Errors in Response Experiments

In the following we analyze the influence of errors on our approach and its accuracy and compare the results with those obtained by using linearized kinetics. We consider a nonlinear kinetic example for which a detailed analytical study is possible. We compare that exact solution with the first-order response theory based on appropriate tracer measurements, and also compare it with the response of the linearized kinetic example. An important interest here is in the effects of error propagations in the analysis due to the application to measurements of poor precision. 

We consider a simple test model which has the advantage that it can be studied analytically even for very large numbers of intermediates, which makes it suitable for the analysis of the interference between experimental errors with the errors due to linearization. This type of model, which is somewhat similar to Eigen s hypercycle model [26], has recently been introduced in connection with a population genetic problem [12]. The model used here is essentially a space-independent, homogeneous version of the model from [12]. We assume that there are two types of chemical species in the system, stable chemicals, A , v = 1,2. and active intermediates X , u = 1,2. and that there is a very large supply of stable species Ay, v = 1,2. and their concentrations ay, v = 1, 2. are assumed to be constant and only the concentrations Xu,u = 1,2. of the active intermediates are variable. We consider that the active intermediates replicate, transform into each other, and disappear through auto-catalytic processes moreover we assume that all active intermediates have the same 

If the kinetic matrix is constant, eq. (12.127) can be solved either analytically by using the Sylvester theorem of the Laplace transformation, or numerically. Finally, the relation between the intrinsic time scale and the laboratory time scale t can be determined from t = u ( )] A constant kinetic matrix is sufficient for 

As an illustration, we start by analyzing a very simple type of response experiment involving the evaluation of self-replication constants. We consider the perturbation of the self-replication of a species by an input flux, whereas the response to this perturbation is given by the variation of the disappearance rate of the species due to the input perturbation. We study two types of excitation (a) an increase of the input flux according to a step function and (b) an excitation of the neutral type, where the total input flux is kept constant but the fraction of a labeled compound in the input flux is varied, also as a step function. We assume that the measured values of the variation of the output flux (disappearance rate) are subject to experimental errors. In order to emulate the two types of response experiments, we solve the evolution equations (12.126) exactly for the two 

The physical interpretation of the results presented in figs 12.2-12.4 is the following. Like any autocatalytic processes, chemical reactions (12.123)-(12.125) lead to saturation effects due to the balance between self-replication and consumption (disappearance) processes. The saturation effects are nonlinear and as a result the experimental errors propagate nonlinearly, which explains the error distortion displayed in fig. 12.2. 

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