Kinetics of biopolymerization on nucleic acid templates

Abstract

SynopsisThe kinetics of biopolymerizat.ion on nucleic acid templates is discussed. The model introduced allows for the simultaneous synthesis of several chains, of a given type, on a common template, e.g., the polyribosome situation. Each growth center [growing chain end plus enzyme(s)] moves one template site at a time, but blocks L adjacent sites.Solutions are found for the probability nj(t) that a template has a growing center t.hat occupies the sites j -L + I, . . . , j a t time t. Two special sets of solutions are considered, the uniform-density solutions, for which nj(t) = n, and the more general steadystate solutions, for which dnj(t)/dt = 0. I n the uniform-density case, there is an upper bound to the rauge of rates of polymerization that can occur. Corresponding to this maximum rate, there is one uniform solution. For a polymerization rate less than this maximum, there are two uniform solutions that give the same rate. In the steady-state case, only 1 , = 1 is discussed. For a steady-state polymerization rate less than the maximum uniform-density rate, the steadystate solutions consist of either one or two regions of nearly uniform density, with the density value(s) assumed in the uniform region(s) being either or both of the uniform-density solutions corresponding to that polymerization rate. For a steady-state polymerization rate eqital to or slightly larger than the maximum uniform-density rate, the steady-state solutions are nearly uniform to the single uniform-density solution for the maximum rate. The boundary conditions (rate of initiation and rate of release of completed chains from the template) govern the choice among the possible solutions, i.e., determine the region(s) of uniformity and the valu4s) assumed in the uniform region(s).

Exact solution of the master equation for the asymmetric exclusion process

“…Repeating the previous step gives det F (B N , A N ; 0) = δ k 1 ,l 1 δ k 2 ,l 2 det F (2) . Iterating this procedure N times finally gives det…”

“…Assuming that the particles were initially at sites y 1 , y 2 it turns out that choosing f (p 1 , p 2 ) = e −ip 1 y 1 −ip 2 y 2 and defining the position of the pole in S 21 by p 1 → p 1 + i0 gives the correct initial condition P (x 1 , x 2 ; 0) = δ x 1 ,y 1 δ x 2 ,y 2 . This gives P (x 1 , x 2 ; t|y 1 , y 2 ; 0) = 1 (2π) 2 2π 0 dp 1 2π 0 dp 2 e −(ǫp 1 +ǫp 2 )t−ip 1 y 1 −ip 2 y 2 e ip 1 x 1 +ip 2 x 2 − 1 − e ip 2 1 − e ip 1 e ip 2 x 1 +ip 1 x 2 (3.11)…”

“…The ASEP (for a definition see below) has been suggested already in 1968 as a model for the kinetics of biopolymerization [2]. Two years later this process was introduced into the mathematical literature [3] where it has received considerable attention in the context of interacting particle systems [4].…”

The Lee-Yang theory of equilibrium and nonequilibrium phase transitions

“…The mean-field approach predicts phase transitions in the steady state as parameters controlling the rate of insertion and extraction of particles at the boundaries are varied [31]. The existence of these phase transitions is confirmed through an exact solution of the ASEP [29][30][31], achieved using a powerful matrix product approach [32,34] which has subsequently been used to solve many generalisations of the ASEP. The details of the matrix product method are not necessary for the following-suffice to say that one ends up calculating a normalisation proportional to (23) through a product of matrices, often of infinite dimension.…”

A mathematical modelling framework for elucidating the role of feedback control in translation termination

“…Kinetic models of biopolymerization on nucleic acid templates were developed early on (MacDonald et al, 1968), (MacDonald and Gibbs], 1969).…”