Strong cooperativity and inhibitory effects in DNA multi-looping processes

Elastic models have been proposed for the study of the physical properties of the DNA chain and the emerging phenomena like cyclisation and looping [6,7], and are the basis for large scale simulations of protein complexes [8]. Within this approach, the elasticity of the bonds between the nucleotid bases determine the physical properties of DNA through its degrees of freedom. An important step toward the understanding of looping phenomena within a physical context was given in [9], where the effect of protein concentration was related to multiprotein bonding positions. An induced phase transition to the loop phase is controlled by the protein concentration. Following this physical analysis, a model of loop formation has been proposed using ideas from statistical mechanics which provides a clear picture of the connection between the protein concentrations, the free energy involved in loop formation [5] and protein binding, as well as the structure of the DNA. The transition between the loop formation phase was reported for the case of a single loop and multiple proteins.

In this Letter, we show that the loop selection process is the result of a strong competition between the different types of loops that can be formed in the same DNA fragment. These loops may appear in DNA segments with several binding configurations, but also in single loop configurations with the possibility of different spatial dispositions of the looped segment [10]. Loop formation entails changes in the structure of the DNA chain which allow distal operators to come into range of a binding protein (see Fig. 1). However, in a scenario where multiple loops may appear, these conformational changes can hamper or even promote additional loop creation once the appearance of a loop has modified the conditions necessary for the formation of additional loops. The possibility of formation of multiple loops becomes manifest through an effective interaction between loops that may for instance affect their size [11].

We focus on the formation of competing types of loops in a segment of DNA assuming that only one loop may be present at the same time in the segment. The conditions necessary for the formation of a loop are either geometrical, where the required operators have been set in positions that are incompatible with additional loop formation, or energetic, where the energy to form an-other loop is not strong enough to undo an existing loop. In general, the most energetically favorable loops will be dominant; however, other loops may also emerge due to the interaction of the proteins binded to the chain during loop formation. As a result, a conformational interaction is induced between potential loops.

Loop formation due to the binding of multiple proteins can be put in a statistical mechanics language by means of a Hamiltonian model which reflects the successive steps intervening in the process [9]. In a DNA segment with 2N binding positions with M different loops, the corresponding Hamiltonian can be written as

Here the set of binary variables σ L,k (=0,1) accounts for the formation of a type L k loop, and the variables σ U,i and σ D,i indicate the binding of a protein monomer at the corresponding position (see Fig. 1). The contributions to the free energy for the formation of a loop are introduced through the coefficients c k (which are independent of the chain length), while the coefficients e k on the other hand, multiply the number of dimers σ U,i σ D,i contributing to loop formation which can be a function of the chain length. Different types of loops may carry different values of c k and e k . The coefficients g U,i and g D,i are associated to the contributions of binding a monomer to the chain. Throughout this work we set g U,i = g D,i = g = g o -1 β ln n, where the protein concentration n is introduced in the Hamiltonian, and the binding contribution g is site independent.

Two-loop interaction-We focus our analysis on the case M = 2 which shows the basic features of loop interactions. An additional study of cases with M > 2 has revealed the absence of important differences in the loop selection mechanism. Changes in the chain due to the formation of a loop L 1 modify the conditions under which another potential configuration of a looped phase L 2 may emerge. This situation can be found in short chains where the deformation of the DNA after the formation of a loop alters t

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