Thermodynamic restrictions on evolutionary optimization of transcription factor proteins

The ability of bacteria to respond within minutes to changes in their environment relies on genetic switches that are controlled by transcription factors. Transcription factors are proteins that-following activation by an environmental change-are able to locate a specific region (the "operator sequence") along the bacterial genome and bind to it, thereby regulating the expression of a gene (or group of genes) adjacent to that region [1]. The number of copies of a transcription factor protein associated with a specific gene varies, but typically it is in the range of 10 2 . Because bacterial genomes have a size in the range of 10 7 sites, a transcription factor must be able to "scan" the DNA for the target site at a rate of 10 5 sites per second or faster in order for at least one of them to reach the target site within seconds. Note that following the search for the target site, the transcription factor still has to bind to the target site to regulate the expression of the gene.

A series of classical papers on the search process [2,3,4] culminated in the work of Berg, Winter and von Hippel (BWH) who showed [5]-for the canonical case of the lac repressor protein of the bacterium E. coli -that the search process takes place not by straightforward 3D diffusion to the target binding site but rather by a slide-jump combination of 1D diffusional sliding along the DNA chain alternating with 3D diffusional jumps between different DNA segments. By restricting part of the search to the 1D “target space”, the binding rate is effectively enhanced with respect to a pure 3D search, while the 3D jumps reduce the repetitive visits to the same sites that characterize purely 1D diffusive searches. This scenario is made possible by a modest, non-specific electrostatic affinity between the transcription factor and duplex DNA. BWH also provided evidence that, under physiological conditions, the search time has a minimum with respect to the strength of this non-specific affinity, which may be the result of evolutionary optimization un-der selective pressure. Subsequent structural studies [6] have shown that the DNA-binding domains of the lac repressor are subject to strong conformational fluctuations when the protein is in contact with non-operator DNA. If the binding domain is in contact with operator sequence DNA then the protein can undergo a large-scale conformational change to a stable structure with direct contacts between the amino-acid side chains and the DNA bases.

It would seem obvious that the delay time between activation and binding of a transcription factor to the operator sequence (“binding time”) is minimized by maximizing the 1D diffusion constant D 1 . However, simply increasing the transport rate will impair the accuracy, or fidelity, with which the protein can distinguish a right from a wrong site. Specifically, if the binding of a transcription factor to the target site is characterized by a certain rate Ω, then the protein is likely to overshoot the target site if the jump rate D 1 /a 2 between sites, with a the spacing between protein binding sites, is large compared to Ω. Similar conflicts between process speed and process fidelity are familiar from DNA duplication and transcription where increased reaction rates increase the number of duplication and transcription errors.

Slutsky and Mirny [7] proposed that conformational fluctuations could ease the conflict between speed and fidelity. If some conformations of the transcription factor are sensitive to the DNA sequence while others are characterized by rapid transport then the transcription factor might be able to scan the genome efficiently by rapidly flipping between the two types of conformations. The aim of this paper is to analyze how close this mechanism can approach limits of search efficiency imposed by fundamental principles of thermodynamics. We will address this question by examining a simple model for the conformational fluctuations, similar to that of Ref. [7], where the transcription factor is allowed to adopt only two conformations (+ and -) when in contact with non-operator After returning to the + state, it restarts the sliding motion. The protein also can desorb from the chain (d) and return to three-dimensional diffusive motion. Following a number of such cycles, the protein lands in the “antenna region” within a distance λ of the target binding site (e). After reaching the target site by one-dimensional diffusion it can undergo a large-scale irreversible conformational transition to the final bound state if it is in thestate (f).

DNA. As illustrated in Fig. 1, in the + state, the protein is less ordered and only loosely associated with the DNA while it can slide along the DNA chain. In the state, the protein is more ordered, closely associated with the DNA and immobile [8]. If the transcription factor is in contact with the target operator sequence then, in addition to these two states, it also can undergo an irreversible conformat

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