DNA promoters and nonlinear dynamics

We investigate the role played by DNA promoters as dynamical activators of transport processes of the RNA polymerase along DNA macromolecules, by introducing an effective potential for the kink of a discrete sine-Gordon chain. At any given moment of its life, a cell of a living organism does not express the totality of its genes (the whole genoma of the organism) but only the genes that are needed for its functioning. This is a consequence of a complicated controlling process which involves specific proteins (RNA polymerase) which selectively bind to specific DNA sequences called promoters, which determine both the site and the frequency of transcription initiation of the genes they control. In procaryotic cells promoters and genes are almost contiguous (with the gene located downstream of it), while in eucaryotic cells the promoter regions are usually located far away from the genes they regulate, so that a mechanism which brings the action of the RNA polymerase (regulatory protein) from the promoter to the gene must be involved. This action could be induced in two different ways. The first one is well known in biology and involves specific protein-protein interactions to fold the doublehelix, bringing the promoter and the gene close to each other. The second one involves the formation of nonlinear excitations which could travel along DNA carrying the regulatory protein from the promoter region to the coding region corresponding to the gene. These nonlinear waves (or solitons) could be seen as conformational excitations of DNA induced by the binding of the RNA polymerase to the promoter. It is known that the binding of the RNA polymerase to the DNA in the promoter region involves first the formation of a "closed complex" (i.e. RNA and the DNA promoter are bound with the hydrogen bonds between bases closed) and then the formation of the so-called "open complex" (in which RNA is still bonded but the hydrogen bonds along DNA are locally open) which extend over 20 base pairs. In previous papers [ 1,2 ] it was suggested that the specificity of the base sequences of natural DNA in promoters may induce dynamical effects that, in turn, can influence the DNA functioning such as gene regulation. The aim of the present contribution is to further extend these studies by deriving an effective potential for a nonlinear wave, which is responsible for the nonlinearity-supported transport mechanism (based on the propagation ofsolitons which are, in fact, kinks of the discrete sine-Gordon equation) in the presence of a nonuniform background corresponding to the DNA sequence of the promoter T7 A~. To this end we model the rotational motion around the sugarphosphate back-bone structure in terms of a chain of pendula, each pendulum representing a specific base 0375-9601/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSDI 0375-9601 ( 94 )00642-3 264 M. Salerno, Yu.S. Kivshar / Physics Letters A 193 (I 994) 263-266 pair. This model was introduced long ago by Englander et al. [ 3 ] and was further improved by many authors [4-8]. Our modification of this pendulum model consists in taking into account the specificity of the base sequence of real DNA using the fact that the hydrogen bond involved in the pairings is double for adenine-thymine (A-T) and triple for guaninecytosine (G-C) . We obtain then a model of nonuniform pendula corresponding to a specific DNA sequence by fixing the ratio between the pendula masses corresponding to A-T and G C pairs to be 5. In spite of the roughness of the model, it was shown in Refs. [ 1,2 ] that it captures some interesting feature of DNA promoters which are in qualitative agreement with biochemical experiments [ 10,11 ]. Thus, for example, promoters can shoot kink solitons towards the left or the right side of the chain. The direction and velocity of these kink solitons may correlate, respectively, to the position and to the frequency of the transcription initiation of the gene, suggesting a fascinating mechanism for genetic activation. We feel compelled however to emphasize some unrealistic feature of the model: it involves local 2zr twists of the backbone structure which, under normal circumstances, would break the helix. We note, however, that the interaction of the RNA polymerase with DNA promoters extends over a large region (about 40 bp) which includes the open complex (about 20 bp) so that it could prevent the breaking of the helix also for large excursion of the angles. To test these ideas, however, more realistic all-atoms models which include the DNA-RNA polymerase binding should be developed. Let us start by introducing the Hamiltonian of the model as N H = Z {' .2 ~I.(~,~ +b~) + 1⁄2kn(~'.+ i ~'~)~ n : l "~-Ifcn(On+l--On)2"~?]n[l--cos(~!ln--On) ]} , ( 1 ) where ~', and 0n are the deflection angles which the two complementary bases form with the line passing between the attaching points of the bases to the backbone double helix and we have used as potential energy between base pairs, a simple periodic function. The parameters k, and/~, denote the back-bone spring constants along the two helixes, In is the moment of inertia of individual bases, N is the number of base pairs in the chain, and q,, is a nonlinear parameter modeling the strength of hydrogen bonds between complementary bases. As mentioned before, we choose the coefficientsfln in Eq. ( 1 ) according to the rule: c~n=2,c~ with 2,,=2 if it refers to A-T or T-A pairs, 2,, = 3 otherwise, with fi a free parameter to be fixed later. For simplicity in the following we consider only uniform restoring forces and uniform moments of inertia along the two strands of DNA, so we fix kn = k, , = K, In = I, n. m = 1 ..... N. The equations of motion obtained from Eq. ( 1 ) are then 1O)n = K(~',+, -2~,~ + ~'n-, ) 1⁄2fin sin (~,~ 0 , , ) , 10~ = K(0~+, -20, , + 0,,_, ) ~n sin (0, qJ,,).