Analytical Expressions and Physics for Single-Cell mRNA Distributions of the lac Operon of E. coli

Abstract

Mechanistic models of stochastic gene expression are of considerable interest, but their complexity often precludes tractable analytical expressions for messenger RNA (mRNA) and protein distributions. The lac operon of Escherichia coli is a model system with regulatory elements such as multiple operators and DNA looping that are shared by many operons. Although this system is complex, intuition suggests that fast DNA looping may simplify it by causing the repressor-bound states of the operon to equilibrate rapidly, thus ensuring that the subsequent dynamics are governed by slow transitions between the repressor-free and the equilibrated repressor-bound states. Here, we show that this intuition is correct by applying singular perturbation theory to a mechanistic model of lac transcription with the scaled time constant of DNA looping as the perturbation parameter. We find that at steady state, the repressor-bound states satisfy detailed balance and are dominated by the looped states; moreover, the interaction between the repressor-free and the equilibrated repressor-bound states is described by an extension of the Peccoud-Ycart two-state model in which both (repressor-free and repressor-bound) states support transcription. The solution of this extended two-state model reveals that the steady-state mRNA distribution is a mixture of the Poisson and negative hypergeometric distributions, which reflects mRNAs obtained by transcription from the repressor-bound and repressor-free states. Finally, we show that the physics revealed by perturbation theory makes it easy to derive the extended two-state model equations for complex regulatory architectures.

Significance

Stochastic models are of interest to understand the sources and/or magnitude of gene expression noise. To identify the appropriate mechanistic model for a system, one must compare the messenger RNA and protein distributions predicted by candidate models with experimental data. However, mechanistic models are quite complex, and most studies have relied on simulations or expressions for mean and variance (rather than the distribution). We show that complex mechanistic models are readily solved by exploiting the disparate timescales in the system. To this end, we solve a detailed model of lac regulation using singular perturbation theory and find readily interpretable expressions with important implications for experiments. Our results reveal the underlying physics, which is applicable to a broad class of gene regulatory architectures.

Introduction

Stochastic models of single-cell gene expression have become progressively more complex. The earliest models had no regulation because they assumed that the promoter and ribosome binding site were always available for initiation, and stochasticity arose from the random initiation of transcription and translation (1, 2, 3). These one-state models were followed by two-state models in which the promoter switched randomly between two states, of which only one was available for initiation of transcription (4, 5, 6). Rigorous comparison of stochastic models with experimental data requires analytical expressions for the distributions of messenger RNA (mRNA) and protein rather than their mean and variance (6). To facilitate such comparison, analytical expressions for the steady-state mRNA and protein distributions have been derived for the one-state and two-state models (1, 4, 5, 6, 7, 8, 9, 10). More general versions of these models have also been solved exactly, e.g., a model including a regulatory feedback loop by Grima et al. and another one including multiple promoter states by Innocentini et al. (11, 12).

Early experiments with unregulated promoters, which used green fluorescent protein intensity as a reporter for protein number, were consistent with the one-state model (13, 14). However, subsequent experiments using single-molecule techniques have led to considerable debate regarding the origin of the stochasticity (15, 16). Specifically, Choi et al. measured the steady-state distributions of LacY obtained when the lac operon of Escherichia coli is uninduced or partially induced. They observed that the size, but not the frequency, of large protein bursts increased dramatically in partially induced cells and explained this result in terms of specific molecular mechanisms known to operate in the lac operon of E. coli. In contrast, So et al. found that essentially the same noise was observed in several operons of E. coli regulated by very different molecular mechanisms. Thus, the central question of this debate is whether the noise arises from gene-specific or genome-wide mechanisms (17, 18, 19, 20).

Attempts to resolve the foregoing question have led to several mechanistic models of stochastic gene expression, taking due account of the molecular mechanisms that regulate gene expression. Some have focused specifically on models of lac expression in E. coli (21, 22, 23, 24), whereas others have studied generic models spanning a wide range of regulatory mechanisms (25). The steady-state mRNA distributions of such models can be determined computationally (21, 22, 23). However, to derive analytical expressions of these distributions, one often appeals to the method of generating functions, which yields a system of nonautonomous first-order ordinary differential equations of dimension equal to the number of states of the operon. It is challenging to solve these equations analytically because mechanistic models invariably entertain more than two states of the operon. Although analytical expressions have been derived for the steady-state mRNA distribution or its first two moments (24, 25, 26, 27, 28), they are quite intricate.

Tractable analytical expressions were obtained for the protein distributions of one- and two-state models by exploiting the vast difference between mRNA and protein lifetimes (1, 6, 7, 29). However, even in mechanistic models of gene expression, there are disparate timescales that can be exploited. In the particular case of the lac operon, DNA looping is much faster than repressor-DNA binding or dissociation (23). In earlier work (30), which was aimed at gaining deeper insights into the data of Choi et al., we exploited the existence of fast DNA looping to derive a simple analytical expression for the steady-state protein distribution in the uninduced and partially induced lac operon of E. coli. This expression was obtained by arguing heuristically that at steady state, the looped states are dominant, the repressor-bound species satisfy detailed balance, and the model reduces to the leaky two-state model, an extension of the two-state model in which the repressor-bound states also permit transcription at a low rate. We confirmed the validity of our heuristic approach by showing that our analytical expressions agreed well with stochastic simulations of the model, but we did not provide rigorous justification for our results.

It turns out that there is a well-developed singular perturbation theory for systematically simplifying stochastic models with disparate timescales (31, 32). Recently, this theory has been used to study microRNA-based feed-forward regulation (33). Here, we use singular perturbation theory to analyze a stochastic model for transcription of the uninduced or partially induced lac operon of E. coli. We show that the results obtained, with the scaled time constant of DNA looping as the perturbation parameter, are identical to those obtained from our heuristic approach (30). The zeroth-order solution shows that the repressor-bound states are predominantly in the looped state. The solution to first order shows that at steady state, the repressor-bound species satisfy detailed balance, and the interactions of the repressor-bound and repressor-free species are described by the leaky two-state model, which extends the Peccoud-Ycart two-state model for mRNAs by allowing transcription from both promoter states. The theory yields simple and physically meaningful expressions for the mean, variance, and generating function of the steady-state mRNA distribution, which is a mixture of the Poisson and negative hypergeometric distributions. In addition, we show that alternative parameter regimes, e.g., operator-repressor dissociation rate being comparable to DNA looping rate, may also result in the leaky two-state model.

Materials and Methods

Model description

The regulation of the lac operon involves a single transcription factor and multiple operators (34, 35). The transcription factor is a repressor protein denoted R, which can bind specifically to three sites on the chromosome, namely the main operator O₁ and two auxiliary operators, O₂ and O₃. Although we shall consider this problem later on, it is convenient to begin by considering the special case of only one auxiliary operator O₂, in which case the lac operon can exist in only four states (Fig. 1). If both operators are free of repressor, the operon is in the repressor-free state, denoted O. If a repressor binds either O₁ or O₂, the operon contains a single repressor; we denote these states by $O_1\cdot R$ and $O_2\cdot R$. Finally, because the repressor is a tetramer, it can bind both operators simultaneously, leading to the formation of a looped state denoted $O_1\cdot R\cdot O_2$. In principle, two repressors can also bind to each of the two operator sites, yielding an operon with two repressors, but experiments show that in wild-type cells, the likelihood of such multirepressor operons is negligibly small (34). Thus, according to our model, the lac operon can only be in one of the four states, namely the repressor-free state O and the three repressor-bound states $O_1\cdot R$, $O_2\cdot R$, and $O_1\cdot R\cdot O_2$.

Figure 1.

The model scheme. The lac operon is assumed to have four states: a repressor-free state, denoted O, in which both the main operator O₁ and the auxiliary operator O₂ are free of repressor; two nonlooped repressor-bound states, $O_1\cdot R$ and $O_2\cdot R$, in which the repressor is bound to O₁ and O₂, respectively; and a looped repressor-bound state $O_1\cdot R\cdot O_2$ in which the repressor is simultaneously bound to O₁ and O₂. To see this figure in color, go online.

The foregoing four states interconvert with the propensities shown in Fig. 1. Let there be N repressors per cell. In subsequent sections, we develop our model and solve it considering a uniform value of N for all cells. In Perturbative Solution Is Close to Solutions of Full and Leaky Two-State Models, we compare our results with simulations in which we allow N to vary stochastically and show that these are are in good agreement for the uninduced as well as the partially induced operon. In Supporting Materials and Methods, Section S1, we justify this observation by appealing to the fact that our model is invalid primarily for the cells that have no repressors, but the proportion of such cells is insignificant because of protection of the repressor upon binding to the operator, which ensures at least one repressor for the majority of cells (also see Fig. S1). The propensities of repressor-operator binding and dissociation depend on the identity of the operator (36, 37). We denote the propensities for binding to O₁ and O₂ by $k_{a_1}N$ and $k_{a_2}N$, respectively, and those for dissociations from O₁ and O₂ by $k_{O_1}$ and $k_{O_2}$, respectively. If the repressor is bound to O₁ or O₂, it can form the looped state $O_1\cdot R\cdot O_2$ by binding to the other operator with propensity $k_{O_1{O}_2}$. Of the four states, only O and $O_2\cdot R$ allow transcription, and transcription from both states occurs with the same propensity v₀. Finally, the mRNA degrades with propensity d₀.

Master equations

We model the kinetic scheme shown in Fig. 1 with chemical master equations (38). The instantaneous state of the system is described by two variables, namely the mRNA copy number m and the state of the operon s, where s is chosen as f, 1, 2, and 12 when the operon is in the states O, $O_1\cdot R$, $O_2\cdot R$, and $O_1\cdot R\cdot O_2$, respectively. We denote the instantaneous and steady-state probabilities of m mRNAs when the operon is in the state s by p_m,s and ${\tilde{p}}_{m,s}$, respectively. The master equations for the model are

$$\frac{d{p}_{m,1}}{dt}=[k_{a_1}N{p}_{m,f}+k_{O_2}{p}_{m,12}-\left(k_{O_1}+k_{O_1{O}_2}\right)p_{m,1}]+d_0[\left(m+1\right)p_{m+\mathrm{1,1}}-m{p}_{m,1}],$$ (1)

$$\frac{d{p}_{m,2}}{dt}=[k_{a_2}N{p}_{m,f}+k_{O_1}{p}_{m,12}-\left(k_{O_2}+k_{O_1{O}_2}\right)p_{m,2}]+d_0[\left(m+1\right)p_{m+\mathrm{1,2}}-m{p}_{m,2}]+v_0\left(p_{m-\mathrm{1,2}}-p_{m,2}\right),$$ (2)

$$\frac{d{p}_{m,12}}{dt}=\left[k_{O_1{O}_2}\left(p_{m,1}+p_{m,2}\right)-\left(k_{O_1}+k_{O_2}\right)p_{m,12}\right]+d_0[\left(m+1\right)p_{m+\mathrm{1,12}}-m{p}_{m,12}]$$ (3)

and

$$\frac{d{p}_{m,f}}{dt}=\left[k_{O_1}{p}_{m,1}+k_{O_2}{p}_{m,2}-\left(k_{a_1}+k_{a_2}\right)N{p}_{m,f}\right]+d_0\left[\left(m+1\right)p_{m+1,f}-m{p}_{m,f}\right]+v_0\left(p_{m-1,f}-p_{m,f}\right)$$ (4)

and the normalization condition is

$$\sum _{m=0}^{\infty }{p}_m=1$$ (5)

where

$$p_m\equiv p_{m,1}+p_{m,2}+p_{m,12}+p_{m,f}$$ (6)

is the marginal mRNA distribution. Our goal is to determine the steady-state marginal mRNA distribution ${\tilde{p}}_m$.

Results and Discussion

Setting up equations for application of singular perturbation theory

Singular perturbation theory exploits the existence of disparate timescales. In the lac operon, the looping propensity $k_{O_1{O}_2}$ is much greater than the propensity for any other reaction (Table 1). It is therefore conceivable that the repressor-bound states undergo looping so rapidly that they equilibrate on the fast timescale $k_{O_1{O}_2}^{-1}$, after which there are relatively slow transitions between the repressor-free and repressor-bound states. To capture this physical argument, it is convenient to replace one of the original fast variables p_m,1, p_m,2, p_m,12 with the slow variable

$$p_{m,b}\equiv p_{m,1}+p_{m,2}+p_{m,12}$$ (7)

which represents the probability of m mRNAs when the operon is repressor-bound. If we replace p_m,12 with p_m,b, Eq. 4 remains unchanged, but the evolution of the repressor-bound states is given by the new equations

$$\frac{d{p}_{m,1}}{dt}=\left[k_{a_1}N{p}_{m,f}+k_{O_2}\left(p_{m,b}-p_{m,1}-p_{m,2}\right)-\left(k_{O_1}+k_{O_1{O}_2}\right)p_{m,1}\right]+d_0\left[\left(m+1\right)p_{m+\mathrm{1,1}}-m{p}_{m,1}\right],$$ (8)

$$\frac{d{p}_{m,2}}{dt}=\left[k_{a_2}N{p}_{m,f}+k_{O_1}\left(p_{m,b}-p_{m,1}-p_{m,2}\right)-\left(k_{O_2}+k_{O_1{O}_2}\right)p_{m,2}\right]+d_0\left[\left(m+1\right)p_{m+\mathrm{1,2}}-m{p}_{m,2}\right]+v_0\left(p_{m-\mathrm{1,2}}-p_{m,2}\right),$$ (9)

and

$$\frac{d{p}_{m,b}}{dt}=\left[\left(k_{a_1}+k_{a_2}\right)N{p}_{m,f}-\left(k_{O_1}{p}_{m,1}+k_{O_2}{p}_{m,2}\right)\right]+d_0[\left(m+1\right)p_{m+1,b}-m{p}_{m,b}]+v_0\left(p_{m-\mathrm{1,2}}-p_{m,2}\right),$$ (10)

and the normalization condition (Eq. 5) becomes

$$\sum _{m=0}^{\infty }\left(p_{m,b}+p_{m,f}\right)=1.$$ (11)

Henceforth, we shall work with Eqs. 4, 8, 9, 10, and 11.

Table 1.

Parameter Values in the Absence of Inducer

Parameter	Value	Percent Error	Reference	Parameter	Value	Percent Error	Reference
$k_{O_1}$	0.0016 s−1	30%	(30)	$k_{O_2}$	0.019 s−1	30%	(30)
$k_{a_1}$	0.0095 s−1	10%	(36)	$k_{a_2}$	0.0041 s−1	10%	(36)
$k_{O_1{O}_2}$	4 s−1	30%	(30)	N	10	N/A	(42)
v0	0.12 s−1	30%	(30)	d0	0.011 s−1	15%	(43)

Percent errors in $k_{a_1}$, $k_{a_2}$, and d₀ are rounded estimates from the corresponding references. The other rate constants were obtained assuming that the repression is 1200. In the experience of our lab, the repression has up to 30% biological variation, which is reflected in the other rate constants. For more details, refer to (30). N/A, not applicable.

We apply singular perturbation theory to these equations by following the method developed by Rawlings and co-workers (31, 32). Multiplying Eq. 4 and Eqs. 8, 9, and 10 with the perturbation parameter $\mathit{ϵ}=\left(d_0/k_{O_1{O}_2}\right)$ and defining τ = d₀t, ${\kappa }_{a_1}=\left(k_{a_1}/d_0\right)$, ${\kappa }_{a_2}=\left(k_{a_2}/d_0\right)$, $k_a=\left(\left(k_{a_1}+k_{a_2}\right)/2\right)$, ${\kappa }_a=\left(k_a/d_0\right)$, ${\kappa }_{O_1}=\left(k_{O_1}/d_0\right)$, ${\kappa }_{O_2}=\left(k_{O_2}/d_0\right)$, ${\kappa }_{O_1{O}_2}=\left(k_{O_1{O}_2}/d_0\right)$, and $m_s=\left(v_0/d_0\right)$ gives the slow equations

$$\mathit{ϵ}\frac{d{p}_{m,1}}{d\tau }=-p_{m,1}+\mathit{ϵ}\left[{\kappa }_{a_1}N{p}_{m,f}+{\kappa }_{O_2}{p}_{m,b}-\left({\kappa }_{O_1}+{\kappa }_{O_2}\right)p_{m,1}-{\kappa }_{O_2}{p}_{m,2}\right]+\mathit{ϵ}\left[\left(m+1\right)p_{m+\mathrm{1,1}}-m{p}_{m,1}\right],$$ (12)

$$\mathit{ϵ}\frac{d{p}_{m,2}}{d\tau }=-p_{m,2}+\mathit{ϵ}\left[{\kappa }_{a_2}N{p}_{m,f}+{\kappa }_{O_1}{p}_{m,b}-{\kappa }_{O_1}{p}_{m,1}-\left({\kappa }_{O_1}+{\kappa }_{O_2}\right)p_{m,2}\right]+\mathit{ϵ}\left[\left(m+1\right)p_{m+\mathrm{1,2}}-m{p}_{m,2}\right]+\mathit{ϵ}{m}_s\left(p_{m-\mathrm{1,2}}-p_{m,2}\right),$$ (13)

$$\mathit{ϵ}\frac{d{p}_{m,b}}{d\tau }=\mathit{ϵ}\left[2{\kappa }_{a}N{p}_{m,f}-\left({\kappa }_{O_1}{p}_{m,1}+{\kappa }_{O_2}{p}_{m,2}\right)\right]+\mathit{ϵ}\left[\left(m+1\right)p_{m+1,b}-m{p}_{m,b}\right]+\mathit{ϵ}{m}_s\left(p_{m-\mathrm{1,2}}-p_{m,2}\right),$$ (14)

and

$$\mathit{ϵ}\frac{d{p}_{m,f}}{d\tau }=\mathit{ϵ}\left({\kappa }_{O_1}{p}_{m,1}+{\kappa }_{O_2}{p}_{m,2}-2{\kappa }_{a}N{p}_{m,f}\right)+\mathit{ϵ}[\left(m+1\right)p_{m+1,f}-m{p}_{m,f}]+\mathit{ϵ}{m}_s\left(p_{m-1,f}-p_{m,f}\right).$$ (15)

Next, we substitute the power-series expansions

$$p_{m,s}\left(\tau \right)=p_{m,s}^{\left(0\right)}\left(\tau \right)+\mathit{ϵ}{p}_{m,s}^{\left(1\right)}\left(\tau \right)+\mathcal{O}\left({\mathit{ϵ}}^2\right),s=\mathrm{1,2},b,f$$ (16)

in Eqs. 11, 12, 13, 14, and 15 and collect terms with coefficients ϵ⁰ and ϵ¹ to determine the zeroth- and first-order relations.

Zeroth-order equations imply that the operon is completely repressed at steady state

Collecting terms with coefficient ϵ⁰ yields the zeroth-order equations

$$p_{m,1}^{\left(0\right)}=0,$$ (17)

$$p_{m,2}^{\left(0\right)}=0,$$ (18)

$$\frac{d{p}_{m,b}^{\left(0\right)}}{d\tau }=2{\kappa }_{a}N{p}_{m,f}^{\left(0\right)}+\left[\left(m+1\right)p_{m+1,b}^{\left(0\right)}-m{p}_{m,b}^{\left(0\right)}\right],$$ (19)

and

$$\frac{d{p}_{m,f}^{\left(0\right)}}{d\tau }=-2{\kappa }_{a}N{p}_{m,f}^{\left(0\right)}+\left[\left(m+1\right)p_{m+1,f}^{\left(0\right)}-m{p}_{m,f}^{\left(0\right)}\right]+m_s\left(p_{m-1,f}^{\left(0\right)}-p_{m,f}^{\left(0\right)}\right),$$ (20)

and the zeroth-order normalization condition

$$\sum _{m=0}^{\infty }\left(p_{m,b}^{\left(0\right)}+p_{m,f}^{\left(0\right)}\right)=1.$$ (21)

It follows from Eqs. 17 to 18 that to zeroth order, the probabilities of the $O_1\cdot R$ and $O_2\cdot R$ states are always zero, and the repressor-bound species are always exclusively in the looped state $O_1\cdot R\cdot O_2$, i.e., $p_{m,b}^{\left(0\right)}=p_{m,12}^{\left(0\right)}$.

Eqs. 19, 20, and 21 immediately yield the steady-state probabilities ${\tilde{p}}_{m,b}^{\left(0\right)}$, ${\tilde{p}}_{m,f}^{\left(0\right)}$. To see this, observe that Eq. 21 implies that $p_{m,b}^{\left(0\right)}$ and $p_{m,f}^{\left(0\right)}$ behave like probabilities. The evolution of these probabilities is governed by Eqs. 19 and 20, which are formally similar to the equations of the two-state model, the only difference being that there is a unidirectional probability flux $2{\kappa }_{a}N{p}_{m,f}^{\left(0\right)}$ from the repressor-free to the repressor-bound state. It follows that ultimately, $p_{m,f}^{\left(0\right)}$ approaches zero, i.e., ${\tilde{p}}_{m,f}^{\left(0\right)}=0$ for all m ≥ 0, and the operon is in the repressor-bound state, i.e, ${\sum }_m{\tilde{p}}_{m,b}^{\left(0\right)}=1$. Now, because ${\tilde{p}}_{m,f}^{\left(0\right)}=0$, there is no transcription from the repressor-free state O at steady state, and because $p_{m,2}^{\left(0\right)}$ is always zero, there is no transcription from the state $O_2\cdot R$ either. It follows that there is no mRNA at steady state, i.e, ${\tilde{p}}_{m,b}^{\left(0\right)}=0$ for all m ≥ 1, and the zeroth-order steady-state probabilities are

$${\tilde{p}}_{m,f}^{\left(0\right)}=0\text{for all }m\ge 0,{\tilde{p}}_{m,b}^{\left(0\right)}=\{\begin{array}{cc}1& \text{if }m=0\\ 0& \text{if }m\ge 1\end{array}.$$ (22)

This is shown rigorously in Appendix A by solving Eqs. 19, 20, and 21 at steady state by the method of generating functions.

The zeroth-order solution (Eq. 22) is physically plausible, for it follows from Eq. 16 that the zeroth-order solution is obtained when $\mathit{ϵ}\equiv d_0/k_{O_1{O}_2}=0$. This relation is satisfied when $k_{O_1{O}_2}$, the propensity for looping, is infinitely large, i.e., the looped state is absorbing. Under this condition, the system must ultimately converge to the looped state, which does not entertain any transcription. The zeroth-order solution is therefore consistent with the physics expected when $k_{O_1{O}_2}\to \infty$. Having said that, one must acknowledge that it is physically impossible for $k_{O_1{O}_2}$ to be infinitely large—it is large but necessarily finite. The first-order terms derived below are corrections to the zeroth-order solution that account for the finiteness of $k_{O_1{O}_2}$. Specifically, we shall find that p_m,1, p_m,2, and ${\tilde{p}}_{m,f}$ are not zero, but small quantities.

Detailed balance of bound species and reduction to leaky two-state model at steady state

Collecting first-order terms yields the first-order equations

$$p_{m,1}^{\left(1\right)}={\kappa }_{a_1}N{p}_{m,f}^{\left(0\right)}+{\kappa }_{O_2}{p}_{m,b}^{\left(0\right)},$$ (23)

$$p_{m,2}^{\left(1\right)}={\kappa }_{a_2}N{p}_{m,f}^{\left(0\right)}+{\kappa }_{O_1}{p}_{m,b}^{\left(0\right)},$$ (24)

$$\frac{d{p}_{m,b}^{\left(1\right)}}{d\tau }=2{\kappa }_{a}N{p}_{m,f}^{\left(1\right)}-\left[\left({\kappa }_{a_1}{\kappa }_{O_1}+{\kappa }_{a_2}{\kappa }_{O_2}\right)N{p}_{m,f}^{\left(0\right)}+2{\kappa }_{O_1}{\kappa }_{O_2}{p}_{m,b}^{\left(0\right)}\right]+\left[\left(m+1\right)p_{m+1,b}^{\left(1\right)}-m{p}_{m,b}^{\left(1\right)}\right]+m_s\left[{\kappa }_{a}N\left(p_{m-1,f}^{\left(0\right)}-p_{m,f}^{\left(0\right)}\right)+{\kappa }_{O_1}\left(p_{m-1,b}^{\left(0\right)}-p_{m,b}^{\left(0\right)}\right)\right],$$ (25)

and

$$\frac{d{p}_{m,f}^{\left(1\right)}}{d\tau }=\left[\left({\kappa }_{a_1}{\kappa }_{O_1}+{\kappa }_{a_2}{\kappa }_{O_2}\right)N{p}_{m,f}^{\left(0\right)}+2{\kappa }_{O_1}{\kappa }_{O_2}{p}_{m,b}^{\left(0\right)}\right]-2{\kappa }_{a}N{p}_{m,f}^{\left(1\right)}+\left[\left(m+1\right)p_{m+1,f}^{\left(1\right)}-m{p}_{m,f}^{\left(1\right)}\right]+m_s\left(p_{m-1,f}^{\left(1\right)}-p_{m,f}^{\left(1\right)}\right)$$ (26)

and the first-order normalization condition

$$\sum _{m=0}^{\infty }\left(p_{m,b}^{\left(1\right)}+p_{m,f}^{\left(1\right)}\right)=0.$$ (27)

Now, the first-order terms (e.g., $p_{m,f}^{\left(1\right)}$, $p_{m,b}^{\left(1\right)}$) are corrections to the zeroth-order solution that do not satisfy the master equations and are not even probabilities, as is evident from Eq. 27. In contrast, the zeroth-order solution (e.g., $p_{m,f}^{\left(0\right)}$, $p_{m,b}^{\left(0\right)}$) and the solution to first order (e.g., $p_{m,f}^{\left(0\right)}+\mathit{ϵ}{p}_{m,f}^{\left(1\right)}$, $p_{m,b}^{\left(0\right)}+\mathit{ϵ}{p}_{m,b}^{\left(1\right)}$) satisfy the master equations with error $\mathcal{O}\left(\mathit{ϵ}\right)$ and $\mathcal{O}\left({\mathit{ϵ}}^2\right)$, respectively. Thus, they represent probabilities, albeit approximate, that may lend themselves to physical interpretation. We show below that at steady state, the solutions to first order, namely ${\tilde{p}}_{m,1}^{\left(0\right)}+\mathit{ϵ}{\tilde{p}}_{m,1}^{\left(1\right)}$, ${\tilde{p}}_{m,2}^{\left(0\right)}+\mathit{ϵ}{\tilde{p}}_{m,2}^{\left(1\right)}$, ${\tilde{p}}_{m,f}^{\left(0\right)}+\mathit{ϵ}{\tilde{p}}_{m,f}^{\left(1\right)}$, and ${\tilde{p}}_{m,b}^{\left(0\right)}+\mathit{ϵ}{\tilde{p}}_{m,b}^{\left(1\right)}$, do provide simple physical interpretations. The first two solutions show that the repressor-bound states satisfy detailed balance, and the last two solutions show that the interactions between the repressor-free and repressor-bound states can be described by the leaky two-state model.

At steady state, Eq. 22 holds, and Eqs. 23, 24, 25, and 26 reduce to the simpler form

$${\tilde{p}}_{m,1}^{\left(1\right)}={\kappa }_{O_2}{\tilde{p}}_{m,b}^{\left(0\right)},$$ (28)

$${\tilde{p}}_{m,2}^{\left(1\right)}={\kappa }_{O_1}{\tilde{p}}_{m,b}^{\left(0\right)},$$ (29)

$$0=\frac{d{\tilde{p}}_{m,b}^{\left(1\right)}}{d\tau }=\left(2{\kappa }_{a}N{\tilde{p}}_{m,f}^{\left(1\right)}-2{\kappa }_{O_1}{\kappa }_{O_2}{\tilde{p}}_{m,b}^{\left(0\right)}\right)+\left[\left(m+1\right){\tilde{p}}_{m+1,b}^{\left(1\right)}-m{\tilde{p}}_{m,b}^{\left(1\right)}\right]+m_s{\kappa }_{O_1}\left({\tilde{p}}_{m-1,b}^{\left(0\right)}-{\tilde{p}}_{m,b}^{\left(0\right)}\right),$$ (30)

and

$$0=\frac{d{\tilde{p}}_{m,f}^{\left(1\right)}}{d\tau }=-\left(2{\kappa }_{a}N{\tilde{p}}_{m,f}^{\left(1\right)}-2{\kappa }_{O_1}{\kappa }_{O_2}{\tilde{p}}_{m,b}^{\left(0\right)}\right)+\left[\left(m+1\right){\tilde{p}}_{m+1,f}^{\left(1\right)}-m{\tilde{p}}_{m,f}^{\left(1\right)}\right]+m_s\left({\tilde{p}}_{m-1,f}^{\left(1\right)}-{\tilde{p}}_{m,f}^{\left(1\right)}\right).$$ (31)

These equations can be solved for ${\tilde{p}}_{m,b}^{\left(1\right)}$ and ${\tilde{p}}_{m,f}^{\left(1\right)}$ by the method of generating functions (Appendix A), but we shall focus hereafter on the physics revealed by these equations.

Eqs. 28 and 29 imply that the repressor-bound states satisfy the principle of detailed balance to order ϵ. To see this, observe that multiplying Eqs. 28 and 29 by ϵ and appealing to the zeroth-order relations $p_{m,1}^{\left(0\right)}=p_{m,2}^{\left(0\right)}=0$, $p_{m,b}^{\left(0\right)}=p_{m,12}^{\left(0\right)}$ yields the equations

$${\tilde{p}}_{m,1}^{\left(0\right)}+\mathit{ϵ}{\tilde{p}}_{m,1}^{\left(1\right)}=\mathit{ϵ}{\kappa }_{O_2}\left({\tilde{p}}_{m,12}^{\left(0\right)}+\mathit{ϵ}{\tilde{p}}_{m,12}^{\left(1\right)}\right)-{\mathit{ϵ}}^2{\kappa }_{O_2}{\tilde{p}}_{m,12}^{\left(1\right)},$$ (32)

$${\tilde{p}}_{m,2}^{\left(0\right)}+\mathit{ϵ}{\tilde{p}}_{m,2}^{\left(1\right)}=\mathit{ϵ}{\kappa }_{O_1}\left({\tilde{p}}_{m,12}^{\left(0\right)}+\mathit{ϵ}{\tilde{p}}_{m,12}^{\left(1\right)}\right)-{\mathit{ϵ}}^2{\kappa }_{O_1}{\tilde{p}}_{m,12}^{\left(1\right)},$$ (33)

which imply that ${\tilde{p}}_{m,1}$, ${\tilde{p}}_{m,2}$, and ${\tilde{p}}_{m,12}$ satisfy the detailed balanced equations ${\kappa }_{O_1{O}_2}{\tilde{p}}_{m,1}={\kappa }_{O_2}{\tilde{p}}_{m,12}$, ${\kappa }_{O_1{O}_2}{\tilde{p}}_{m,2}={\kappa }_{O_1}{\tilde{p}}_{m,12}$ to order ϵ. Note that application of the same argument to Eqs. 28 and 29, but without replacing ${\tilde{p}}_{m,b}^{\left(0\right)}$ with ${\tilde{p}}_{m,12}^{\left(0\right)}$, shows that to order ϵ, we also have the relations ${\kappa }_{O_1{O}_2}{\tilde{p}}_{m,1}={\kappa }_{O_2}{\tilde{p}}_{m,b}$ and ${\kappa }_{O_1{O}_2}{\tilde{p}}_{m,2}={\kappa }_{O_1}{\tilde{p}}_{m,b}$, and the marginal probabilities of the $O_1\cdot R$ and $O_2\cdot R$ states are ${\sum }_m{\tilde{p}}_{m,1}={\kappa }_{O_2}/{\kappa }_{O_1{O}_2}$ and ${\sum }_m{\tilde{p}}_{m,2}={\kappa }_{O_1}/{\kappa }_{O_1{O}_2}$, respectively. We shall use these relations later.

At steady state, the occurrence of detailed balance among the repressor-bound states is physically plausible because under this condition, the repressor-bound states form a quasi-closed subsystem in the sense that they have almost no interaction with the rest of the system (consisting of the repressor-free state). To see this, observe that at steady state, p_m,1, p_m,2, and p_m,f are small, which implies that the probability fluxes from the repressor-bound states to the repressor-free state (due to repressor dissociation) and the probability fluxes from the repressor-free state to the repressor-bound state (due to repressor binding) are negligibly small compared to the probability fluxes between the repressor-bound states (due to looping). The restriction to steady state is necessary for the validity of this conclusion, for it is only under this condition that Eqs. 23 and 24 reduce to Eqs. 28 and 29. Far from the steady state, under conditions in which the probability of the repressor-free state is significant, the repressor-bound states do not form a quasi-closed system because p_m,1 and p_m,2 depend on probability fluxes from both $p_{m,b}^{\left(0\right)}$ and $p_{m,f}^{\left(0\right)}$. At steady state, however, they form a quasi-closed subsystem, and such subsystems attain quasi-equilibrium, characterized by detailed balance among the states of the subsystem (39).

Eqs. 30 and 31 imply that to order ϵ, the steady-state probabilities of the repressor-bound and repressor-free states ${\tilde{p}}_{m,b}$, ${\tilde{p}}_{m,f}$ are described by the leaky two-state model. To see this, observe that multiplying Eqs. 30 and 31 by ϵ and appealing to the zeroth-order solutions (Eq. 22) yields the relations

$$0=\frac{d}{d\tau }\left({\tilde{p}}_{m,b}^{\left(0\right)}+\mathit{ϵ}{\tilde{p}}_{m,b}^{\left(1\right)}\right)=\left[2{\kappa }_{a}N\left({\tilde{p}}_{m,f}^{\left(0\right)}+\mathit{ϵ}{p}_{m,f}^{\left(1\right)}\right)-2{\kappa }_{O_1}{\kappa }_{O_2}\mathit{ϵ}\left({\tilde{p}}_{m,b}^{\left(0\right)}+\mathit{ϵ}{\tilde{p}}_{m,b}^{\left(1\right)}\right)\right]+\left[\left(m+1\right)\left({\tilde{p}}_{m+1,b}^{\left(0\right)}+\mathit{ϵ}{\tilde{p}}_{m+1,b}^{\left(1\right)}\right)-m\left({\tilde{p}}_{m,b}^{\left(0\right)}+\mathit{ϵ}{\tilde{p}}_{m,b}^{\left(1\right)}\right)\right]+m_s{\kappa }_{O_1}\mathit{ϵ}\left[\left({\tilde{p}}_{m-1,b}^{\left(0\right)}+\mathit{ϵ}{\tilde{p}}_{m-1,b}^{\left(1\right)}\right)-\left({\tilde{p}}_{m,b}^{\left(0\right)}+\mathit{ϵ}{\tilde{p}}_{m,b}^{\left(1\right)}\right)\right]+{\mathit{ϵ}}^2\left[2{\kappa }_{O_1}{\kappa }_{O_2}{\tilde{p}}_{m,b}^{\left(1\right)}-m_s{\kappa }_{O_1}\left({\tilde{p}}_{m-1,b}^{\left(1\right)}-{\tilde{p}}_{m,b}^{\left(1\right)}\right)\right],$$ (34)

$$0=\frac{d}{d\tau }\left({\tilde{p}}_{m,f}^{\left(0\right)}+\mathit{ϵ}{\tilde{p}}_{m,f}^{\left(1\right)}\right)=\left[2{\kappa }_{O_1}{\kappa }_{O_2}\mathit{ϵ}\left({\tilde{p}}_{m,b}^{\left(0\right)}+\mathit{ϵ}{\tilde{p}}_{m,b}^{\left(1\right)}\right)-2{\kappa }_{a}N\left({\tilde{p}}_{m,f}^{\left(0\right)}+\mathit{ϵ}{p}_{m,f}^{\left(1\right)}\right)\right]+\left[\left(m+1\right)\left({\tilde{p}}_{m+1,f}^{\left(0\right)}+\mathit{ϵ}{\tilde{p}}_{m+1,f}^{\left(1\right)}\right)-m\left({\tilde{p}}_{m,f}^{\left(0\right)}+\mathit{ϵ}{\tilde{p}}_{m,f}^{\left(1\right)}\right)\right]+m_s\left[\left({\tilde{p}}_{m-1,f}^{\left(0\right)}+\mathit{ϵ}{\tilde{p}}_{m-1,f}^{\left(1\right)}\right)-\left({\tilde{p}}_{m,f}^{\left(0\right)}+\mathit{ϵ}{\tilde{p}}_{m,f}^{\left(1\right)}\right)\right]-{\mathit{ϵ}}^2\left(2{\kappa }_{O_1}{\kappa }_{O_2}{\tilde{p}}_{m,b}^{\left(1\right)}\right),$$ (35)

which imply that to order ϵ, the steady-state probabilities ${\tilde{p}}_{m,b}$ and ${\tilde{p}}_{m,f}$ satisfy the equations

$$0=\frac{d{\tilde{p}}_{m,b}}{dt}=\left(k_1{\tilde{p}}_{m,f}-k_0{\tilde{p}}_{m,b}\right)+d_0\left[\left(m+1\right){\tilde{p}}_{m+1,b}-m{\tilde{p}}_{m,b}\right]+\lambda v_0\left({\tilde{p}}_{m-1,b}-{\tilde{p}}_{m,b}\right),$$ (36)

$$0=\frac{d{\tilde{p}}_{m,f}}{dt}=-\left(k_1{\tilde{p}}_{m,f}-k_0{\tilde{p}}_{m,b}\right)+d_0\left[\left(m+1\right){\tilde{p}}_{m+1,f}-m{\tilde{p}}_{m,f}\right]+v_0\left({\tilde{p}}_{m-1,f}-{\tilde{p}}_{m,f}\right),$$ (37)

where

$$\begin{array}{l}\lambda \equiv {\kappa }_{O_1}\mathit{ϵ}=\frac{k_{O_1}}{k_{O_1{O}_2}}\ll 1,\\ k_0\equiv 2{\kappa }_{O_1}{\kappa }_{O_2}\mathit{ϵ}{d}_0=\frac{2k_{O_1}{k}_{O_2}}{k_{O_1{O}_2}}=2\lambda k_{O_2}\ll 1,\\ k_1\equiv 2{\kappa }_{a}N{d}_0=2k_{a}N.\end{array}$$ (38)

Eqs. 36, 37, and 38 describe the steady states of the leaky two-state model (Fig. 2), in which transitions occur between repressor-bound and repressor-free states with propensities k₀ and k₁, but the repressor-bound state also allows leaky transcription with a low propensity λv₀.

Figure 2.

The leaky two-state model. Like the two-state model, the operon switches randomly between the repressor-free and the repressor-bound states with propensities k₀ and k₁, and the repressor-free state permits transcription with propensity v₀. However, unlike the two-state model, the leaky two-state model also permits transcription from the repressor-bound state with a small propensity λv₀. To see this figure in color, go online.

Eqs. 36, 37, and 38 can also be derived from physical arguments that serve to reveal the origin of the expressions in Eq. 38. To this end, consider the rates of the various processes occurring in the repressor-free state and the pool of repressor-bound states. At steady state, transcription from the repressor-free state occurs at the rate

$${\tilde{p}}_{m,f}{v}_0,$$ (39)

and transitions from the repressor-free to the repressor-bound state, which can occur by two mutually exclusive pathways, namely association of cytosolic repressor with O₁ or O₂, occur at the rate

$${\tilde{p}}_{m,f}\times k_{a_1}N+{\tilde{p}}_{m,f}\times k_{a_2}N=2k_{a}N{\tilde{p}}_{m,f}=k_1{\tilde{p}}_{m,f}.$$ (40)

The corresponding expressions for the repressor-bound state follow from two properties of repressor-bound species that were obtained from perturbation theory, namely, they are dominated by the looped species, i.e., p_m,b ≈ p_m,12, and satisfy detailed balance at steady state, i.e., $k_{O_1{O}_2}{\tilde{p}}_{m,1}\approx k_{O_2}{\tilde{p}}_{m,12}$, $k_{O_1{O}_2}{\tilde{p}}_{m,2}\approx k_{O_1}{\tilde{p}}_{m,12}$. Hence, at steady state, transcription from the repressor-bound state occurs at the rate

$${\tilde{p}}_{m,2}{v}_0\approx \left(\frac{k_{O_1}}{k_{O_1{O}_2}}{\tilde{p}}_{m,b}\right)v_0=\lambda v_0{\tilde{p}}_{m,b}$$ (41)

and transitions from the repressor-bound to the repressor-free state, which can occur by two mutually exclusive pathways—namely dissociation of the repressor from $O_1\cdot R$ or $O_2\cdot R$—occur at the rate

$${\tilde{p}}_{m,1}{k}_{O_1}+{\tilde{p}}_{m,2}{k}_{O_2}=\left(\frac{k_{O_2}}{k_{O_1{O}_2}}{\tilde{p}}_{m,b}\right)k_{O_1}+\left(\frac{k_{O_1}}{k_{O_1{O}_2}}{\tilde{p}}_{m,b}\right)k_{O_2}=\frac{2k_{O_1}{k}_{O_2}}{k_{O_1{O}_2}}{\tilde{p}}_{m,b}=k_0{\tilde{p}}_{m,b}.$$ (42)

It follows from Eqs. 39, 40, 41, to 42 that the master equations describing the steady state of the repressor-bound and repressor-free states are given by Eqs. 36, 37, and 38. More importantly, this approach yields physical insights into the origin of the expressions in Eq. 38. In Steady States of the Wild-Type lac Operon Also Follow the Leaky Two-State Model, we shall exploit this physical approach to derive the leaky two-state model for the wild-type lac operon, which has two auxiliary operators. Furthermore, in Appendix C, we show that more general sets of parameter values (e.g., where $k_{O_1}$ is of the same order as $k_{O_1{O}_2}$) may also result in the leaky two-state model.

Perturbative solution is close to solutions of full and leaky two-state models

In the previous section, we showed that the perturbative solution to first order satisfies to order ϵ Eqs. 4 and 8, 9, 10, and 11 of the full model, as well as Eqs. 36 and 37 and Eq. 11 of the leaky two-state model. It follows that if ϵ is sufficiently small, the perturbative solution to first order is a good approximation to the solutions of the full model as well as the leaky two-state model, and hence, the leaky two-state model is a good approximation to the full model. However, because we do not know exactly how small ϵ should be for the validity of this conclusion, it is necessary to verify that the particular value of ϵ = 0.00275 derived from Table 1 is indeed sufficiently small.

We confirmed that the steady-state marginal mRNA distribution obtained from the perturbative solution to first order is essentially identical to that obtained from the full model. It is shown in Appendix A that the generating function for the steady-state marginal mRNA distribution obtained from the perturbative solution to first order is

$$\tilde{g}\left(z\right)=1+\lambda m_s\left(z-1\right)+\frac{{\zeta }_0}{{\zeta }_1}\sum _{m=0}^{\infty }\frac{{\left\{m_s\left(z-1\right)\right\}}^{m+1}}{{\left(1+{\zeta }_1\right)}_m\left(m+1\right)},$$ (43)

where (⋅)_m is the rising factorial, $m_s\equiv v_0/d_0$ is the mean number of mRNAs synthesized per mRNA lifetime in the unregulated operon (1, 2), and ${\zeta }_0\equiv k_0/d_0$, ${\zeta }_1\equiv k_1/d_0$ denote the mean number of transitions per unit mRNA lifetime between the repressor-bound and repressor-free states. We compared the steady-state marginal mRNA distributions obtained from Eq. 43 with those obtained from simulations of the full model using the optimized direct method implementation of Gillespie’s stochastic simulation algorithm (10⁶ realizations run for 36,000 s with constant N and initial state defined to be the looped operator state and no mRNAs present in the system) (40). Fig. 3, a shows the steady-state distributions calculated with the parameter values in Table 1 for the uninduced lac operon (obtained in the absence of the inducer). Under this condition, the approximate and simulated results agree well, but it remains to check whether they also agree in the presence of the inducer. In principle, the inducer can bind to cytosolic repressor, which decreases the association propensities $k_{a_1}N$ and $k_{a_2}N$, as well as operator-bound repressor, which increases the dissociation propensities $k_{O_1}$, $k_{O_2}$. However, because the inducer has a much higher affinity for the cytosolic repressor, only $k_{a_1}N$ and $k_{a_2}N$ decrease in the presence of small inducer concentrations (15). Fig. 3, b and c show that the steady-state distributions obtained from simulations and perturbation theory agree well even when $k_{a_1}N$ and $k_{a_2}N$ are decreased 10- and 20-fold. The perturbative solution to first order is therefore a good approximation to the solution of the full model when the lac operon is uninduced or partially induced.

Figure 3.

Perturbative solution is in agreement with the solutions of the full and the leaky two-state models. The steady-state mRNA distributions obtained from the perturbative solution (squares) agree with the solutions of the full model with constant N (gray lines), full model with stochastically varying N (triangles), and the leaky two-state model (circles). Distributions in the three panels correspond to (a) the uninduced operon (parameter values in Table 1), (b) partially induced operon with mean of the counts of cytosolic repressors in cells reduced 10-fold, and (c) partially induced operon with the mean of the counts of cytosolic repressors in cells reduced 20-fold. 95% binomial confidence intervals on the triangles (obtained using the Wilson method) show that the differences between the solutions of the full model with stochastically varying N and the perturbative solutions are not statistically significant for the majority of data points (error bars overlap squares). To see this figure in color, go online.

Although good agreement is obtained with both fully repressed and partially induced cells, there are grounds for questioning the results in the latter case. Indeed, because the repressor number is a discrete parameter, reducing it 20-fold implies that a significant fraction of the cells have no repressor, and in such cells, our approximate solutions are not valid because the looping rate is not faster than all other rates—in fact, looping does not occur at all. It is therefore necessary to check the validity of our approximation in a model taking due account of the discrete nature of repressors. If we assume random formation of cytosolic repressor by de novo synthesis and dissociation of bound repressor and random loss of cytosolic repressor by cell division and operator binding, we obtain a stochastic model of repressor dynamics that can be used to explore the effect of not only the discrete nature of the repressor but also that of cell-to-cell variation of the repressor number. A detailed description of this stochastic model is given in the Supporting Materials and Methods. Here, it suffices to observe that the mRNA distributions obtained by simulating the full model with stochastic repressor dynamics are no different from those obtained by simulating the full model with constant repressor number N, provided the parameter values of the former are chosen such that the mean repressor number equals N (Fig. 3). Thus, stochastic repressor dynamics have no effect on the steady-state mRNA distribution in both fully repressed and partially induced cells. To understand this, observe that the assumption that all cells have the same repressor number N is tantamount to assuming that the distribution of repressors in cells is a Dirac delta function centered at N. We find that the steady-state repressor distribution obtained in the presence of stochastic repressor dynamics is similar to a Dirac delta function—the repressor distribution decays rapidly when the repressor number deviates from the mean of the distribution (Fig. S1). Because the repressor distributions are similar in the absence and presence of stochastic repressor dynamics, so are the corresponding mRNA distributions. The stochastic model of the repressor dynamics also shows that both fully repressed and partially induced cells are unlikely to have no repressors because in both cases, a repressor is almost always bound to the operator, which ensures that each daughter cell inherits at least one repressor upon cell division (Fig. S1). This ensures that even in partially induced cells with a significantly reduced repressor number, there is good agreement between the full model with and without stochastic repressor dynamics.

The steady-state marginal mRNA distributions obtained from the perturbative solution to first order also agree with the corresponding distributions obtained from the solution of the leaky two-state model. It is shown in Appendix B that the generating function of the steady-state marginal mRNA distribution for the leaky two-state model is

$$\tilde{h}\left(z\right)=e^{\lambda m_s\left(z-1\right)}\times {}_{1}F_1\left({\zeta }_0;{\zeta }_0+{\zeta }_1;(1-\lambda )m_s(z-1)\right),$$ (44)

where ₁F₁(; ;) denotes Kummer’s hypergeometric function of the first kind. Fig. 3 shows that the steady-state marginal mRNA distributions derived from Eq. 44 are esentially identical to those derived from the generating function in Eq. 43 for the perturbative solution to first order. It turns out that in this case, we can provide additional analytical justification for the closeness of the two distributions. To see this, observe that Eq. 38 implies that if ϵ is small, so are λ and ζ₀. Because $\lambda \ll 1$, Eq. 44 can be rewritten as

$$\tilde{h}\left(z\right)\approx \left[1+\lambda m_s\left(z-1\right)\right]\times \left[1+\stackrel{\infty }{\sum _{m=0}}\frac{{\left({\zeta }_0\right)}_{m+1}}{{\left({\zeta }_0+{\zeta }_1\right)}_{m+1}}\times \frac{{\left\{m_s\left(z-1\right)\right\}}^{m+1}}{\left(m+1\right)!}\right],$$

and because ${\zeta }_0\ll 1,{\zeta }_1$, we have (ζ₀)_{m + 1} ≈ ζ₀m! and (ζ₀ + ζ₁)_{m + 1} ≈ ζ₁(ζ₁ + 1)_m, which imply that $\tilde{h}\left(z\right)\approx \tilde{g}\left(z\right)$. As the generating functions of the steady-state mRNA distributions obtained from the leaky two-state model and the perturbative solution to first order are essentially identical, so are the corresponding distributions.

It is noteworthy that even though leaky two-state models can yield bimodal distributions (41), we did not observe them in our simulations. We shall explain this result in the next section.

The leaky two-state model yields simple expressions and physical insights

It is shown in Appendix B that the steady-state marginal probability of the leaky two-state model ${\tilde{p}}_m$ can be derived from the generating function (Eq. 44). However, we gain more physical insight by confining our attention to the latter. Indeed, the generating function (Eq. 44) is a mixture of Poisson and negative hypergeometric distributions, which shows that the steady-state mRNA distribution is derived from two different subpopulations of mRNA. Because λ is the marginal probability of the state O₂ · R, the Poissonian function represents the mRNA produced by transcription from the operon state O₂ · R. On the other hand, because $\lambda \ll 1$, the hypergeometric function is essentially identical to the generating function for the two-state model (5) and represents the mRNA produced by transcription from the free state O. We show below that although λ is small, transcription from $O_2\cdot R$ cannot be ignored because it provides most of the mRNA in the uninduced lac operon.

It follows from Eq. 44 that the mean μ(m) and variance σ²(m) of the marginal mRNA distribution are

$$\mu \left(m\right)=m_s\left[\lambda +\left(1-\lambda \right)\frac{{\zeta }_0}{{\zeta }_0+{\zeta }_1}\right]\approx m_s\left(\lambda +\frac{{\zeta }_0}{{\zeta }_1}\right),$$ (45)

$${\sigma }^2\left(m\right)=\mu \left(m\right)+{\left\{m_s\left(1-\lambda \right)\right\}}^2\frac{{\zeta }_0}{{\zeta }_0+{\zeta }_1}\frac{{\zeta }_1}{{\zeta }_0+{\zeta }_1}\frac{1}{1+{\zeta }_0+{\zeta }_1}\approx \mu \left(m\right)+{\left(\frac{m_s}{{\zeta }_1}\right)}^2{\zeta }_0,$$ (46)

where the approximate results are obtained by appealing to the relations $\lambda \ll 1$ and ${\zeta }_0\ll 1\ll {\zeta }_1$. Because λ and ζ₀/(ζ₀ + ζ₁) ≈ ζ₀/ζ₁ are the marginal probabilities of the states $O_2\cdot R$ and O, Eq. 45 shows that the mean consists of mRNA obtained by transcription from both $O_2\cdot R$ and O. However, in the uninduced lac operon, most of the mRNA is obtained by transcription from $O_2\cdot R$ because the parameter values in Table 1 imply that in the absence of inducer, λ = 4 × 10⁻⁴ is very small, but ζ₀/ζ₁ = 1 × 10⁻⁴ is even smaller. This result is consistent with the data of Choi et al., who observed that in uninduced cells, 80% of the LacY molecules were derived from transcription of $O_2\cdot R$ (15). It also explains the absence of bimodal distributions in our simulations. Indeed, the leaky two-state model yields modes at m_sλ and m_s if the sojourn time of the system in both the bound and free states (i.e., 1/k₀ and 1/k₁, respectively) is larger than or on the same order of magnitude as 1/d₀, which is the timescale for the mRNA numbers to reach steady state (12). In terms of normalized rates, this means that the criteria for bimodality are ${\zeta }_0\ll 1$ and ${\zeta }_1\ll 1$. However, we have modeled the lac operon close to the fully repressed condition, in which case ${\zeta }_0\ll 1$ but ${\zeta }_1\gg 1$. Hence, we obtain unimodal distributions, with one mode corresponding to the bound states. Eq. 46 shows that the variance also contains two terms, but only the second term reflects the variance due to regulation because the first term appears even in one-state models, which have no regulation (1). Moreover, the variance due to regulation is entirely due to transcription from the repressor-free state O. We show below that transcriptions from $O_2\cdot R$ are so short-lived that they are averaged out on the slow timescale.

Although Eqs. 45 and 46 provide simple expressions for the mean and variance, noisy gene expression is often characterized in terms of the size and frequency of bursts of mRNAs or proteins. In particular, Choi et al. suggested that in the lac operon, transcriptions are allowed by two types of repressor dissociation (15):

• 1)A partial dissociation occurs when a repressor in the looped state dissociates from O₁ but not O₂, thus leading to the state $O_2\cdot R$ which allows transcription. However, because the repressor is still in the neighborhood of O₁, it usually rebinds to O₁ rapidly. The short duration of a partial dissociation is unlikely to yield multiple mRNAs. We call this a transcriptional leakage.
• 2)A complete dissociation occurs when the repressor dissociates from both operators and enters the cytosol. In this case, the repressor may take a relatively long time to rebind to O₁, thus allowing a transcriptional burst, i.e., production of multiple mRNAs. Thus, complete dissociations lead to transcription from the operon state O and yield transcriptional bursts.

It is convenient that Eqs. 45 and 46 can be rewritten in terms of the size and frequency of the transcriptional bursts and leakages. To see this, observe that during partial dissociations, O₂ is repressor-free for the time period $1/k_{O_1{O}_2}$, so that the mean size of the leakages is

$$b_p=\frac{v_0}{k_{O_1{O}_2}}.$$ (47)

On the other hand, detailed balance at steady state implies that the mean frequency of partial dissociations per cell cycle is

$$a_p=\frac{{\sum }_{m=0}^{\infty }{\tilde{p}}_{m,12}{k}_{O_1}}{d_0}=\frac{{\sum }_{m=0}^{\infty }{\tilde{p}}_{m,2}{k}_{O_1{O}_2}}{d_0}\approx \frac{{\sum }_{m=0}^{\infty }\left(\lambda {\tilde{p}}_{m,b}\right)k_{O_1{O}_2}}{d_0}\approx \frac{\lambda k_{O_1{O}_2}}{d_0}.$$ (48)

It follows that a_pb_p equals m_sλ, the first term of Eq. 45. Similarly, during complete dissociations, the operon is repressor-free for the time period 1/k₁, so that the mean size of bursts is

$$b_c=\frac{v_0}{k_1}=\frac{m_s}{{\zeta }_1}$$ (49)

and the mean frequency of the complete dissociations per cell cycle is

$$a_c=\frac{\sum _{m=0}^{\infty }{\tilde{p}}_{m,b}{k}_0}{d_0}\approx \frac{k_0}{d_0}={\zeta }_0,$$ (50)

which implies that a_cb_c equals m_sζ₀/ζ₁, the second term of Eq. 45, and $a_c{b}_c^2$ equals ζ₀(m_s/ζ₁)², the second term of Eq. 46. It follows that Eqs. 45 and 46 can be rewritten as

$$\mu \left(m\right)=a_p{b}_p+a_c{b}_c,$$ (51)

$${\sigma }^2\left(m\right)=\mu \left(m\right)+a_c{b}_c^2.$$ (52)

These equations show that both leakages and bursts contribute to the mean, but leakages contribute to the variance only to the extent that they contribute to the mean, whereas bursts contribute an additional term $a_c{b}_c^2$. This is because the leakages occur on the fast timescale $1/k_{O_1{O}_2}$ and are averaged out on the slow timescale.

Steady states of the wild-type lac operon also follow the leaky two-state model

It was shown in Detailed Balance of Bound Species and Reduction to Leaky Two-State Model at Steady State that the leaky two-state model corresponding to the simplified model with only one auxiliary operator (Fig. 1) could also be derived by appealing to two physical properties of the repressor-bound states at steady state, namely, they satisfy detailed balance and are dominated by the looped states. Now, the wild-type lac operon contains two auxiliary operators O₂ and O₂, which results in two additional significant states of the operon, namely $O_3\cdot R$ and the looped state $O_3\cdot R\cdot O_1$ (34, 35). We shall now appeal to the foregoing physical properties of repressor-bound states to show that at steady state, the wild-type lac operon also follows Eqs. 36 and 37, but λ, k₀, and k₁ are now given by the expressions

$$\lambda =\frac{k_{O_1}/k_{O_2}}{k_{O_1{O}_2}/k_{O_2}+k_{O_1{O}_3}/k_{O_3}},k_0=3\lambda k_{O_2},k_1=3k_{a}N,$$ (53)

where $k_{O_3}$ denotes the propensity of repressor dissociation from O₃; $k_a=\left(k_{a_1}+k_{a_2}+k_{a_3}\right)/3$ is the mean of propensities $k_{a_1}$, $k_{a_2}$, $k_{a_3}$ for binding to O₁, O₂, O₃, respectively; and $k_{O_1{O}_3}$ denotes the propensity of $O_3\cdot R\cdot O_1$ loop formation. To see this, observe that at steady state, transcription from the repressor-free state occurs at the rate ${\tilde{p}}_{m,f}{v}_0$, and transitions from the repressor-free to the repressor-bound state occur at the rate

$${\tilde{p}}_{m,f}{k}_{a_1}N+{\tilde{p}}_{m,f}{k}_{a_2}N+{\tilde{p}}_{m,f}{k}_{a_3}N=3k_{a}N{\tilde{p}}_{m,f}=k_1{\tilde{p}}_{m,f}.$$ (54)

To determine the corresponding rates for the repressor-bound state, note that the repressor-bound states are predominantly in the looped state, i.e.,

$${\tilde{p}}_{m,b}\equiv {\tilde{p}}_{m,1}+{\tilde{p}}_{m,2}+{\tilde{p}}_{m,3}+{\tilde{p}}_{m,12}+{\tilde{p}}_{m,13}\approx {\tilde{p}}_{m,12}+{\tilde{p}}_{m,13},$$ (55)

and satisfy the principle of detailed balance, i.e.,

$$\begin{array}{l}{k}_{O_1{O}_2}{\tilde{p}}_{m,1}=k_{O_2}{\tilde{p}}_{m,12},\\ k_{O_1{O}_2}{\tilde{p}}_{m,2}=k_{O_1}{\tilde{p}}_{m,12},\\ k_{O_1{O}_3}{\tilde{p}}_{m,1}=k_{O_3}{\tilde{p}}_{m,13},\\ k_{O_1{O}_3}{\tilde{p}}_{m,3}=k_{O_1}{\tilde{p}}_{m,13}.\end{array}$$ (56)

Eqs. 55 and 56 can be solved to obtain

$${\tilde{p}}_{m,12}=\left(\frac{k_{O_1{O}_2}/k_{O_2}}{k_{O_1{O}_2}/k_{O_2}+k_{O_1{O}_3}/k_{O_3}}\right){\tilde{p}}_{m,b},$$ (57)

$${\tilde{p}}_{m,13}=\left(\frac{k_{O_1{O}_3}/k_{O_3}}{k_{O_1{O}_2}/k_{O_2}+k_{O_1{O}_3}/k_{O_3}}\right){\tilde{p}}_{m,b},$$ (58)

$${\tilde{p}}_{m,1}=\left(\frac{1}{k_{O_1{O}_2}/k_{O_2}+k_{O_1{O}_3}/k_{O_3}}\right){\tilde{p}}_{m,b},$$ (59)

$${\tilde{p}}_{m,2}=\left(\frac{k_{O_1}/k_{O_2}}{k_{O_1{O}_2}/k_{O_2}+k_{O_1{O}_3}/k_{O_3}}\right){\tilde{p}}_{m,b},$$ (60)

and

$${\tilde{p}}_{m,3}=\left(\frac{k_{O_1}/k_{O_3}}{k_{O_1{O}_2}/k_{O_2}+k_{O_1{O}_3}/k_{O_3}}\right){\tilde{p}}_{m,b}.$$ (61)

It follows from Eqs. 59, 60, to 61 that transcription from the repressor-bound state occurs at the rate

$${\tilde{p}}_{m,2}{v}_0=\left(\frac{k_{O_1}/k_{O_2}}{k_{O_1{O}_2}/k_{O_2}+k_{O_1{O}_3}/k_{O_3}}{\tilde{p}}_{m,b}\right)v_0=\lambda v_0{\tilde{p}}_{m,b},$$ (62)

and transitions from the repressor-bound to the repressor-free state occur at the rate

$${\tilde{p}}_{m,1}{k}_{O_1}+{\tilde{p}}_{m,2}{k}_{O_2}+{\tilde{p}}_{m,3}{k}_{O_3}=3\lambda k_{O_2}{\tilde{p}}_{m,b}=k_0{\tilde{p}}_{m,b}.$$ (63)

These rate expressions imply that at steady state, the master equations describing the interactions between the repressor-bound and repressor-free states of the wild-type lac operon are also given by Eqs. 36 and 37, albeit with different parameter values given by Eq. 53. This result provides rigorous justification for our earlier work concerned with the steady state protein distributions in the wild-type lac operon (30). Moreover, it shows that the two physical facts inferred from perturbation theory, namely that repressor-bound states are dominated by the looped state and satisfy detailed balance, are sufficient for extending the model to more complex scenarios.

Conclusions

Intuition suggests that because of fast DNA looping, the repressor-bound states of the lac operon are predominantly in the looped state and satisfy detailed balance. In earlier work, we derived these results by a heuristic approach and showed that the resulting steady-state protein distributions were in good agreement with stochastic simulations of the full model (30). We have shown above that singular perturbation theory provides a rigorous justification for the foregoing approximations. More precisely, application of perturbation theory shows that at steady state,

• 1.To zeroth order, the operon is always in a looped state; in particular, the probabilities of the free and nonlooped repressor-bound states are zero.
• 2.To first order, the probabilities of the repressor-bound states are in detailed balance, and the dynamics reduce to those of the leaky two-state model, a variant of the Peccoud-Ycart two-state model in which the bound state also permits transcription at a small, but significant, rate.

The leaky two-state model equations can be easily solved to obtain simple and physically meaningful expressions for the mean, variance, and generating function of the mRNA distribution. Our analytical result complements the existing studies on lac operon, which have relied on computational approaches for insights (21, 22). Recently, Cao and Grima have shown that the leaky two-state model serves as a good approximation for a wide range of gene regulatory networks incorporating feedback loops (41). We have shown that this approximation is valid for repressed or partially induced lac operon as well. The physics inferred from perturbation theory can also be used to derive the reduced (leaky two-state) model equations for more complex regulatory architectures which obviates the need for going through the intricate procedures of perturbation theory. This is a useful tool that is likely to facilitate the analysis of complex mechanistic models with auxiliary operators and DNA looping.

Author Contributions

A.N. designed the research. K.C. and A.N. performed the research. K.C. analyzed the data and performed simulations. K.C. and A.N. wrote the article.

Editor: Stanislav Shvartsman.

Appendix A: Steady-State Solution of Full Model to First Order

Because we wish to derive the steady-state marginal probability $p_m\equiv p_{m,b}+p_{m,f}$, it is convenient to replace either p_m,b or p_m,f with p_m. We have chosen to replace p_m,b with p_m, i.e., we shall solve for the zeroth- and first-order approximations of p_m and p_m,f (instead of p_m,b and p_m,f) at steady state. If p_m has the perturbation expansion $p_m=p_m^{\left(0\right)}+\mathit{ϵ}{p}_m^{\left(1\right)}+\mathcal{O}\left({\mathit{ϵ}}^2\right),$ the perturbation expansions of p_m,b and p_m,f imply that $p_m^{\left(0\right)}=p_{m,b}^{\left(0\right)}+p_{m,f}^{\left(0\right)}$ and $p_m^{\left(1\right)}=p_{m,b}^{\left(1\right)}+p_{m,f}^{\left(1\right)}$.

Zeroth-order solution

Adding Eqs. 19 and 20 yields

$$\frac{d{p}_m^{\left(0\right)}}{d\tau }=\left[\left(m+1\right)p_{m+1}^{\left(0\right)}-m{p}_m^{\left(0\right)}\right]+{\nu }_0\left(p_{m-1,f}^{\left(0\right)}-p_{m,f}^{\left(0\right)}\right).$$ (64)

We shall obtain steady-state solutions of Eqs. 20, 21, and 64 by the method of generating functions. To this end, let ${\tilde{p}}_{m,f}^{\left(0\right)}$, ${\tilde{p}}_m^{\left(0\right)}$ denote the steady states of these equations, and let ${\tilde{g}}_f^{\left(0\right)}\left(z\right)\equiv {\sum }_m{z}^m{\tilde{p}}_{m,f}^{\left(0\right)}$, ${\tilde{g}}^{\left(0\right)}\left(z\right)\equiv \sum z^m{\tilde{p}}_m^{\left(0\right)}$ denote the corresponding generating functions. Multiplying both sides of Eqs. 20 and 21 and Eq. 64 with z^m and summing over m gives at steady state

$$\frac{d{\tilde{g}}_f^{\left(0\right)}}{dz}=\left(\frac{{\zeta }_1}{1-z}+m_s\right){\tilde{g}}_f^{\left(0\right)},$$ (65)

$$\frac{d{\tilde{g}}^{\left(0\right)}}{dz}=m_s{\tilde{g}}_f^{\left(0\right)},$$ (66)

where ${\zeta }_1\equiv \left(2k_{a}N/d_0\right),m_s\equiv \left(v_0/d_0\right)$, and the boundary condition

$${\tilde{g}}^{\left(0\right)}\left(1\right)=1.$$ (67)

To solve these equations, note that integrating Eq. 65 yields ${\tilde{g}}_f^{\left(0\right)}\left(z\right)=\left(C_1{e}^{m_{s}z}/{\left(1-z\right)}^{{\zeta }_1}\right)$, where C₁ is a constant of integration. Because ${\tilde{g}}_f^{\left(0\right)}\left(1\right)={\sum }_m{\tilde{p}}_{m,f}^{\left(0\right)}\le 1$, C₁ must be zero, which implies that ${\tilde{g}}_f^{\left(0\right)}\left(z\right)=0.$ Then, it follows from Eqs. 66 to 67 that

$${\tilde{g}}^{\left(0\right)}\left(z\right)=1$$ (68)

and ${\tilde{g}}_b^{\left(0\right)}\left(z\right)\equiv {\sum }_m{z}^m{\tilde{p}}_{m,b}^{\left(0\right)}={\tilde{g}}^{\left(0\right)}\left(z\right)-{\tilde{g}}_f^{\left(0\right)}\left(z\right)=1$. But ${\tilde{g}}_f^{\left(0\right)}\left(z\right)=0$ and ${\tilde{g}}_b^{\left(0\right)}\left(z\right)=1$ imply that to zeroth order, the steady-state probabilities are

$${\tilde{p}}_{m,f}^{\left(0\right)}=0\text{for all }m,$$ (69)

$${\tilde{p}}_{m,b}^{\left(0\right)}=\{\begin{array}{cc}1& \text{if }m=0\\ 0& \text{if }m>0\end{array}.$$ (70)

First-order solution

To solve for the generating functions of ${\tilde{p}}_{m,f}^{\left(1\right)}$ and ${\tilde{p}}_m^{\left(1\right)}$, add Eqs. 30 and 31 to get

$$0=\frac{d{\tilde{p}}_m^{\left(1\right)}}{d\tau }=\left[\left(m+1\right){\tilde{p}}_{m+1}^{\left(1\right)}-m{\tilde{p}}_m^{\left(1\right)}\right]+m_s\left({\tilde{p}}_{m-1,f}^{\left(1\right)}-{\tilde{p}}_{m,f}^{\left(1\right)}\right)+m_s{\kappa }_{O_1}\left({\tilde{p}}_{m-1,b}^{\left(0\right)}-{\tilde{p}}_{m,b}^{\left(0\right)}\right).$$ (71)

Now, define ${\tilde{g}}_f^{\left(1\right)}\left(z\right)\equiv {\sum }_m{z}^m{\tilde{p}}_{m,f}^{\left(1\right)}$, ${\tilde{g}}^{\left(1\right)}\left(z\right)\equiv {\sum }_m{z}^m{\tilde{p}}_m^{\left(1\right)}$ and multiply both sides of Eqs. 31 and 71 with z^m, and summing over all values of m gives the equations

$$\left(z-1\right)\frac{d\left(\mathit{ϵ}{\tilde{g}}_f^{\left(1\right)}\right)}{dz}=\left[m_s\left(z-1\right)-{\zeta }_1\right]\left(\mathit{ϵ}{\tilde{g}}_f^{\left(1\right)}\right)+{\zeta }_0,$$ (72)

$$\frac{d\left(\mathit{ϵ}{\tilde{g}}^{\left(1\right)}\right)}{dz}=m_s\left(\lambda +\mathit{ϵ}{\tilde{g}}_f^{\left(1\right)}\right),$$ (73)

where ${\zeta }_0\equiv \frac{2\lambda k_{O_2}}{d_0}$, and the boundary condition

$${\tilde{g}}^{\left(1\right)}\left(1\right)=0.$$ (74)

Solving Eq. 72 under the constraint of bounded ${\tilde{g}}_f^{\left(1\right)}\left(1\right)=\sum _m{\tilde{p}}_{m,f}^{\left(1\right)}$ yields

$$\mathit{ϵ}{\tilde{g}}_f^{\left(1\right)}\left(z\right)=\frac{{\zeta }_0}{{\zeta }_1}\times {}_{1}F_1\left(1;{\zeta }_1+1;m_s\left(z-1\right)\right),$$ (75)

which implies that

$$\mathit{ϵ}{\tilde{p}}_{m,f}^{\left(1\right)}=\frac{{\zeta }_0}{{\zeta }_1}\frac{m_s^m}{{\left({\zeta }_1+1\right)}_m}{}_{1}F_1\left(1+m;{\zeta }_1+1+m;-m_s\right)$$

for all m ≥ 0. Now, it follows from Eqs. 73 to 74 that

$$\mathit{ϵ}{\tilde{g}}^{\left(1\right)}\left(z\right)=m_s{\int }_1^z\left(\lambda +\mathit{ϵ}{\tilde{g}}_f^{\left(1\right)}\left(z\right)\right)dz=m_s\left[\lambda \left(z-1\right)+{\int }_1^z\mathit{ϵ}{\tilde{g}}_f^{\left(1\right)}\left(z\right)dz\right].$$ (76)

Substituting the series for the hypergeometric function in Eq. 75 and integrating term-by-term yields

$$\mathit{ϵ}{\tilde{g}}^{\left(1\right)}\left(z\right)=m_s\lambda \left(z-1\right)+\frac{{\zeta }_0}{{\zeta }_1}\sum _{m=0}^{\infty }\frac{{\left[m_s\left(z-1\right)\right]}^{m+1}}{{\left({\zeta }_1+1\right)}_m\left(m+1\right)},$$ (77)

where (⋅)_m denotes the Pochhammer symbol (rising factorial). It follows from Eq. 77 that

$$\mathit{ϵ}{\tilde{p}}_0^{\left(1\right)}=-m_s\lambda +\frac{{\zeta }_0}{{\zeta }_1}\sum _{m=0}^{\infty }\frac{{\left(-m_s\right)}^{m+1}}{{\left(1+{\zeta }_1\right)}_m\left(m+1\right)},$$

and Eq. 76 implies that

$$\mathit{ϵ}{\tilde{p}}_1^{\left(1\right)}=m_s\left(\lambda +\mathit{ϵ}{\tilde{p}}_{0,f}^{\left(1\right)}\right),\mathit{ϵ}{\tilde{p}}_m^{\left(1\right)}=m_s\frac{\mathit{ϵ}{\tilde{p}}_{m-1,f}^{\left(1\right)}}{m}$$

for all m ≥ 2. Given these expressions for ${\tilde{p}}_{m,f}^{\left(1\right)}$ and ${\tilde{p}}_m^{\left(1\right)}$, we can determine ${\tilde{p}}_{m,b}^{\left(1\right)}={\tilde{p}}_m^{\left(1\right)}-{\tilde{p}}_{m,f}^{\left(1\right)}$.

It follows from Eqs. 68 to 77 that to first order, the generating function for the steady-state marginal distribution of mRNA is

$$\tilde{g}\left(z\right)={\tilde{g}}^{\left(0\right)}\left(z\right)+\mathit{ϵ}{\tilde{g}}^{\left(1\right)}\left(z\right)=1+m_s\lambda \left(z-1\right)+\frac{{\zeta }_0}{{\zeta }_1}\sum _{m=0}^{\infty }\frac{{\left\{m_s\left(z-1\right)\right\}}^{m+1}}{{\left({\zeta }_1+1\right)}_m\left(m+1\right)}.$$

Appendix B: Steady-State Solution of Leaky Two-State Model

Once again, it is convenient to replace p_m,b by the marginal probability $p_m\equiv p_{m,b}+p_{m,f}$, which evolves in accordance with the equation

$$\frac{d{p}_m}{dt}=d_0\left[\left(m+1\right)p_{m+1}-m{p}_m\right]+\lambda v_0\left[\left(p_{m-1}-p_{m-1,f}\right)-\left(p_m-p_{m,f}\right)\right]+v_0\left(p_{m-1,f}-p_{m,f}\right),$$ (78)

obtained by adding Eqs. 36 and 37. We shall solve for the steady states ${\tilde{p}}_{m,f}$, ${\tilde{p}}_m$ of Eqs. 37 and 78. To this end, let ${\tilde{h}}_f\left(z\right)\equiv \sum _m{z}^m{\tilde{p}}_{m,f}$, $\tilde{h}\left(z\right)\equiv \sum z^m{\tilde{p}}_m$, multiply both equations by $z^m$, and sum over m to obtain at steady state

$$\left(z-1\right)\frac{d{\tilde{h}}_f}{dz}=[m_s\left(z-1\right)-\left({\zeta }_0+{\zeta }_1\right)]{\tilde{h}}_f+{\zeta }_0\tilde{h},$$

$$\left(z-1\right)\frac{d\tilde{h}}{dz}=m_s\left(z-1\right)\left(1-\lambda \right){\tilde{h}}_f+\lambda m_s\left(z-1\right)\tilde{h}.$$

Moreover, because ${\sum }_m{p}_m=1$, $\tilde{h}\left(z\right)$ satisfies the boundary condition $\tilde{h}\left(1\right)=1$. The above system of two first-order differential equations can be reduced to the single second-order differential equation

$$\left(z-1\right)\frac{d^2\tilde{h}}{d{z}^2}+\left[\left({\zeta }_0+{\zeta }_1\right)-\left(1+\lambda \right)m_s\left(z-1\right)\right]\frac{d\tilde{h}}{dz}-m_s\left[{\zeta }_0+\lambda {\zeta }_1-m_s\lambda \left(z-1\right)\right]\tilde{h}=0.$$

If we define v = m_s(1 − λ)(z − 1) and $\tilde{f}\left(v\right)=\text{exp}\left\{-\lambda m_s\left(z-1\right)\right\}\tilde{h}\left(z\right){|}_{z=1+v/\left(1-\lambda \right)m_s}$, the above equation reduces to Kummer’s standard differential equation

$$v\frac{d^2\tilde{f}}{d{v}^2}+\left({\zeta }_0+{\zeta }_1-v\right)\frac{d\tilde{f}}{dv}-{\zeta }_0\tilde{f}=0.$$

Because $\tilde{f}\left(0\right)=1$, we obtain the solution $\tilde{f}\left(v\right)={}_{1}F_1\left({\zeta }_0;{\zeta }_0+{\zeta }_1;v\right)$, which can be rewritten as

$$\tilde{h}\left(z\right)=e^{\lambda m_s\left(z-1\right)}\times {}_{1}F_1\left({\zeta }_0;{\zeta }_0+{\zeta }_1;\left(1-\lambda \right)m_s\left(z-1\right)\right)$$

by returning to the original variables. Differentiating this equation m times yields

$$\frac{d^m}{d{z}^m}\tilde{h}\left(z\right)=m_s^m{e}^{\lambda m_s\left(z-1\right)}\stackrel{m}{\sum _{k=0}}\left(\begin{array}{c}m\\ k\end{array}\right){\lambda }^{m-k}{\left(1-\lambda \right)}^k\frac{{\left({\zeta }_0\right)}_k}{{\left({\zeta }_0+{\zeta }_1\right)}_k}{}_{1}F_1[{\zeta }_0+k;{\zeta }_0+{\zeta }_1+k;\left(1-\lambda \right)m_s\left(z-1\right)],$$

which implies that the steady-state marginal distribution of mRNA is

$${\tilde{p}}_m=\frac{m_s^m}{m!}{e}^{-\lambda m_s}\stackrel{m}{\sum _{k=0}}\left(\begin{array}{c}m\\ k\end{array}\right){\lambda }^{m-k}{\left(1-\lambda \right)}^k\frac{{\left({\zeta }_0\right)}_k}{{\left({\zeta }_0+{\zeta }_1\right)}_k}{}_{1}F_1\left({\zeta }_0+k;{\zeta }_0+{\zeta }_1+k;-m_s\left(1-\lambda \right)\right).$$

These results are consistent with the full time-dependent solution of the leaky two-state model (41).

Appendix C: Other Parameter Combinations May Also Result in Leaky Two-State Model

Let us consider a hypothetical scenario in which $k_{O_1}$ is comparable to $k_{O_1{O}_2}$. In this case, intuition suggests that the repressor-bound states are predominantly in the looped and $O_2\cdot R$ states. Once again, we follow the perturbation-theoretic approach outlined above to derive reduced equations. Multiplying Eq. 4 and Eqs. 8, 9, and 10 with the redefined perturbation parameter $\mathit{ϵ}=\left(d_0/k_{O_1}+k_{O_1{O}_2}\right)$ and defining ${\lambda }_1=\left(k_{O_1}/k_{O_1}+k_{O_1{O}_2}\right)$ and ${\lambda }_2={\lambda }_1=\left(k_{O_1{O}_2}/k_{O_1}+k_{O_1{O}_2}\right)$ gives the slow equations

$$\mathit{ϵ}\frac{d{p}_{m,1}}{d\tau }=-p_{m,1}+\mathit{ϵ}\left({\kappa }_{a_1}N{p}_{m,f}+{\kappa }_{O_2}{p}_{m,12}\right)+\mathit{ϵ}\left[\left(m+1\right)p_{m+\mathrm{1,1}}-m{p}_{m,1}\right],$$ (79)

$$\mathit{ϵ}\frac{d{p}_{m,2}}{d\tau }=-p_{m,2}-{\lambda }_1{p}_{m,1}+{\lambda }_1{p}_{m,b}+\mathit{ϵ}\left({\kappa }_{a_2}N{p}_{m,f}-{\kappa }_{O_2}{p}_{m,2}\right)+\mathit{ϵ}\left[\left(m+1\right)p_{m+\mathrm{1,2}}-m{p}_{m,2}\right]+\mathit{ϵ}{m}_s\left(p_{m-\mathrm{1,2}}-p_{m,2}\right),$$ (80)

$$\mathit{ϵ}\frac{d{p}_{m,b}}{d\tau }=-{\lambda }_1{p}_{m,1}+\mathit{ϵ}\left(2{\kappa }_{a}N{p}_{m,f}-{\kappa }_{O_2}{p}_{m,2}\right)+\mathit{ϵ}\left[\left(m+1\right)p_{m+1,b}-m{p}_{m,b}\right]+\mathit{ϵ}{m}_s\left(p_{m-\mathrm{1,2}}-p_{m,2}\right),$$ (81)

$$\mathit{ϵ}\frac{d{p}_{m,f}}{d\tau }={\lambda }_1{p}_{m,1}+\mathit{ϵ}\left({\kappa }_{O_2}{p}_{m,2}-2{\kappa }_{a}N{p}_{m,f}\right)+\mathit{ϵ}\left[\left(m+1\right)p_{m+1,f}-m{p}_{m,f}\right]+\mathit{ϵ}{m}_s\left(p_{m-1,f}-p_{m,f}\right).$$ (82)

Next, we substitute the power-series expansions

$$p_{m,s}\left(\tau \right)=p_{m,s}^{\left(0\right)}\left(\tau \right)+\mathit{ϵ}{p}_{m,s}^{\left(1\right)}\left(\tau \right)+\mathcal{O}\left({\mathit{ϵ}}^2\right),s=\mathrm{1,2},b,f,$$ (83)

in Eqs. 79, 80, 81, and 82 and collect terms with coefficients ϵ⁰ and ϵ¹ to determine the zeroth- and first-order relations as follows. The zeroth-order equations are

$$p_{m,1}^{\left(0\right)}=0,$$ (84)

$$p_{m,2}^{\left(0\right)}={\lambda }_1{p}_{m,b}^{\left(0\right)}.$$ (85)

Note that in this case, the zeroth-order equations reveal a partially induced operon as expected. The first-order equations are

$$p_{m,1}^{\left(1\right)}={\kappa }_{a_1}N{p}_{m,f}^{\left(0\right)}+{\kappa }_{O_2}{\lambda }_2{p}_{m,b}^{\left(0\right)},$$ (86)

$$\frac{d{p}_{m,2}^{\left(0\right)}}{d\tau }=-p_{m,2}^{\left(1\right)}-{\lambda }_1{p}_{m,1}^{\left(1\right)}+{\lambda }_1{p}_{m,b}^{\left(1\right)}+{\kappa }_{a_2}N{p}_{m,f}^{\left(0\right)}-{\kappa }_{O_2}{\lambda }_1{p}_{m,b}^{\left(0\right)}+\left[\left(m+1\right)p_{m+\mathrm{1,2}}^{\left(0\right)}-m{p}_{m,2}^{\left(0\right)}\right]+m_s\left(p_{m-\mathrm{1,2}}^{\left(0\right)}-p_{m,2}^{\left(0\right)}\right)$$ (87)

$$\frac{d{p}_{m,b}^{\left(0\right)}}{d\tau }=-{\lambda }_1{p}_{m,1}^{\left(1\right)}+2{\kappa }_{a}N{p}_{m,f}^{\left(0\right)}-{\kappa }_{O_2}{\lambda }_1{p}_{m,b}^{\left(0\right)}+\left[\left(m+1\right)p_{m+1,b}^{\left(0\right)}-m{p}_{m,b}^{\left(0\right)}\right]+m_s{\lambda }_1\left(p_{m-1,b}^{\left(0\right)}-p_{m,b}^{\left(0\right)}\right),$$ (88)

$$\frac{d{p}_{m,f}^{\left(0\right)}}{d\tau }={\lambda }_1{p}_{m,1}^{\left(1\right)}-2{\kappa }_{a}N{p}_{m,f}^{\left(0\right)}+{\kappa }_{O_2}{\lambda }_1{p}_{m,b}^{\left(0\right)}+\left[\left(m+1\right)p_{m+1,f}^{\left(0\right)}-m{p}_{m,f}^{\left(0\right)}\right]+m_s\left(p_{m-1,f}^{\left(0\right)}-p_{m,f}^{\left(0\right)}\right).$$ (89)

Using Eq. 86 in Eqs. 88 and 89, at steady state

$$0=\frac{d{\tilde{p}}_{m,b}^{\left(0\right)}}{d\tau }=\left({\kappa }_{a_1}{\lambda }_2+{\kappa }_{a_2}\right)N{\tilde{p}}_{m,f}^{\left(0\right)}-{\kappa }_{O_2}{\lambda }_1\left(1+{\lambda }_2\right){\tilde{p}}_{m,b}^{\left(0\right)}+\left[\left(m+1\right){\tilde{p}}_{m+1,b}^{\left(0\right)}-m{\tilde{p}}_{m,b}^{\left(0\right)}\right]+m_s{\lambda }_1\left({\tilde{p}}_{m-1,b}^{\left(0\right)}-{\tilde{p}}_{m,b}^{\left(0\right)}\right),$$ (90)

$$0=\frac{d{\tilde{p}}_{m,f}^{\left(0\right)}}{d\tau }=-\left({\kappa }_{a_1}{\lambda }_2+{\kappa }_{a_2}\right)N{\tilde{p}}_{m,f}^{\left(0\right)}+{\kappa }_{O_2}{\lambda }_1\left(1+{\lambda }_2\right){\tilde{p}}_{m,b}^{\left(0\right)}+\left[\left(m+1\right){\tilde{p}}_{m+1,f}^{\left(0\right)}-m{\tilde{p}}_{m,f}^{\left(0\right)}\right]+m_s\left({\tilde{p}}_{m-1,f}^{\left(0\right)}-{\tilde{p}}_{m,f}^{\left(0\right)}\right).$$ (91)

The above equations are formally identical to equations for a leaky two-state model and can be solved using the generating function method. Notably, in this case, the leaky two-state model is reflected in the zeroth-order terms because of partial induction of the operon.