# 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 ^{14}C labelled
arginine. The higher the count, the better the conditions.

The factors are coded at five levels as follows.

level | -1.7 | -1 | 0 | 1 | 1.7 | |

factor 1 | enzyme (mg protein) | 3.2 | 6.0 | 10.0 | 14.0 | 16.8 |

factor 2 | arginine (pmoles) | 860 | 1000 | 1200 | 1400 | 1540 |

factor 3 | pH | 6.6 | 7.0 | 7.5 | 8.0 | 8.4 |

The results of the experiments are as follows.

Factor 1 | Factor 2 | Factor 3 | Counts |
---|---|---|---|

1 | 1 | 1 | 4930 |

1 | 1 | -1 | 4810 |

1 | -1 | 1 | 5128 |

1 | -1 | -1 | 4983 |

-1 | 1 | 1 | 4599 |

-1 | 1 | -1 | 4599 |

-1 | -1 | 1 | 4573 |

-1 | -1 | -1 | 4422 |

1.7 | 0 | 0 | 4891 |

-1.7 | 0 | 0 | 4704 |

0 | 1.7 | 0 | 4566 |

0 | -1.7 | 0 | 4695 |

0 | 0 | 1.7 | 4872 |

0 | 0 | -1.7 | 4773 |

0 | 0 | 0 | 5063 |

0 | 0 | 0 | 4968 |

0 | 0 | 0 | 5035 |

0 | 0 | 0 | 5122 |

0 | 0 | 0 | 4970 |

0 | 0 | 0 | 4925 |

#### Questions

- Using a model of the form

**y**= b_{0}+ b_{1}x_{1}+ b_{2}x_{2}+ b_{3}x_{3}+ b_{11}x_{12}+ b_{22}x_{22}+ b_{33}x_{32}+ b_{12}x_{1}x_{2}+ b_{13}x_{1}x_{3}+ b_{23}x_{2}x_{3}

Set up the design matrix**D**.

- 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?

- Determine the coefficients of the model using the pseudo-inverse

**b**= (**D**'**D**)^{-1}**D**'**y**

where**y**is the vector of responses.

- 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.

- 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.

- 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.

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

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

- 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.