Protein-DNA target search relies on Quantum Walk

Protein-DNA interactions play a fundamental role in all life systems. A critical issue of such interactions is given by the strategy of protein search for specific targets on DNA. The mechanisms by which the protein are able to find relatively small cognate sequences, typically 15-20 base pairs (bps) for repressors, and 4-6 bps for restriction enzymes among the millions of bp of non-specific chromosomal DNA have hardly engaged researcher for decades. Recent experimental studies have generated new insights on the basic processes of protein-DNA interactions evidencing the underlying complex dynamic phenomena involved, which combine three-dimensional and one-dimensional motion along the DNA chain. It has been demonstrated that protein molecules spend most of search time on the DNA chain with an extraordinary ability to find the target very quickly, in some cases, with two orders of magnitude faster than the diffusion limit. This unique property of protein-DNA search mechanism is known as facilitated diffusion. Several theoretical mechanisms have been suggested to describe the origin of facilitated diffusion. However, none of such models currently has the ability to fully describe the protein search strategy.In this paper, we suggest that the ability of proteins to identify consensus sequence on DNA is based on the entanglement of π-π electrons between DNA nucleotides and protein amino acids. The π-π entanglement is based on Quantum Walk (QW), through Coin-position entanglement (CPE). First, the protein identifies a dimer belonging to the consensus sequence, and localize a π on such dimer, hence, the other π electron scans the DNA chain until the sequence is identified. By focusing on the example of recognition of consensus sequences by EcoRV or EcoRI, we will describe the quantum features of QW on protein-DNA complexes during search strategy, such as walker quadratic spreading on a coherent superposition of different vertices and environment-supported long-time survival probability of the walker. We will employ both discrete- or continuous-time versions of QW. Biased and unbiased classical Random Walk (CRW) has been used for a long time to describe Protein-DNA search strategy. QW, the quantum version of CRW, have been widely studied for its applications in quantum information applications. In our biological application, the walker (the protein) resides at a vertex in a graph (the DNA structural topology). Differently to CRW, where the walker moves randomly, the quantum walker can hop along the edges in the graph to reach other vertices entering coherently a superposition across different vertices spreading quadratically faster than CRW analogous evidencing the typical speed up features of the QW. When applied to protein-DNA target search problem, QW gives the possibility to achieve the experimental diffusional motion of proteins over diffusion classical limits experienced along DNA chains exploiting quantum features such as CPE and long-time survival probability supported by environment. In turn, we come to the conclusion that, under quantum picture, the protein search strategy does not distinguish between one-dimensional (1D) and three-dimensional (3D) case.SignificanceMost biological processes are associated to specific protein molecules binding to specific target sequences of DNA. Experiments have revealed a paradoxical phenomenon that can be synthesized as follows: proteins generally diffuse on DNA very slowly, but they can find targets very fast overwhelming two orders of magnitude faster than the diffusion limit. This paradox is known as facilitated diffusion. In this paper, we demonstrate that the paradox is solved by invoking the quantum walk picture for protein search strategy. This because the protein exploits quantum properties, such as long-time survival probability due to coherence shield induced by environment and coin-position entanglement to identify consensus sequence, in searching strategy. To our knowledge, this is the first application of quantum walk to the problem of protein-DNA target search strategy.