Weak transcription factor clustering at binding sites can facilitate information transfer from molecular signals

TL;DR

The paper addresses how transcription factor clustering at binding sites modulates information transfer from the Bicoid morphogen to downstream cell fate decisions in the Drosophila embryo. It employs two complementary models: a Hill-function framework capturing weak cooperativity and a mechanistic Ising-like clustering model that includes surrounding DNA regions to study $I(C;s)$ and $I(C;x)$. The key finding is that weak clustering can maximize functionally relevant information, with the measurement timescale $\tau$ determining whether clustering acts as a switch-like enhancer for short times or yields maximal developmental information for longer times; clustering can also approach the information bottleneck (IB) bound under realistic sensor constraints. These results reconcile observed clustering with information-optimization principles and offer concrete predictions for experiments measuring binding-site occupancy and downstream gene expression.

Abstract

Transcription factor concentrations provide signals to cells that allow them to regulate gene expression to make correct cell fate decisions. Calculations for noise bounds in gene regulation suggest that clustering or cooperative binding of transcription factors decreases signal-to-noise ratios at binding sites. However, clustering of transcription factor molecules around binding sites is frequently observed. We develop two complementary models for clustering transcription factors at binding site sensors that allow us to study information transfer from a signal, the morphogen Bicoid, to a variable relevant to development, namely future cell fates. We find that weak cooperativity or clustering can allow for maximal information transfer, especially about the relevant variable. The timescale of measurement is crucial for predicting the optimal clustering strength: for short measurements, clustering allows for the implementation of a switch, while for long measurements, weak clustering allows the sensor to access maximal developmental information provided in a nonlinear signal. Finally, we find that clustering not only facilitates information maximization about the relevant variable, but also can allow the binding site sensors to achieve optimality in a related optimization goal, the information bottleneck (IB) bound. While the measurement time restricts the region on the information plane that is accessible, changes in clustering in conjunction with changes in the binding energy can shift the binding site along the optimal bound, and towards an optimal trade-off between obtaining information about the signal and obtaining relevant information.

Figures (10)

• Figure 1: Signal processing in the fly embryo. A) The normalized maternal Bicoid gradient Gregor2007a along the embryonal head-to-tail axis, which regulates the four gap genes; inset shows hetereogeneous Bicoid concentration inside a single nucleus (airy-scan image with experimental design described in Ref. Munshi_2024). B) The normalized concentration of the four gap gene proteins (hunchback, giant, Krüppel, knirps in red, yellow, blue, green respectively) in nc 14 as a function of embryonal position x, scaled to embryo length L, exemplifying the developing body segments Petkova. C) hb RNA expression in nc 13 as a function of embryonal position Littleetal, when hb expression is mostly regulated through the proximal hbP2 enhancer region. D) Sketch of real the hbP2 region Park_2019, containing ca. 6 binding sites, and our simplification with equidistant binding sites of equivalent binding strength.
• Figure 2: Information transfer through a sensor with cooperative binding sites. A) Sketch of cooperative binding of transcription factors (TFs) to six binding sites. B) Mean occupation according to the Hill function for different parameters $h$ and $k$. C) The information $I(C;x)$ (and $I(C;s)$) as a function of $h$ and $k$ shows a well-defined maximum (cross) in parameter space. We use $k_\textrm{off} = 1/$s. D) The maximum possible information $I(C;x)$ and $I(C;s)$ over all values of $k$ for different $h$ (maxima indicated by vertical lines). For the longer measurement time $\tau=600$ s (orange), the maximal information value per $h$ is higher than for $\tau=10$ s (blue), and maximal values are reached for lower $h$.
• Figure 3: Information transfer through a binding site or DNA region with individual clustering signal molecules. A) Sketch of model binding sites (solid) and surrounding sites on DNA (dashed; DNA region), with on- and off-rates for transcription factors. B) Steady-state occupation $\bar{C}$ of the binding site region (no surrounding sites) for $J=0$ and $\varepsilon_b\approx 14.1$, and $J=7$ and $\varepsilon_b\approx8.2$ as a function of signal concentration, $s$, with the standard deviation (shaded region) for measurement time $\tau=10$ min, and best-fit Hill function (dotted line), with $h\approx 1$ and $h\approx 4$, respectively. C) Standard deviation $\sigma_{C}$ from simulations of the binding site region matches the analytical solution from the master equation (which can be approximated by Eq. 7 as shown in Appendix VI C, Fig 7A), decreasing from its instantaneous value to a $1/\sqrt{\tau}$-scaling for longer $\tau$. Parameter values are $\mu =-14.0$, $J=3.5$ and $\varepsilon_b\approx 11.1$, with error bars over 5 independent repeats of the calculation, each based on 20 simulations. D) Sketch of the simulation procedure: for all $\mu$ corresponding to signal concentrations $s(x)$ along the embryonal axis, we perform Gillespie simulations and average occupation over time $\tau$ to obtain $C$. For each $s$, we estimate $P(C|s)$ based on histograms of $C$ from 100 repeats of the simulation. We calculate $I(C;s)$ and $I(C;x)$ from this distribution for each $J$, using a modified parameter $\varepsilon_b$ to retain half maximum occupation at $x\sim 0.47$. E+F) Mutual information in an instantaneous measurement: Clustering improves both $I(C;s)$ and $I(C;x)$, with a maxima for clustering strengths $J\approx 3$ and $J\approx 5$, respectively. For high $J$, the occupation of additional sites along the DNA can further increase the information. G+H) Mutual information in a long measurement ($\tau=600$ s): Clustering decreases $I(C;s)$. The information $I(C;x)$ also decreases with clustering, but with minor information loss (5% of the maximum) up until intermediate clustering strengths of $J=6$ (shaded area).
• Figure 4: Binding site sensors that incorporate cooperativity and clustering can theoretically achieve the information bottleneck optimization goal. A) Sensors from the mechanistic model for instantaneous (blue) and long measurement times (orange) for changing clustering strength $J$ (color shade) on the information plane, with optimal IB bound from data in black. B) Different binding site sensors from the Hill function model, parameterized by $h$ and $k$, are optimal for different constraints: grey circles and white crosses mark paths with a (variable) constraint on $h$ and a (variable) constraint on $I(C;s)$ (IB goal), respectively. C) Information plane with IB bound in black and a selection of exemplary binding site sensors (blue crosses and orange plusses for $\tau=10$ s and $\tau=600$ s, respectively) show that randomly chosen binding sites are well below the bound. The sensors that optimize $I(C;x)$ for a given $I(C;s)$ in our parameter ranges of $h$ and $k$ are shown with larger circles; color shade indicates the value of $h$. D) Selected binding site sensors (1,2) from panel C representing the sensors along the IB bound (top), and sensors that represent point (3) in panel C (bottom, for $\tau=10$ s in blue, $60$ s in grey and $600$ s in orange). E) Optimal values of $h$, $k$ and $\tau$ for binding site sensors fitted to $P(C|s)$ from IB optimization, for three values on the bound in Fig.4C (grey arrows) show that a variety of binding site sensors are consistent with the IB bound.
• Figure 5: $I(C;x)$ and $I(C;s)$ as a function of $h$ and $k$ for two representative measurement times. A) $I(C;x)$ and $I(C;s)$ for $k_{\rm on} = 10$ s$^{-1}$ nM$^{-h}$ constant and varying $k_{\rm off}$; the maxima (cross) are always at the boundaries of parameterization space. B) Longer measurements with $\tau = 600$ s show a similar behaviour to $\tau=10$ s, with a single optimum (cross). The left panels are replotted from the main with an increased range of $k$ values. C left and middle) The optimized information $I(C;x)$ and $I(C;s)$ over all $k$ and $h$ as a function of measurement time $\tau$, with $I(C;x)$ reaching its maximal bound $I(s,x)$ at approximately $\tau\sim 1000s$. C right) The value of $h$ at the optimum of $I(C;x)$ decreases with measurement time $\tau$.