Eukaryotic gene regulation at equilibrium, or non?

Abstract

Models of transcriptional regulation that assume equilibrium binding of transcription factors have been less successful at predicting gene expression from sequence in eukaryotes than in bacteria. This could be due to the non-equilibrium nature of eukaryotic regulation. Unfortunately, the space of possible non-equilibrium mechanisms is vast and predominantly uninteresting. The key question is therefore how this space can be navigated efficiently, to focus on mechanisms and models that are biologically relevant. In this review, we advocate for the normative role of theory-theory that prescribes rather than just describes-in providing such a focus. Theory should expand its remit beyond inferring mechanistic models from data, towards identifying non-equilibrium gene regulatory schemes that may have been evolutionarily selected, despite their energy consumption, because they are precise, reliable, fast, or otherwise outperform regulation at equilibrium. We illustrate our reasoning by toy examples for which we provide simulation code.

Keywords: Gene regulation; Modeling; Non-equilibrium regulation; Optimization.

Figure 1

Reaction cycles as the fundamental “unit” of NEQ models highlight key signatures of time-irreversibility. (a) Three biological examples of irreversible cycles. (Left) A walking myosin on an actin filament. (Middle) An enzymatic cycle. (Right) Gene promoter progression towards activation. (b) A class of models that can continuously transform from EQ to NEQ is parametrized by a unique “reversibility” parameter α for a given size (N = 3 states), at constant mean state occupancies (P_i = 1/N) and residence times (T_i = T₁ = 1min). (c) Two stochastic realizations, for a fully reversible (α = 0, left) and fully irreversible cycle (α = 1, right). (Top) Individual state occupancies. (Bottom) Winding number for each realization, i.e., the number of counter-clockwise cycle completions. Irreversibility leads to a clear temporal ordering of reactions, as highlighted by the progression of the winding number. (d) Entropy production Σ or dissipation (Σ times temperature) as a function of the current J along the cycle. The current J is the slope of the mean winding number (black dash line in C), whose magnitude is equal to the inverse of the period of the cycle. Both J and Σ monotonically increase as the cycle approaches full irreversibility (α = 1). When α = 1, the current is maximal, J_max = 1/(NT₁) and the entropy production tends to infinity. The presence of currents and entropy production are hallmarks of NEQ reaction schemes. (e) Residence time distribution P(T_{2,3}) for the combined states 2 and 3, i.e., time spent in 2 and/or 3 before ending in 1. T_{2,3} is phase-type distributed and its shape depends on α, changing from exponential-like (α = 0) to “peaked” (α = 1). Peaked residence time distributions are another signature of strongly irreversible processes. (Inset) residence time distribution P(T₁) for individual states. T₁ is exponentially distributed and does not depend on α.

Figure 2

Two toy models of gene regulation with functional advantages when operating out-of-equilibrium (NEQ) versus in equilibrium (EQ). (a) (Left) High-specificity model implements kinetic proof-reading through an irreversible transition with rate k_q leading to the active state (expression E is equal to active state occupancy). The two other states denote TF bound/unbound to the DNA. Ratio k₋/k_q determines the strength of the proof-reading. When k_q→∞, the model reduces to the simple two state model (unbound and active) at equilibrium. (Right) High-sensitivity model with N = 3 TF binding sites (8 occupancy states) and higher-order cooperativity ${\text{a}}_{\text{i}}^{\text{n}}$ between TFs (with a₂ ≥ a₁ ≥ 1). The a_i are interpreted as cooperativities as they reduce the TF unbinding rates in presence of n other bound TFs. Expression E is “all-or-nothing”, equal to the occupancy of the all-bound state (active state). Detailed balance holds only when a₁ = a₂; the ratio a₂/a₁ moves the model out-of-equilibrium by controlling the degree of asymmetry among cooperativities. For both models, the TF binding rate k₊ is assumed to be proportional to the TF concentration. (b) Induction curves. (Left) Induction curves for EQ (blue) and NEQ (orange) proof-reading model with specific TF binding site (k₋ = 1) and unspecific site (k₋^NS = 10²). Both models lead to similar induction curves for specific sites, but NEQ model’s expression plateaus at k_q/(k₋ + k_q) for non-specific binding whereas the EQ model still plateaus at 1. (Right) Induction for EQ (blue) and NEQ (blue to orange) asymmetric cooperativity models (with same residence time T_A). The NEQ models achieve a vast range of sensitivities, defined as the slope at half-maximum expression. (c) Optimal operating regime of the two models. (Left) Specificity S, defined as the ratio of expression from specific and non-specific TF binding sites, as a function of the proof-reading ratio k₋/k_q, with TF concentration adjusted to hold E = 0.5, shows an optimal regime (10⁻³ < k₋/k_q < ≃ 1, gray region) where the specificity of a NEQ model clearly outperforms the EQ limit, k_q→∞. (Right) Hill coefficient H, defined as the log-derivative of the expression curve at half maximum, as a function of the cooperativity ratio a₂/a₁, with E = 0.5, and fixed active state residence time, shows an optimal regime (1 < a₂/a₁ < f(a₁), gray region) where the sensitivity of a NEQ model is larger than the EQ limit, a₁ = a₂. (d) Only a fraction of regulatory phenotype space is accessible at fixed average expression. (Left) All NEQ models outperform the EQ limit (black curve), though the maximal specificity increase is limited. (Right) A fraction of NEQ models outperforms the EQ limit (black curve), with some breaking the limit of H = N = 3 set by Hill-type regulation.

Figure 3

Normative approach helps us navigate a complex NEQ gene regulatory model. (a) Scheme of the Monod-Wyman-Changeux-like (MWC) model (here with N = 1 TF binding site, simplified from [44]) for a putative eukaryotic enhancer, describing TF un/binding (with rates k₋ and k₊) and Mediator un/binding (with rates κ₋ and κ₊), TF-Mediator interaction (parameter α), and a proof-reading step (with rate k_link) by the formation of a link between the TF and the Mediator. For k_link→∞, this model reduces to classic equilibrium MWC. (b) Stochastic realization of the model for N = 3 TF binding sites in EQ regime (blue, k_link→∞) and NEQ regime (orange). (Top) Occupancy of the mediator-bound (expressing) state as a function of time. (Bottom) Protein counts simulated from the active state of the NEQ model show bursts due to slow activation dynamics, measured by the noise parameter Φ [44]. (c) Accessible regulatory phenotypes of the model at fixed average expression, such as specificity S, propagated noise Φ (left colorbar) and sensitivity H (right colorbar), as a function of TF residence time T_TF. Black line corresponds to the EQ limit. Φ is in trade-off with S (high specificity implies high noise), while T_TF is in trade-off with H (high sensitivity implies high TF residence time). Thus, it is difficult to optimize all regulatory phenotypes at once. The star stands for one possibly relevant model that provides good improvement in S, high H, low Φ and low T_TF. (d) Varying k_link and α enables sampling the regulatory space in (C). The colorbar reflects the magnitude of an arbitrary utility function (cf. [67]) of the combined phenotypes (increasing with S and H, decreasing with Φ and T_TF). Gray curves represent equi-phenotype lines. Most parameter values lead to functionally unattractive models (blue region). Only a small subspace (orange region around the star) of models are functionally relevant as they simultaneously optimize multiple phenotypes. (e) By first optimizing the regulatory phenotypes, the normative approach can restrict the model or parameter space prior to inference, making model construction and sub-sequent inference of NEQ models more tractable.