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The first summand describes the activation of gene expression in terms of a Hill function [ 27], where Vmax > 0 denotes the maximal expression rate, K > 0 the protein concentration corresponding to the half-maximal expression rate, and β > 0 describes the cooperativity of promoter activation.

With this framework, we are interested in elucidating the general network features that can keep the activity of the transcriptional center to be compensated (or invariant) to parallel changes in the maximal expression rate of the network genes.

Same as in the one-dimensional marginal distribution, it can be seen that the variability in protein concentration is mainly dominated by the variability in the maximal expression rate Vmax.

We computed the density for a state space extended according to (2) by the cooperativity β, by the maximal expression rate Vmax and by both β and Vmax, i.e., with extended state space variables (x, β)', (x, Vmax)' and (x, β, Vmax)', respectively.

Example 3 (parameter estimation) Reconsider the ODE model (7) of autoregulated gene expression from Example 1. Assume that we want to estimate the maximal expression rate Vmax of the protein X based on an observation ξ(T) = 5 (in [ mol volume ]) at time T = 20.

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(C and D ) Scatter plots comparing fitted Kd and maximal transcription rate (Tx rate) in relation to maximal expression (C ) and in relation to simulated spatial range of expression (D ).

The plasma concentration of E7820 was defined as cp, and Emax,I, IC50 and γ were the maximal drug effect, the plasma concentration at half maximal effect, and the shape exponent of the Hill equation describing the drug effect on the α2-integrin expression rate, respectively.

Moreover, the rate-limited expression of endoplasmic reticular (ER) molecular chaperones is strongly associated with the maximal expression of exogenous proteins by BEVS [ 23].

Distinct maximal expression patterns of CgIκB1 in hemocytes and CgIκB2 in the gonad were observed.

Lentivirus delivery from the bridge produced maximal expression at the implant and a gradient of expression rostrally and caudally.

The most appropriate condition for maximal expression of McoA-laccase was 60 °C and pH 2.2.

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