Information integration and energy expenditure in gene regulation

Summary

The quantitative concepts used to reason about gene regulation largely derive from bacterial studies. We show that this bacterial paradigm cannot explain the sharp expression of a canonical developmental gene in response to a regulating transcription factor (TF). In the absence of energy expenditure, with regulatory DNA at thermodynamic equilibrium, information integration across multiple TF binding sites can generate the required sharpness but with strong constraints on the resulting “higher-order cooperativities”. Even with such integration there is a “Hopfield barrier” to sharpness, represented, for n TF binding sites, by the Hill function with Hill coefficient n. If, however, energy is expended to maintain regulatory DNA away from thermodynamic equilibrium, as in kinetic proofreading, this barrier can be breached and greater sharpness achieved. Our approach is grounded in fundamental physics, leads to testable experimental predictions and suggests how a quantitative paradigm for eukaryotic gene regulation can be formulated.

eTOC

The physical principles governing gene regulation in bacteria can't explain the sharpness of gene expression in eukaryotes; energy use and information integration have to be taken into account as well.

Introduction

The molecular machinery which transcribes DNA into RNA is general purpose. Deciding which gene to transcribe requires regulatory DNA sequence information, which is interpreted by sequence-specific, DNA-binding transcription factors (TFs). Quantitative measurements of TF-DNA and TF-TF interactions in bacteria (Ptashne, 2004), together with analysis of the underlying physics (Ackers et al., 1982), have introduced fundamental quantitative concepts like “affinity” and “cooperativity” to explain the regulated recruitment of RNA polymerase to a gene. This bacterial paradigm has been widely used to interpret experimental results even outside the bacterial domain. However, eukaryotic transcription differs considerably from bacterial transcription, which raises the question of whether the bacterial paradigm is sufficient to explain how eukaryotic genes are regulated.

Bacterial TF sequence motifs have an average length of 16 base pairs, while those in eukaryotes are only half as long (Wunderlich and Mirny, 2009), suggesting that eukaryotes depend on combinatorial integration of many small packets of information. Such information integration might be implemented through nucleosomes or by multi-protein co-regulators such as Mediator or CBP/p300 that make multiple contacts between TFs and the transcriptional machinery (Spitz and Furlong, 2012). Also, while bacterial gene regulation appears not to require energy from donors like ATP, making it reasonable to assume that it takes place at thermodynamic equilibrium, eukaryotic gene regulation depends on energy expenditure to reorganise chromatin, displace nucleosomes, post-translationally modify regulatory proteins and methylate DNA. This qualitative appreciation of eukaryotic complexity has been difficult to translate into rigorous, well-defined concepts and new kinds of experiments which can explain the role of these molecular mechanisms in gene regulation.

Quantitative models grounded in physics can fill this critical gap. The physics-based “thermodynamic formalism” that was developed for bacteria, which assumes that regulation takes place at thermodynamic equilibrium, has been codified (Bintu et al., 2005) and applied to gene regulation in Drosophila, yeast and human cells (Segal and Widom, 2009; Sherman and Cohen, 2012). However, the molecular complexity that is found in eukaryotes, especially that which implements the information integration and energy expenditure described above, has not been incorporated into these models.

Questions about the sufficiency of the bacterial paradigm have been accumulating (Coulon et al., 2013) but the absence of a compelling example and the lack of appropriate concepts have made it easy to fall back on what is familiar. Here, we present a compelling example of insufficiency and introduce appropriate quantitative concepts, rigorously based on the underlying physics, with which to reason about eukaryotic gene regulation.

We bring together three ingredients, which exemplify a general approach to the problem. First, we focus on a property of gene regulation that can be described quantitatively; second, we identify a biological system in which that property has been measured; and, third, we exploit a mathematical framework that allows us to analyse both equilibrium and non-equilibrium systems.

The quantitative property on which we focus is the sharpness of gene expression in response to a TF, or the extent to which a small change in TF concentration can lead to a larger change in gene expression. Sharpness has been investigated in several biological systems but is particularly evident in developmental patterning. The zygotic gap gene hunchback (hb) is expressed in an anterior region of the early Drosophila embryo under regulation by the maternal morphogen Bicoid (Bcd) (Figure 1A). The expression levels of Hb and Bcd proteins are related to each other in a way that is closely approximated by a simple algebraic expression (Gregor et al., 2007)

Figure 1.

Sharpness in development and cooperativity mechanisms. A. Top, Drosophila embryo stained for Hb expression. Bottom, plot adapted from Figure 4A of Gregor et al. (2007), showing mean ± standard error of Hb and Bcd from several embryos (blue) and a fit to the Hill function ℋ₅ (red). B. Examples of indirect, long-distance cooperativity, adapted from Figure 1 of Spitz and Furlong (2012). C. Top, pairwise cooperativity between two sites. Bottom, higher-order cooperativity of order three.

$$\frac{[\text{Hb}]}{{[\text{Hb}]}_{\mathit{max}}}\approx \frac{x^5}{1+x^5}.$$ (1)

Here, the concentration of Hb, denoted [Hb], is normalised to its maximal level, while x denotes Bcd concentration normalised to the value at which half-maximal Hb expression is reached, so that x = [Bcd]/[Bcd]_0.5. Eq. 1 describes the hb gene regulation function, which expresses quantitatively how the output of hb depends on [Bcd].

The expression in Eq. 1 is a Hill function, ℋ_a(x) = x^a/(1+x^a), whose Hill coefficient, a, has the value a = 5. Increasing Hill coefficients imply increasing sharpness. In Eq. 1, the sharpness represented by a = 5 reflects the precision with which individual nuclei use Bcd to determine their position along the anterior-posterior axis and create the tight boundary between Hb “on” and Hb “off.

Gregor et al explain how the sharpness in Eq. 1 arises by saying that it is “consistent with the idea that Hb transcription is activated by cooperative binding of effectively five Bcd molecules” (Gregor et al., 2007). This reflects the conventional bacterial paradigm, in which sharpness is accounted for at thermodynamic equilibrium by pairwise cooperativity between TFs, whereby TF binding at one site influences the affinity of TF binding at another site (Ptashne, 2004). With n binding sites and pairwise cooperativity, it is widely believed, as Gregor et al suggest, that sharpness corresponding to a Hill coefficient of n can be achieved, without requiring any expenditure of energy.

To examine this idea, we use a recently-introduced mathematical framework which generalises the thermodynamic formalism to accommodate mechanisms that expend energy (Ahsendorf et al., 2014). We show that for regulatory DNA at thermodynamic equilibrium with only pairwise cooperativity, the experimentally-measured sharpness described in Eq. 1 cannot be biochemically realised, no matter how many TF binding sites are present. The widely-held belief that the bacterial paradigm can be extrapolated in this way is not rigorously justified. We believe this is a compelling example of its insufficiency.

Information integration through nucleosomes or co-regulators could yield indirect, long-distance forms of cooperativity (Figure 1B), which could link multiple TF binding sites. To account for this, we introduce the concept of “higher-order cooperativity” at thermodynamic equilibrium (Figure 1C). If such cooperativities are present, greater levels of sharpness become possible. With n binding sites and higher-order cooperativities, a Hill coefficient of n or more still remains out of reach but a Hill coefficient less than n can be achieved. Furthermore, not just the Hill coefficient but also the overall shape of the gene regulation function (GRF) can match what is found experimentally: with enough binding sites, GRFs can be found that are statistically indistinguishable in shape from the Hill functions in Eq. 1. However, these GRFs lie on the edge of what can be biochemically achieved and impose stringent quantitative constraints on the mechanisms responsible for higher-order cooperativity.

Higher-order cooperativities improve sharpness but reveal fundamental barriers to what can be achieved without the expenditure of energy. The existence of such barriers was first suggested in Hopfield's work on kinetic proofreading (Hopfield, 1974; Ninio, 1975). He showed, in effect, that if a biochemical system operates at thermodynamic equilibrium then physics imposes a barrier to how well a given information processing task can be accomplished. (In his case, the task was achieving fidelity in transcription and translation.) The only way to bypass this barrier is to expend energy and maintain the system away from equilibrium. Kinetic proofreading is one way to do this.

We identify here a “Hopfield barrier” for sharpness in gene regulation. With n binding sites, a Hill coefficient of n sets the Hopfield barrier; at thermodynamic equilibrium, no GRF can reach it, even with higher-order cooperativities. If, however, energy is expended to maintain regulatory DNA away from equilibrium then much greater sharpness can be achieved.

Results

The rationale for the model

We introduce a mathematical model for analysing gene regulation. As with all models, the conclusions depend on the assumptions (Gunawardena, 2014). Our assumptions are guided by the example of hb but the model is general and not restricted to this example. The anterior expression pattern of hb is believed to be regulated by at least 3 enhancers (Perry et al., 2011). Both the classical P2 enhancer, which is promoter proximal, and a shadow enhancer, located ∼3kb upstream, drive broad anterior patterns early in embryo development. Later, the central stripe enhancer drives expression near the middle of the embryo. Bcd is a transcriptional activator for the P2 and shadow enhancers; the stripe enhancer is also targetted by transcriptional repressors. The stripe enhancer has no effect on sharpness early in nuclear cycle 14 (Perry et al., 2012), when the data were acquired on which Eq. 1 is based (Gregor et al., 2007).

Accordingly, we focus on a single TF, binding to a specified but arbitrary number of sites and functioning solely as a transcriptional activator. TF binding sites can be anywhere on the genome and are not assumed to be confined to a single enhancer; thus, our analysis is not limited to the 5-7 Bcd binding sites thought to be present in the hb P2 enhancer.

Molecular mechanisms other than TF binding and unbinding, such as nucleosomes or co-regulators, are not directly represented in our model but their influence is captured through their effects on rate constants and the dependence of these constants on the state of DNA (“microstate”—see below). This permits general conclusions to be drawn without knowing the specific mechanisms at work in a particular gene but does not allow us to assign mechanisms to the effects we find. Other molecular features, such as post-transcriptional mechanisms or network effects like feedback, could influence sharpness but these are not thought to be relevant for Bcd regulation of hb. Addressing such features in future work may yield further insights into sharpness.

A graph-based model of gene regulation

We recently introduced a graph-based “linear framework” for modelling gene regulation (Ahsendorf et al., 2014). We use this to formulate a general model of a gene responding to a TF, called T, binding as a monomer to a number, n, of sites. Oligomerisation of a TF in solution can contribute to gene-expression sharpness but this is not thought to be significant in Bcd regulation of hb (Lebrecht et al., 2005; Gregor et al., 2007) and we do not consider it here. In this section and the next, we discuss the quantitative details of how T binds and unbinds, how cooperativity is defined, and how T influences transcription.

The model consists of the labelled, directed graph, G_n (Figure 2A). The vertices of G_n represent the microstates, or patterns of T bound to DNA, with the binding sites labelled by the numbers 1, ⋯, n The edges represent binding or unbinding of T from the microstates. Each edge has a label describing the rate of the corresponding reaction. The label on a binding edge is the product of the concentration of T, [T], and an on-rate for binding, a_i,s, where i is the binding site and S is the subset of sites at which T is already bound. Subsets are denoted {i₁, ⋯, i_k}, where the site indices, i₁, ⋯, i_k are drawn from the numbers 1, ⋯, n. The label on an unbinding edge is an off-rate, b_i,s′ where S′ is the subset of sites to which T is bound and i is one of the sites in S′.

Figure 2.

Linear framework model. A. The graph G₃ showing the 8 microstates and the associated labelled, directed edges, with middle-layer labels omitted for clarity. B. The essential parameters at thermodynamic equilibrium. The association constants K_i,s have units of (concentration)⁻¹ while k_i and ω_i,S are non- dimensional. S ∪ {i} is the set in which site i has been added to the sites in S. C. The higher-order cooperativity ω_i,S measures whether binding of T to site i, when T is already bound to the sites in S, shows reduced affinity (ω_i,S < 1), unchanged affinity (ω_i,S = 1), or enhanced affinity (ω_i,S > 1), as compared to binding to site i when no other sites are bound.

Importantly, the on-rates, a_i,s′ and the off-rates, b_i,s′, can depend on the site of binding or unbinding, i, as well as on the pattern of existing binding to a subset of sites, S or S′. This reflects the potential influence of background mechanisms, such as nucleosomes or co-regulators, and allows higher-order cooperativities to be introduced below.

The linear framework describes how such a graph gives rise to a stochastic master equation for the probabilities of the microstates. As in the thermodynamic formalism, we make the basic assumption that regulatory DNA is at steady state. However, unlike the thermodynamic formalism, the linear framework allows steady-state probabilities to be calculated irrespective of whether or not the system is at thermodynamic equilibrium (Ahsendorf et al., 2014); see the Experimental Procedures and the Supplemental Information (SI).

Higher-order cooperativities and the exchange formula

Higher-order cooperativity between multiple TF binding sites may be important in gene regulation but thermodynamic formalism models have usually been limited to pairwise cooperativity. This is not a fundamental limitation but arises from technical difficulties with the Principle of Detailed Balance, which imposes algebraic constraints on higher-order cooperativities (SI) that have not been worked out within the thermodynamic formalism. Detailed balance, or “microscopic reversibility”, is a fundamental requirement arising from the time-reversal symmetry of the laws of physics (Mahan, 1975). The constraints are a serious obstacle because they mean that the numerical values of higher-order cooperativities cannot be chosen independently. Thus it is important to determine these constraints (Eq. 2) and to thereby identify a subset of cooperativities whose numerical values are independent (Eq. 3).

If the regulatory system described by G_n can reach thermodynamic equilibrium, the relevant parameters are the association constants K_i,s (Figure 2B), of which there are n2ⁿ⁻¹ (SI). To define higher-order cooperativities at equilibrium, we compare the binding of T to site i when T is already bound at the sites in S (K_i,S) to the binding of T to site i when T not bound elsewhere (K_i,∅, where ∅ denotes the empty set). This yields a non-dimensional higher-order cooperativity, ω_i,S = K_i,S/K_i,∅, whose value indicates whether or not there is positive or negative cooperativity or independence (Figure 2C). To non-dimensionalise the remaining association constants, we define k_i = K_i,∅/K_1,∅,

The number of sites in S is called the order of ω_i,S and denoted #S; it specifies how many sites collaborate to influence binding. Pairwise cooperativity corresponds to order 1. Thermodynamic formalism models set ω_i,S = 1 for #S>1. In this case, detailed balance reduces to a symmetry requirement on pairwise cooperativities, ω_i,{j} = ω_j,{i} (see Eq. 2). With only pairwise cooperativity, there are only n(n−1)/2 parameters, instead of n2ⁿ⁻¹, which greatly simplifies thermodynamic formalism calculations.

We prove that, because of detailed balance, higher-order cooperativities must satisfy the “exchange formula” (SI)

$${\omega }_{i,S\cup \{j\}}{\omega }_{j,S}={\omega }_{j,S\cup \{i\}}{\omega }_{i,S}$$ (2)

which summarises the algebraic constraints among the cooperativities. Here, i, j are sites that are not in S while the notation S ∪ {v}, for v = i or v = j, denotes the addition of v to the sites in S. We further prove that, if we retain only those ω_i,S for which i is less than all the sites in S (abbreviated i < S), then the parameters

$$K_{1,\varnothing },k_i(i>1),{\omega }_{i,S}(i<S),$$ (3)

of which there are 2ⁿ − 1, are algebraically independent and all the K_i,s can be calculated from them using Eq. 2 (SI). We can thus vary the parameters in Eq. 3 independently and be confident that detailed balance holds. These fundamental results provide the basis for the equilibrium calculations that follow.

Equilibrium gene regulation functions (GRFs)

To calculate a GRF, we make the same basic assumption as in the thermodynamic formalism and consider the overall rate of transcription to be an average over the steady-state probabilities of the microstates. For this, we must specify the rate of transcription in each microstate, about which surprisingly little is known for eukaryotic genes. As explained above, we assume that T acts as a transcriptional activator (SI), so that the binding of T does not reduce the expression level. We consider three expression strategies which work for any number of sites (Figure 3A): all-or-nothing, in which transcription only occurs when all sites are bound; one-or-more, in which transcription occurs when at least one site is bound; and average-binding, in which transcription is proportional to the number of bound sites.

Figure 3.

Gene expression strategies and shape measures. A. The graph G₃, with pairs of reversible edges as unlabelled single lines for clarity, illustrating three expression strategies for a transcriptional activator. Each microstate is annotated with a number describing the corresponding rate of gene expression, with the maximal rate normalised to 1. B. Plot of a hypothetical GRF (black), together with its derivative (red) showing steepness (ρ) and position (γ), as defined in Eq. 6. The derivative may have multiple local maxima and ρ and γ are defined at the global maximum.

It is computationally infeasible to explore all expression strategies but these three strategies broadly sample the spectrum of possibilities (SI). All-or-nothing and one-or-more are extreme opposites, while average-binding is an intermediate strategy. All-or-nothing is widely used in thermodynamic formalism models and average-binding corresponds to the “fractional saturation” used in models of protein allostery (Monod et al., 1965; Mirny, 2010).

The level of protein expression after normalisation to its asymptotic maximum is a rational function of x = [T], denoted f_n(X), which has the form, for the all-or-nothing strategy (SI),

$$f_n(x)=\frac{C_n{x}^n}{1+c_{1}x+\cdots +c_n{x}^n}.$$ (4)

The coefficients c_k are given (SI) by a sum of products

$$c_k=\left(\sum _{1\le i_1<\cdots <i_k\le n}\right(\prod _{j=1}^k{k}_{i_j}{\omega }_{i_j,\{i_{j+1},\cdots ,i_k\}}\left)\right){(K_{1,\varnothing })}^k.$$ (5)

which involves only the independent parameters in Eq. 3 and allows higher-order cooperativity of any order up to the maximum of n − 1. For the other strategies (Figure 3A), only the numerator of Eq. 4 changes (SI). The GRFs discussed in this paper are strictly increasing functions (SI). For the all-or-nothing strategy at equilibrium they also appear to be sigmoidal (“S-shaped”), so that the derivative of the GRF has only a single maximum, but this is not so in general (Figure 3B).

The algebraic form of the Hill function, ℋ_a(x) = x^a/(1 + x^a), is closest to that of the GRF for the all-or-nothing strategy in Eq. 4 (SI). If a < n it is clear that the former cannot algebraically resemble the latter because the degrees of their respective denominator polynomials are different. If a = n, algebraic resemblance is only possible, and then only approximately, if the parameters in the GRF are given implausible numerical values (SI). We see that Hill functions are not GRFs. However, the question posed by Eq. 1 is whether a GRF can match the shape of a Hill function. This is a more delicate problem.

Position and steepness as quantitative measures of shape

To determine the match between a GRF and a Hill function, we introduce two quantitative measures of shape. We first normalise the concentration scale of x = [T], in a similar way to Eq. 1, by taking x_0.5 to be the concentration at which f_n is half-maximal, so that f_n(x_0.5) = 0.5, and setting y = x/x_0.5. The normalised GRF is g_n(y) = f_n(yx_0.5), where y and g_n(y) are now both non-dimensional quantities. Note that for ℋ_a(x), x_0.5 = 1, so that Hill functions are already normalised.

To quantify shape, we take the maximum derivative, ρ(g_n) (“steepness”), and the position of the maximum derivative γ(g_n) (“position”),

$$\rho (g_n)=\underset{y\ge 0}{max}\frac{d{g}_n}{\mathit{\text{dy}}},\gamma (g_n)=z\text{such that}\frac{d{g}_n}{\mathit{\text{dy}}}{|}_{y=z}=\rho (g_n),$$ (6)

which are also non-dimensional quantities (Figure 3B). The advantage of γ and ρ is that they can be calculated from g_n, in contrast to a numerical fit to a Hill function, which is subject to statistical noise. Two GRFs with the same γ and ρ (“matched”) are not identical but have similar sharpness. Considering only one measure of sharpness, such as ρ, can be misleading (below).

Impact of higher-order cooperativity on sharpness

We first determined the position and steepness of a GRF in the all-or-nothing strategy with n = 5 sites, allowing higher-order cooperativity of any order. Although γ and ρ depend only on the coefficients c_k in Eq. 5, these aggregated parameters have no biochemical meaning and we seek instead to understand how γ and ρ depend on the affinities and cooperativities in Eq. 3, which are defined in terms of molecular interactions. Because of normalisation, γ and ρ do not depend on K_1,∅ (SI), so we set K_1,∅ = 1 in units of (concentration)⁻¹ and chose k_i and ω_i,S to be in the range [10⁻³, 10⁻³] by random logarithmic sampling. We believe this range is generous but most of our results do not depend on it (below and SI).

A sample of 10⁵ GRFs chosen in this way reveals that position and steepness are not independent but are constrained within a crescent-shaped region in which the highest steepness is found at the extremes of position (Figure 4A). On the left of the region, high steepness occurs only for very low position and the resulting GRFs are highly degenerate: when these GRFs are fitted to Hill functions, they yield a Hill coefficient of a = 1 (inset at top and caption). The marginal distribution (top) shows that these degenerate GRFs account for nearly half of all GRFs in this parameterisation. Degeneracy underscores the importance of considering γ together with ρ.

Figure 4.

Position and steepness for n = 5 sites. A. Probability density function (blue points) of (γ(g₅), ρ(g₅)) obtained by random sampling, with the respective marginal distributions (top and right). The inset (top) shows the GRF of the marked point, annotated with the value a obtained by fitting to ℋ_a. The gray line marks the boundary of the position-steepness region, obtained by a biased sampling algorithm (SI). The magenta line is the locus of (γ(ℋ_a), ρ(ℋ_a) for varying a (the “Hill line”), with the integer values of a marked by magenta crosses and numbers. B. Higher-order cooperativities (left), plotted on a logarithmic scale, for the GRFs closest in (γ, ρ) distance to the integer Hill coefficients (magenta crosses) in panel A, with the Hill coefficient annotated on the left (magenta). The corresponding curve (right) is annotated with the value a obtained by fitting to ℋ_a. C. Position-steepness boundaries with the parameter range [10^−p, 10^p] for varying p. At the top are the higher-order cooperativities (left) and curve (right) for the marked GRF closest to ℋ_a, within the p = 5 region, plotted as in panel B. D. Position-steepness regions for all three expression strategies; see also Figure 6 and Figure S1.

The upper edge of the crescent-shaped region has low probability. We therefore used, for Figure 4 and those that follow, a biased sampling algorithm to identify the boundary of the region (SI), with the same parameter range but with the GRFs filtered so that γ(g_n) lies in the interval [0.5γ(ℋ_n)]. This focusses on the GRFs of interest and avoids the degeneracy near γ(g_n) = 0. We found the gray boundary in Figure 4A.

The right-hand edge of the gray boundary coincides with that of the randomly sampled region in a series of line segments. Strikingly, the “Hill line” on which Hill functions are located (magenta curve) lies just to the right of this boundary. We see that a GRF cannot have greater position than a Hill function of the same steepness, so that the Hill functions define a barrier. The line segments on the boundary of the region touch the Hill line at their corners and these occur, surprisingly, at exactly the integer values, 2, 3, 4, of the Hill coefficient. When the GRFs which are closest to these points are fitted to Hill functions, the estimated Hill coefficients correspond very closely to the integer values (Figure 4B, right). Such a close correspondence in fitted shape is unexpected in view of the lack of algebraic resemblance between GRFs and Hill functions, as discussed above. The emergence of bona fide GRFs which closely match the shape of Hill functions with integer Hill coefficients is intriguing in view of the coefficient 5 found in Eq. 1. However, this shape matching requires high levels of positive and negative higher-order cooperativity of all orders (Figure 4B, left).

The boundary of the position-steepness region lies below ℋ₅ and approaches it at the tip of a cusp. Changing the parameter range does not alter the line segments in the boundary but the tip of the cusp approaches closer to ℋ₅ as the range is increased (Figure 4C). The GRF closest to ℋ₅ has a fitted Hill coefficient close to 5 (top, right), although not as close as for integer values less than 5. This still requires high levels of positive and negative higher-order cooperativity of all orders (top, left).

Each expression strategy reveals a different trade-off between position and steepness (Figure 4D). The Hill line also presents a barrier to the one-or-more strategy but from the opposite side, while the average- binding strategy straddles the Hill line. For the one-or-more strategy, the position-steepness region approaches at cusps the Hill functions whose coefficients are integers less than 5 (Figure 4D) but, in contrast to the all-or-nothing strategy, the region does not touch the Hill line and the integer-valued Hill functions are not closely matched to the nearest GRFs unless the parameter range is increased (data not shown). The barrier presented by the Hill line seems, therefore, to act differently in the all-or-nothing and one-or-more strategies. Irrespective of the expression strategy, ℋ₅ offers a barrier to all strategies with n = 5 sites: each region lies below it and only approaches it at the tip of a cusp as the parameter range is increased.

The features found above are reproduced for different numbers of sites (Figure S1 for n = 7 sites).

Pairwise cooperativity alone permits limited sharpness

Thermodynamic formalism models have typically been limited to pairwise cooperativity. We restricted ourselves to pairwise cooperativity by setting ω_i,S = 1 for #S > 1. We found that the position-steepness region for the all-or-nothing strategy increases initially with increasing n but then shrinks in extent and no GRF approaches close to ℋ₅ (Figure 5A). The one-or-more and average-binding strategies do not even get close to ℋ₃ (Figure 5B).

Figure 5.

Pairwise cooperativity only. Each panel uses the colour code in the centre and shows the Hill line in magenta. A. Position-steepness regions for the all-or-nothing strategy. B. Position-steepness regions for the one-or-more (top) and average-binding (bottom) strategies. The biased sampling algorithm had to be modified to find the average-binding regions (SI).

Thus, in contrast to common assumptions, pairwise cooperativity alone is insufficient for sharp responses in eukaryotic genes. None of the expression strategies considered here can account for ℋ₅ in Eq. 1 with only pairwise cooperativity, no matter how many sites are available.

Non-equilibrium GRFs exceed the equilibrium sharpness barriers

If the system is maintained away from thermodynamic equilibrium by energy expenditure, detailed balance no longer holds. The non-equilibrium GRF for the all-or-nothing strategy then takes the form (SI)

$$f_n^{ne}(x)=\frac{d_n{x}^n+\cdots +d_{2^n-1}{x}^{2^n-1}}{e_0+e_{1}x+\cdots +e_{2^n-1}{x}^{2^n-1}},$$ (7)

where the coefficients of the highest order term, x²ⁿ⁻¹, in the numerator and the denominator are equal, so that d_2ⁿ−1 = e_2ⁿ−1. GRFs for the other strategies differ only in the numerator (SI). The denominator of Eq. 7 shows a striking increase in degree, from n to 2ⁿ − 1, in comparison to that of the equilibrium f_n in Eq. 4, despite the number of sites being the same.

The parameters in Figure 2B are no longer meaningful away from equilibrium and the coefficients d_i and e_i in Eq. 7 are expressions in the rate constants a_i,S and b_i,S′ . For reasons discussed below, the largest number of sites that we can feasibly analyse is n = 3 (SI).

We took non-dimensional parameters a_i,S/a_1,∅ and b_i,S′/b_1,{1} in the range [10⁻², 10²], deliberately restricting the range so that, if the system were at equilibrium, it would be comparable with the previous equilibrium analysis (SI). Because of normalisation, the steepness and position of $g_n^{ne}$ are independent of the values of a_1,∅ and b_1,{1} (SI), so we set a_1,∅ = 1 in their respective units. For each expression strategy, we found (Figure 6) that the non-equilibrium position-steepness region is much enlarged (black boundary) compared to the corresponding equilibrium region for the same number of sites (blue boundary). The non-equilibrium regions now include the Hill line up to ℋ₃ and the all-or-nothing region can reach as far as ℋ₅ if the parameter range is increased (Figure S2A).

Figure 6.

The non-equilibrium case for n = 3 sites. Position-steepness regions for all expression strategies, showing the equilibrium (blue) and non-equilibrium (black) boundaries. The horizontal scale for the one-or-more strategy is extended. See also Figure S2.

The limitation to n = 3 sites arises from loss of detailed balance, which leads to a dramatic increase in the complexity of the coefficients in Eq. 7 (SI; see Discussion). This complexity is algebraic, not numerical. To compute position-steepness regions, cooperativities are treated as symbols whose numerical values are assigned by sampling. Symbolic calculation of the GRF is extremely expensive away from equilibrium but numerical calculation of individual GRFs presents no particular difficulty.

Symbolic treatment of parameters is also informative because it also reveals the structure of the non-equilibrium GRF (Eq. 7). The denominator of this GRF increases in degree exponentially with n but the denominator of the equilibrium GRF (Eq. 4) increases only linearly. This discrepancy arises from loss of detailed balance (SI). With n = 3 sites, the non-equilibrium position-steepness region comfortably exceeds the equilibrium region (Figure 6). Because of the exponential increase in the degree of the GRF denominator, when there are n = 5 sites, or however many sites are relevant for Bcd regulation of hb, the discrepancy in the position-steepness regions will be even greater and the non-equilibrium region will extend well beyond ℋ₅.

In confirmation of this, we used numerical parameter values to find a non-equilibrium GRF on n = 5 sites, with parameters in the same range, [10⁻², 10²] as in Figure 6, whose position and steepness match that of ℋ₆ (Figure S2B). If the parameter range is increased to [10⁻³, 10³] then there is a GRF on 4 sites whose position and steepness match that of ℋ_5.7 (Figure S2B). Being away from equilibrium makes it much easier to achieve the sharpness required for Eq. 1.

Discussion

Eukaryotic gene regulation lies at the nexus of many of the central issues in modern biology, including multi-cellular development (Davidson, 2006), the evolution of complexity (Carroll, 2008), cellular reprogramming (Takahashi and Yamanaka, 2016) and synthetic biology (Keung et al., 2015). The extraordinary molecular complexity implicated in such regulation continues to present a formidable challenge. It has made it difficult to see the wood for the trees, to discern general principles and to unravel how different molecular mechanisms contribute to specific forms of information processing.

In this paper, we have presented compelling evidence that the bacterial paradigm, upon which it has been so convenient to default, is not sufficient for reasoning about eukaryotic genes and we have introduced appropriate quantitative concepts for doing so. We have done this by taking seriously the lessons of the bacterial paradigm itself. The paradigm relied on analysing the physics of interaction between TFs and DNA. What we have done here is to update this foundation for two of the key processes that influence gene-expression sharpness in eukaryotes: information integration and energy expenditure.

While information integration is often acknowledged, it has not been defined sufficiently clearly to know how to find it, making experimental analysis problematic. We have introduced the concept of “higher-order cooperativity”, ω_i,S (Figure 2B), for a system at thermodynamic equilibrium, as a measure of how the affinity of TF binding at site i is influenced by the presence of TF bound at the sites in S. This is a precisely defined quantity that allows experimentally-testable hypotheses to be framed.

Different TFs often work together and the definition of higher-order cooperativities that we have given here for homotypic interactions of a single TF can be readily extended to heterotypic interactions between different TFs. Such cooperativities could arise from nucleosomes (Mirny, 2010; Voss et al., 2011) or from co-regulators like Mediator and CBP/p300 (Borggrefe and Yue, 2011; Wang et al., 2013). Mediator is especially provocative as a potential mechanism of higher-order cooperativity. Mediator has around 30 subunits and the Med1 subunit alone interacts with up to 20 different TFs (Borggrefe and Yue, 2011). Some TFs interact with multiple subunits and the overall Mediator complex exhibits different conformations, which suggests how local information may be globally integrated (Nussinov et al., 2013). Our analysis shows that if gene expression follows an all-or-nothing strategy, high levels of positive and negative higher-order cooperativity of all orders are needed to yield high levels of sharpness (Figure 4B, C). Experimental measurements will show whether co-regulators like Mediator or CBP/p300 can meet these requirements.

When energy is expended during gene regulation, much higher sharpness can be achieved for the same number of TF binding sites (Figure 6). The concept of a “Hopfield barrier” offers a way to articulate this rigorously. With n binding sites, the Hopfield barrier to sharpness is set by the Hill function ℋ_n. whose steepness cannot be exceeded by any equilibrium GRF (Figure 4D). For GRFs in the all-or-nothing strategy, the Hill line itself forms a Hopfield barrier (Figure 4C). However, both these barriers are readily breached away from equilibrium (Figure 6 and Figure S2).

There are many routes through which energy can be expended, including chromatin reorganisation, nucleosome displacement, protein post-translational modification and DNA methylation. Experiments which perturb these routes and assay the impact on sharpness can bring to light which energy expending mechanisms are particularly relevant. For the specific example of hb regulation by Bcd, we note that Bcd binds to Drosophila Mediator in a way that affects early embryonic patterning (Park et al., 2001; Bosveld et al., 2008) and that Bcd also binds to the Sin3/Rp3 histone-deacetylase complex (Singh et al., 2005). The impact of these interactions on sharpness appears not to have been previously studied. The early Drosophila embryo provides an unrivalled experimental context for testing the hypotheses made here and this is now work in progress.

Real-time studies have already confirmed the importance of non-equilibrium kinetics in gene regulation (Voss et al., 2011; Hammar et al., 2014). Larson and colleagues have also argued for the importance of a non-equilibrium perspective (Coulon et al., 2013). Our results strongly endorse this but an important challenge lies ahead. When a system is at thermodynamic equilibrium, detailed balance implies that any path to a microstate can be used to calculate the steady-state probability of the microstate. The history of the system is irrelevant. Away from equilibrium, detailed balance no longer holds and all possible paths to a microstate must be examined to calculate its steady-state probability. Non-equilibrium systems are history dependent. The resulting combinatorial explosion results in a profound increase in algebraic complexity, which manifests itself in the striking difference between the equilibrium GRF in Eq. 4 and the non-equilibrium GRF in Eq. 7. It is this which leads to the breaching of the Hopfield barrier. We suspect that further insights into non-equilibrium gene regulation are concealed within this algebraic complexity. We are only just learning how to uncover them (Ahsendorf et al., 2014).

The concepts introduced here encourage us to examine other forms of genetic information processing. If energy expenditure is important, what is it buying? What could not be achieved if regulatory DNA is at thermodynamic equilibrium? Such questions can be answered by quantifying each information processing task, as we have done here for sharpness (Eq. 6), and developing experimental systems in which it can be measured. We may look forward in this way to a quantitative classification of the kinds of information processing that genes undertake and an understanding in molecular terms of how energy expenditure breaks the corresponding Hopfield barriers.

We note two further implications of the present paper. First, Hill functions emerge in an unexpected light. When Archibald Vivian (A.V.) Hill first introduced them in 1913, he recognised that they had no biochemical justification and were only a convenient fit to the data on oxygen binding to haemoglobin (Hill, 1913). The empirical nature of Hill functions has been repeatedly pointed out (Engel, 2013; Weiss, 1997). We were all the more surprised, therefore, to find that, when the Hill coefficient is an integer, there are bona fide GRFs which are statistically indistinguishable from Hill functions (Figure 4B). In this sense, the Hill functions appear to be closer to biochemistry than Hill, or anyone else, could have imagined. We hope to clarify the mathematical reasons for this in subsequent work.

Second, we have exploited mathematics differently here to what is sometimes expected of it. We have relied on data to frame the question but we have not fitted any mathematical models to data. In recent years, experimental biologists have become more comfortable with the idea that theory can follow experiment, as a way to analyse and understand data. Our colleague Rob Phillips calls this “Figure 7 theory” (Phillips, 2015). Here, we have used mathematics to introduce concepts and to determine the limits of what can be expected, thereby providing a foundation for designing new kinds of experiments. Experiment will follow theory. This is “Figure 1 theory” (Phillips, 2015). We believe it has much to recommend it as we face the daunting molecular complexity of eukaryotic gene regulation. We need to think about how such regulation works using concepts which are not just based on intuition and induction but are also grounded in the underlying physics, which is the bedrock on which all biology rests. This is an old lesson (Gunawardena, 2013; Bialek, 2015). If we have lost sight of it in the press of mastering the molecular details, now is the time to revisit it and construct a new paradigm for eukaryotic gene regulation.

Experimental Procedures

The mathematical model and the results presented here are based on the “linear framework” for gene regulation (Ahsendorf et al., 2014); see the SI for full details. The framework starts from a labelled, directed graph, 𝒢, which gives rise to a stochastic master equation

$$\frac{\mathit{\text{du}}}{\mathit{\text{dy}}}=L(\mathcal{G}).u$$

for the vector of microstate probabilities, u = (u₁, _⋯ u_n )^t. Here, L(𝒢) is the Laplacian matrix of 𝒢 and N is the number of microstates in the graph. For the graph G_n used here, N = 2ⁿ, where n is the number of TF binding sites. Provided 𝒢 is strongly connected, which is the case for G_n, the steady state, u*, at which $\left(\frac{\mathit{\text{du}}}{\mathit{\text{dt}}}\right){|}_{u=u^{\ast }}=0$, is one dimensional. A basis vector may be calculated in terms of the edge labels in one of two ways, depending on whether the system is at equilibrium or not. If the system reaches thermodynamic equilibrium, $u_i^{\ast }$ may be calculated by choosing any path of reversible edges from the reference vertex 1 to i and taking the product, over all reversible edges in the path, of the ratio of the label on the forward edge, in the direction from 1 to i, to the label on the reverse edge. The Principle of Detailed Balance ensures that this result is independent of the chosen path because of the cycle condition: on any cycle of reversible edges, the product of the labels going clockwise around the cycle equals the product of the labels going counter-clockwise. The cycle condition leads to the exchange formula in Eq. 2, which allows the algebraically independent set of parameters in Eq. 3 to be chosen. For G_n, a path of reversible edges can be chosen from 1, the vertex with no sites bound, to i such that $u_i^{\ast }$ is expressed in terms of the independent parameters. Away from equilibrium, $u_i^{\ast }$ has to be calculated using the Matrix-Tree Theorem as a sum, over all directed spanning trees rooted at i, of the product of the labels on the edges of each spanning tree. Once u* is known, the state-state probability of microstate i is given by

$$Pr(i)=\frac{u_i^{\ast }}{u_1^{\ast }+\cdots +u_N^{\ast }}.$$

For a system that reaches thermodynamic equilibrium, the denominator in this formula is the partition function of equilibrium statistical mechanics but the formula holds equally for a system away from thermodynamic equilibrium with u* calculated as above. The gene regulation function for mRNA production rate as output is defined as an average over the steady-state probabilities,

$$\frac{d}{\mathit{\text{dt}}}[\text{mRNA}]=\sum _{1\le i\le N}r(i)Pr(i).$$

The expression rate in microstate i, given by r(i), depends on the gene expression strategy being followed, as specified in Figure 3. To obtain protein level as output, it is assumed that mRNA is linearly degraded and that steady-state protein level is proportional to steady-state mRNA level. The proportionality constants are absorbed in the normalisation that underlies the definitions of position and steepness in Eq. 6.

In the equilibrium GRFs, the quantities $u_i^{\ast }$ depend on paths of reversible edges from 1 to i which can incur up to n factors of x = [T], so that the degree of the denominator polynomial in Pr(i) is n (Eq. 4). In contrast, in the non-equilibrium GRFs, the quantities $u_i^{\ast }$ depend on directed spanning trees rooted at i, which each have N − 1 edges and can incur up to N − 1 factors of x, so that the degree of the denominator polynomial becomes 2ⁿ − 1(Eq. 7).

The numerical results presented in Figures 4, 5 and 6 are obtained by a biased sampling algorithm in which the boundary of the position-steepness region is found by successive approximation. An initial region is found by independently selecting parameter values for GRFs by logarithmic random sampling within the specified range, calculating the (γ, ρ) coordinates of these GRFs and determining the enclosing boundary. This initial boundary is then successively improved by randomly altering GRFs on the current boundary until the area of the region ceases to increase. The details are given in the SI, along with the tests that were used to confirm convergence and to check the numerical accuracy of the results.

Highlights.

• Gene regulation is understood quantitatively in terms of a bacterial paradigm.

• This paradigm cannot account for sharpness of gene expression in development.

• Information integration or energy expenditure can explain the sharpness.

• Hill functions form a “Hopfield barrier” for sharpness at thermodynamic equilibrium.