Approximate solutions to the response time problems of transcription autoregulatory gene networks

The response time is the time required by a gene expression machinery to generate half of its steady state protein level. Response time decides how fast a gene regulatory motif can response to an external adverse signal. The differential kinetic rate equations associated with the response time of the autoregulatory gene networks are generally nonlinear and they are not solvable analytically. We derive analytical approximations for the response time of unregulated, positive and negative autoregulated gene expression systems. We further show that our approximations are accurately in line with the numerical simulations over wide range of parameter values. Results suggest a linear increase in the response times with respect to changes in the ratio of the lifetimes of protein-product and mRNA in all the unregulated, positive and negative autoregulated gene expression systems. There exists of an optimum affinity condition for the promoter-protein product interactions of the positive autoregulatory motif at which the response time attains a maximum. We argue that such maximum response-time originates mainly from the unbinding dynamics of the protein-product from the respective promoter.