Transcription in Molecular Biochemistry: An Idealized Physical-Mathematical Model

Abstract

Transcription in molecular biochemistry has originated various established physicochemical descriptions: Brownian-ratchet and energy-landscape models for elongation/pausing, kinetic schemes, thermodynamic promoter-binding models, structural-mechanistic approaches and other related ones. A new idealized physical-mathematical model for transcription, complementary to those existing models, is proposed here: it emphasizes a minimal reaction-diffusion/Smoluchowski formulation well suited for spatial-temporal descriptions. The model describes how, in the presence of a molecular chain (the template), two initially unbound small molecules (free monomers) become covalently bound to each other, forming a new molecular chain (the transcript) that copies the template. The process occurs under the combined action of a promoter and a terminator, attached to the ends of the template, and of a moving enzyme in a non-equilibrium state. The enzyme is represented by its active site and is subject to a constant effective force, interpreted as arising from energy released in internal transitions of free monomers and transferred to the enzyme (molecular-motor action). All entities evolve in an aqueous medium at rest, in thermal equilibrium at (approximately) room temperature. The new coarse-grained dynamics is based upon successively simplifying and approximate non-equilibrium non-linear coupled Smoluchowski equations: (i) containing a new quantum correction that enables the bond-forming chemical reaction generating the transcript, and (ii) implementing an effective detachment mechanism for the transcript, the enzyme, and the template at termination. Other novelties in the new model are approximate mean first-passage times (τ's ) for the three basic processes (transcript formation, its overall motion during binding and enzyme motion). Approximate theoretical formulae are given: τ_b, τ_c,1 and τ_E,1, respectively, as integrals with integrands dominated by exponentials. For suitable effective interactions, estimates indicate that the integrals yielding τ_b, τ_c,1 and τ_E,1 be, in orders of magnitude, about 10−2 s and, so, about known typical durations of transcription (per pair of new bound monomers in the transcript). A generalization to longer templates and transcripts is outlined.