Living cells need to distinguish persistent signals from transient ones. There is few work on studying persistent detection, in the context of cell signalling, from a stochastic signal point of view. This paper aims to address this gap. This paper considers a persistence detection problem defined over a reaction pathway consisting of three species: an inducer, a transcription factor (TF) and a gene, where the inducer can activate the TF and an active TF can bind to the gene promoter. We model the pathway using  chemical master equation so the counts of bound promoters over time is a stochastic signal. We consider the problem of using the continuous-time stochastic signal of the counts of bound promoters to infer whether the inducer signal is persistent or not. We use statistical detection theory to derive the solution to this detection problem, which is to compute the log-likelihood ratio of observing a persistent signal to a transient one. We then show that, if the input is persistent, then the positive part of the log-likelihood ratio can be approximately computed by using the continuous-time signals of the number of active TF molecules and the number of bound promoters  . Finally, we show how we can use a coherent feedforward loop  to approximately compute this log-likelihood ratio.
Living cells need to distinguish persistent signals from transient ones. Given that the cellular environment is stochastic, it is necessary to understand how cells make decision from a stochastic point of view. However, there is few work on studying persistence detection, in the context of cell signalling, from a stochastic signal point of view. This paper aims to address this gap. This paper considers a persistence detection problem defined over a reaction pathway consisting of three species: an inducer, a transcription factor (TF) and a gene, where the inducer can activate the TF and an active TF can bind to the gene promoter. We model the pathway using the chemical master equation so the counts of bound promoters over time is a stochastic signal. We consider the problem of using the continuous-time stochastic signal of the counts of bound promoters to infer whether the inducer signal is persistent or not. We use statistical detection theory to derive the solution to this detection problem, which is to compute the log-likelihood ratio of observing a persistent signal to a transient one. We then show that, under certain conditions, this log-likelihood can be approximately computed by using the continuous-time signals of the number of active TF molecules and the number of bound promoters when the input is persistent . Finally, we show that the the coherent feedforward loop can be used to approximately compute the mean log-likelihood ratio.
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