Theory on the dynamic memory in the transcription-factor-mediated transcription activation

We develop a theory to explain the origin of the static and dynamical memory effects in transcription-factor-mediated transcription activation. Our results suggest that the following inequality conditions should be satisfied to observe such memory effects: (a) τ(L)≫max(τ(R),τ(E)), (b) τ(LT)≫τ(T), and (c) τ(I)≥(τ(EL)+τ(TR)) where τ(L) is the average time required for the looping-mediated spatial interactions of enhancer-transcription-factor complex with the corresponding promoter--RNA-polymerase or eukaryotic RNA polymerase type II (PolII in eukaryotes) complex that is located L base pairs away from the cis-acting element, (τ(R),τ(E)) are respectively the search times required for the site-specific binding of the RNA polymerase and the transcription factor with the respective promoter and the cis-regulatory module, τ(LT) is the time associated with the relaxation of the looped-out segment of DNA that connects the cis-acting site and promoter, τ(T) is the time required to generate a complete transcript, τ(I) is the transcription initiation time, τ(EL) is the elongation time, and τ(TR) is the termination time. We have theoretically derived the expressions for the various searching, looping, and loop-relaxation time components. Using the experimentally determined values of various time components we further show that the dynamical memory effects cannot be experimentally observed whenever the segment of DNA that connects the cis-regulatory element with the promoter is not loaded with bulky histone bodies. Our analysis suggests that the presence of histone-mediated compaction of the connecting segment of DNA can result in higher values of looping and loop-relaxation times, which is the origin of the static memory in the transcription activation that is mediated by the memory gene loops in eukaryotes.