How the DNA sequence affects the Hill curve of transcriptional response

The Hill coefficient is often used as a direct measure of the cooperativity of binding processes. It is an essential tool for probing properties of reactions in many biochemical systems. Here we analyze existing experimental data and demonstrate that the Hill coefficient characterizing the binding of transcription factors to their cognate sites can in fact be larger than one – the standard indication of cooperativity – even in the absence of any standard cooperative binding mechanism. By studying the problem analytically, we demonstrate that this effect occurs due to the disordered binding energy of the transcription factor to the DNA molecule and the steric interactions between the different copies of the transcription factor. We show that the enhanced Hill coefficient implies a significant reduction in the number of copies of the transcription factors which is needed to occupy a cognate site and, in many cases, can explain existing estimates for numbers of the transcription factors in cells. The mechanism is general and should be applicable to other biological recognition processes.

💡 Research Summary

The paper challenges the conventional interpretation of Hill coefficients greater than one as a direct signature of cooperative binding in transcription factor (TF)–DNA interactions. By re‑examining a broad set of experimental data—including bacterial LacI and AraC systems as well as eukaryotic factors such as Gcn4—the authors observe that the apparent Hill coefficient (H) varies widely across different DNA target sequences, sometimes reaching values far above the canonical cooperative threshold even when no known cooperative mechanism (e.g., dimerization, allosteric coupling) is present.

To explain this paradox, the authors construct a statistical‑mechanical model that incorporates two physically realistic ingredients. First, they treat the binding energy of a TF to its cognate site as a random variable drawn from a Gaussian distribution whose variance reflects sequence‑dependent variations (base composition, methylation, nucleosome positioning, etc.). This “energy disorder” means that some sites are intrinsically higher‑affinity than others, and the ensemble average of the Boltzmann factor ⟨e^{‑βE}⟩ replaces a single deterministic dissociation constant. Second, they impose a steric exclusion rule: each TF occupies a finite physical volume on the DNA, so only one TF can bind to a given location and neighboring sites are blocked when a TF is already bound. This steric interaction creates a saturation constraint that becomes significant at moderate TF concentrations.

When these two effects are combined, the resulting binding isotherm deviates from the simple Michaelis–Menten form and takes on a Hill‑like shape with an effective exponent n that is not a true cooperativity parameter but rather an emergent property of the disorder‑induced heterogeneity and steric crowding. Analytically, the effective binding constant can be written as K_eff ≈ ⟨e^{‑βE}⟩^{‑1}, and the occupancy follows

θ(