginine. The higher the count, the better the conditions.

The factors are coded at five levels as follows.

 level-1.7 -1011.7
factor 1enzyme (mg protein)3.2 6.010.014.016.8
factor 2arginine (pmoles)860 1000120014001540
factor 3pH6.6 7.07.58.08.4

The results of the experiments are as follows.

Factor 1Factor 2Factor 3Counts
1114930
11-14810
1-115128
1-1-14983
-1114599
-11-14599
-1-114573
-1-1-14422
1.7004891
-1.7004704
01.704566
0-1.704695
001.74872
00-1.74773
0005063
0004968
0005035
0005122
0004970
0004925

Questions

  1. Using a model of the form
    y = b0 + b1x1 + b2x2 + b3x3 + b11x12 + b22x22 + b33x32 + b12x1x2 + b13x1x3 + b23x2x3
    Set up the design matrix D.

  2. How many degrees-of-freedom are required for the model? How many are available for replication and so how many left to determine the significance of the lack-of-fit?

  3. Determine the coefficients of the model using the pseudo-inverse
    b = (D'D)-1D'y
    where y is the vector of responses.

  4. Determine the 20 predicted responses by
    y = D.b
    and so the overall sum of square residual error, and the root mean square residual error. Express the latter error as a percentage of the average measurement.

  5. Determine the sum of square replicate error, and so, from question 4, the sum of square lack-of-fit error. Divide the sum of square residual, lack-of-fit and replicate errors by their appropriate degrees of freedom and so construct a simple ANOVA table with these three errors, and compute the F-ratio.

  6. Determine the variance of each of the 10 parameters in the model as follows. Compute the matrix (D'D)-1 and take the diagonal elements for each parameter. Multiply these by the mean square residual error obtained in question 5 above.

  7. Calculate the t-statistic for each of the 10 parameters in the model, and so determine which are most significant.

  8. Select the intercept and five other most significant coefficients and determine a new model. Calculate the new sum of squares residual error, and comment.

  9. Using partial derivatives, determine the optimum conditions for the enzyme a essay using coded values of the three factors. Convert these to the raw experimental conditions.