A mathematical model for the analysis of the turnover of protein mixtures I. General mathematical formalism.

In tracer experiments which are mostly performed to determine the rate of protein turnover the treatment of protein degradation as a first-order reaction leads to a monoexponential function as a representation of the decrease of the initial labelling. In a protein mixture, however, the assumption of a single turnover constant is no more admissible. The analysis of curves by a sum of exponential functions is only a makeshift and is level less appropriate if protein degradation is only a partial process in a complex model. -- As in the cell (or, more generally, in a biological object) a multitude of proteins with very different turnover constants is present, continuous distributions of protein synthesis rates and turnover constants are proposed. The hypothesis is made that the distributions of the two quantities are independent of each other. A number of arguments lead to a gamma distribution for the turnover constants. On this basis a model for protein turnover consisting of a homogeneous pool (soluble amino acid) and an inhomogeneous one (proteins) is proposed and mathematically described by a system of integro-differential equations which is not analytically solvable in the general case. The distribution of the relative protein amount in dependence on the turnover constant as well as quantities characterizing the inhomogeneous protein pool are discussed. The analysis of a case of full dependence between synthesis and degradation rates shows the usefulness of the model also beyond its original range of validity.