Reduction of dimension

In the full model for gene expression in the presence of decoy binding sites includes, one keeps track of (at least) two separate reaction species: the free protein and the bound protein. Dimensional reduction can be achieved if we assume that the interaction between the two, i.e. the binding and the unbinding, is so fast as to be at any time at a quasi steady state. The concentrations (the unit of concentration is arbitrary and can be set to 1 protein molecule per cell) x_f of free protein, y_f of free decoy binding sites, and x_b of the protein-binding site complex then satisfy (1) where k_b is the dissociation constant for the interaction (the ratio of dissociation and association rate constants).

The total concentration x of protein, both bound and free, and the total decoy binding site concentration y are given by (2) The total binding site concentration does not change over time, i.e. y is a parameter. The other quantities will change in time, due to production and decay mechanisms described later.

Multiplying the second equation of (2) by x_f and then using (1), we obtain for the bound protein level a relationship (3) allowing us to determine the concentration of bound protein from that of free protein. The total protein concentration x comprises both free and bound molecules, (4) The inverse relationship to (4), (5) can be used to determine the concentration of free protein if the total protein concentration is given. Formulae (4) and (5) represent our dimensional reduction: it suffices to know the amount of free protein to determine the total protein level; conversely, the total protein level determines the proportions of bound and free molecules.

The rate of protein degradation c = c(x) is given by the sum of the degradation rate of free protein and that of bound protein, (6) in which γ_f and γ_b are the degradation rate constants for the free and bound protein, and x_f depends on x as determined in (5).

We shall assume that γ_f = 1, which can be achieved by measuring time in the units of the lifetime of free protein. With this specification, the parameter γ_b is interpreted as the ratio between degradation rate constants for bound and free protein; hence, γ_b = 1 as long as the interaction with decoy binding sites does not affect the propensity of protein to be degraded; γ_b = 1 will also hold if proteins are stable and the decline in their concentration is due to cell growth. If γ_b < 1 (γ_b > 1), binding decoy sites diminishes (enhances) the propensity of protein molecules for degradation.

The master equation

Proteins are transcribed in randomly-timed bursts, each burst leading to the synthesis of an exponentially distributed amount of protein, with the average burst size given by a parameter b [3]. The overall protein dynamics will be hybrid, with deterministic decay occurring with rate (6) being balanced by the stochastic burst-like production.

The probability density function p(x, t) of observing the protein at concentration x at time t satisfies the master equation [10, 22, 23] (7) where a(x) is the stochastic transcription rate, which may depend on the present level x of protein, owing to transcriptional autoregulation of the protein. The functional form of a(x) can be deduced from the details of molecular interaction between the protein and its gene.

We focus on the case of non-cooperative positive autoregulation, which is appropriate for the HIV-1 regulatory protein Tat [24, 25] as well as for the experimental system involving tTA driven by 1×tetO promoter [15], assuming the form of (8) in which a₀ is the basal transcription rate (per protein lifetime), a₁ gives the difference between the up-regulated and the basal rates, and k_p is the dissociation constant for the interaction between the protein and its binding site at the promoter. The concentration x_f of free protein is obtained as a function of the total protein concentration x according to the expression (5).

Any decrease in free protein concentration resulting from the interaction between the protein and the functional binding sites is neglected because (eukaryotic) transcription factors are typically present at hundreds or thousands of copies, while a typical gene would be present at 1–2 copies and be regulated by a handful of functional sites. This would hold in the tTA system [15], where a single copy of a gene encoding the tTA protein is driven by a promoter containing a single TetO binding site (for the tTA protein), and would also extend to the HIV-1 regulatory protein Tat in the case of an infection by a single copy (or a small number) of the virus.

The stationary solution p(x, t) = p(x) to (7) can be shown to satisfy a simpler equation (see S1 Supporting Information), (9) from which, by direct integration, (10) where κ is a normalisation constant; such a form has been disclosed in previous studies [10, 22].

For the specific choices of the burst rate (8) and decay rate (6), Equation (10) reduces to a closed expression (see S1 Supporting Information for details) (11) where (12) (13) (14) (15) The free protein level x_f is understood in (11) to be a function of x, as given by expression (5). The formula (11) for the stationary distribution is valid as long as some special cases, namely that of k_p = k_b, or k_p = k_b+γ_b y, or γ_b y = 0, are avoided; results for these special cases are given in S1 Supporting Information.

One can argue that the activity of a gene is determined by the concentration x_f of free proteins, rather than the total protein concentration x. Therefore, it is of interest to express the above results in terms of the probability density function $\tilde{p}(x_{\mathrm{\text{f}}})$ of x_f. The transformation rule implies that (16) in which p(x(x_f)) can be evaluated using the expressions (11), in which, this time round, the free protein concentration x_f takes the role of the independent variable while x is understood to be a function of x_f, as given by (4).

Should one prefer to measure the protein concentration on a logarithmic scale, i.e. using z = log₁₀(x), the probability density function $\breve{p}(z)$ of z is given by (17) which is defined for any value of z. To measure the free protein concentration on the logarithmic scale, the transformation should be applied to $\tilde{p}(x_{\mathrm{\text{f}}})$ instead of p(x).

In Fig. 1, top, we show the probability density function of the total protein concentration on a logarithmic scale for changing levels of decoy binding site concentration (0, 50, 80 and 120 sites per cell). Following previous studies [15], the rate of decay for protein molecules that are bound to decoy binding sites is set to be equal to the decay rate for free protein (γ_b = 1), and the protein is assumed to bind to decoy sites much more tightly than to its promoter (k_b = 1 ≪ k_p = 500, cf. [15]). The transcription rate is set to start at a low basal value of a₀ = 0.75 bursts per protein lifetime and can be increased by a₁ = 25 bursts per lifetime in case of full promoter occupancy, and the mean burst size is set to b = 50. Such choices ensure that at the basal rate of expression, a large fraction of protein molecules is sequestered away by decoy binding sites; on the other hand, the relatively high regulable rate ensures that the positive feedback loop is established once there are enough protein copies. Indeed, the figure suggests that adding decoy binding sites leads to a bimodal distribution of protein level, the lower mode corresponding to the basal expression scenario and the higher mode corresponding to the upregulated scenario.

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Fig 1. Protein level distributions with and without decoy binding sites.

Top: Probability density function of the total protein concentration on a logarithmic scale. The number y of binding sites is varied between 0 and 100. The decay rate constants γ_f and γ_b for free and bound protein are both set to one; the dissociation constant k_b for the interaction with binding sites is set to one molecule; the dissociation constant k_p for the interaction with the promoter is set to 500 molecules; the basal transcription rate is 0.75 bursts per protein lifetime and the regulable transcription rate is 25 bursts per lifetime. The mean burst size is 50 molecules. Middle and bottom: The gene-expression response to changes in a₁ is graded in the absence of decoy binding sites (y = 0, middle); it is binary if decoy binding sites are present (y = 100, bottom). The other model parameters are chosen as follows: γ_f = γ_b = 1, k_b = 1, k_p = 500, a₀ = 0.75, b = 50.

https://doi.org/10.1371/journal.pone.0120555.g001

Fig. 1, middle and bottom, illustrates the response of protein distribution to changes in the dose of an external repressor in the absence (Fig. 1, middle) or in the presence (Fig. 1, bottom) of decoy binding sites. For simplicity, we assume that the effect of the repressor is in reducing the efficiency of the autoregulation loop, i.e. reducing a₁. Such an effect is easy to model with the current mathematical model; at the same time, it is equivalent to more complex repressing mechanisms, such as the protein-protein interaction between the dox repressor and tTA protein [15]. While in the absence of decoy binding sites the response of the protein distribution is graded (Fig. 1, middle), in their presence the response is binary (Fig. 1, bottom).

Exploring the parameter space

As has been illustrated on specific examples in the previous section, our stochastic model allows for a bimodal distribution of protein level, especially if decoy binding sites are present. In this section we seek to delineate the regions of the parameter space for which bimodality, as opposed to unimodality, is observed. In doing so we follow a strategy that was previously used in [26] to explore the parameter space of a stochastic gene expression model in the absence of decoy binding site. We extend this strategy by considering bimodality not only on linear, but also on logarithmic scale.

Bimodality of protein distribution can occur only if there are multiple local extrema of p(x), i.e. multiple solutions to the equation dp(x)/dx = 0. Differentiating the product on the left-hand side of (9) and regrouping the terms yields (18) implying that for local extrema the bracketed term on the right-hand side of (18) must vanish.

For simplicity, we shall assume that γ_b = 1, i.e. that protein molecules decay with the same rate constant whether they are free or bound to decoy sites. The general case of unequal decay poses challenges that are technical, but not fundamental in their nature, and can be treated by the same methods as used below for γ_b = 1.

We recall that the gene response function a and of the decay rate c satisfy (19) Therefore, the derivative dp/dx is equal to zero at a given total protein level x if for the associated free protein level x_f we have (20) The number of local extrema changes if the right-hand side of (20), as a function of x_f, crosses the value zero tangentially, i.e. if (21) holds in addition to (20). For any fixed level of x_f, Equations (20) and (21) specify the combinations of parameter values (a₀, a₁, k_p, b, y, and k_b) for which a transition from unimodality to bimodality in p(x) occurs at x corresponding to the fixed value of x_f. As the value of x_f is varied, the parameter values satisfying (20) and (21) span the hyper-surface in the parameter space that separates the region in which bimodality is observed from that in which unimodality occurs.

Instead of grappling with the 6-dimensional geometry (5-dimensional after a suitable scaling) of the parameter space, it is fruitful to try to understand one of its 2-dimensional cross-sections, obtained by prescribing the values for all but two parameters. Hence, if we fix the values of the regulable transcription rate a₁, mean burst size b, decoy binding site concentration y and the dissociation constant k_b for the binding to decoy sites, we can use (20) and (21) to determine the curves that delineate in the (a₀, k_p)-cross-section of the parameter space the regions of bimodality and unimodality.

Before doing that, we need to clarify an important point about bimodality itself. If a random variable has a distribution which is bimodal, the distribution of a non-linear transformation of the variable, even if it be a monotone one, need not be bimodal (and vice versa). As gene-expression measurements are often reported on a logarithmic scale, it is of interest to examine the conditions for bimodality of the probability density function $\breve{p}(z)$ of the log-scaled total protein concentration z = log₁₀(x).

The relationship (17) between the linear-scale and logarithmic-scale densities p(x) and $\breve{p}(z)$ can be expressed in terms of the linear variable x as (22) implying that (23) in which we used (18) to express the derivative dp/dx.

Assuming, as before, that free and bound protein decay with the same rate constant, we have c = x, and the last two terms of the bracketed expression in (23) cancel each other; therefore the derivative $\mathrm{\text{d}}\breve{p}/\mathrm{\text{d}}x$ vanishes at a given protein level x, and so does the derivative $\mathrm{\text{d}}\breve{p}/\mathrm{\text{d}}z$ at the associated log-scaled level z = log₁₀(x), if (24) holds for the free protein concentration x_f. The number of local extrema of $\breve{p}$ changes if the left-hand side of (24) crosses zero tangentially, i.e. if, in addition to (24), the derivative of the left-hand side of (24) with respect to x_f vanishes. Since the left-hand side of (24) differs only by an additive constant from that of (20), the condition of tangential crossing is given by (21).

Equations (24) and (21), in which x_f is allowed to vary arbitrarily, specify a hyper-surface in the parameter space which encloses the region of bimodality of the logarithmic-scale density. Since any solution to (20)–(21) can be turned by means of the translation a₀ → a₀−1 into a solution to (24)–(21), the hyper-surface is a translation, by 1 to the left along the a₀ axis, of the hyper-surface enclosing the region of bimodality of the linear-scale density.

In Figs. 2 and 3 we show two cross-sections of the parameter space obtained by fixing the regulable transcription rate a₁ = 25, the dissociation constant for decoy site binding k_b = 1, the mean burst size b = 50, and the number of decoy binding sites to either y = 0 (Fig. 2) or y = 100 (Fig. 3). The two remaining model parameters, the basal transcription rate a₀ and the dissociation constant for promoter binding k_p are varied over realistic ranges 0 < a₀ < 4 and 0 < k_p < 1500. Nine parameter sets from either cross-section are selected for which the linear-scale and logarithmic densities are shown. Alternative versions of Fig. 3 obtained by choosing different parameter values can be found in S1 Fig.

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Fig 2. Bimodality and unimodality for the linear-scale and logarithmic-scale densities of total protein concentration in the absence of decoy binding sites.

The basal transcription rate a₀ (per protein lifetime) and the dissociation constant for promoter binding k_p (measured in the units of molecule number) are varied, the others being prescribed fixed values, namely the regulable transcription rate a₁ is set to 25 bursts per protein lifetime, the number y of decoy binding sites is set to zero and the mean burst size b is equal to 50 molecules.

https://doi.org/10.1371/journal.pone.0120555.g002

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Fig 3. Bimodality and unimodality for the linear-scale and logarithmic-scale densities of total protein concentration in the presence of decoy binding sites.

The basal transcription rate a₀ (the average number of basal bursts per protein lifetime) and the dissociation constant for promoter binding k_p (number of molecules required for promoter half-occupancy) are varied, the others being prescribed fixed values: the regulable transcription rate a₁ is set to 25 bursts per protein lifetime, there are y = 100 decoy binding sites to which the proteins can bind at dissociation constant k_b = 1 molecule; the burst size has the mean of b = 50 molecules and the proteins are degraded with unit rate whether they are bound or not, i.e. γ_f = γ_b = 1.

https://doi.org/10.1371/journal.pone.0120555.g003

A rough picture of the cross-sections can be fleshed out by referring back to the physical interpretation of our model. If the basal rate a₀ is low, any time the protein concentration decreases by chance to levels that are significantly lower than the dissociation constant for promoter binding, its autoregulation loop will be switched off for long periods of time, so that there would be a high probability of finding a low number of protein copies in the cell. Restarting the autoregulation loop will be more difficult in the presence of decoy binding sites, which will sequester the proteins produced in the first few bursts of basal expression. Higher levels of basal rate will make it easier for the protein levels to escape this trap, and with high probability a large number of proteins will be present in the cell.

On the other hand, if the dissociation constant for promoter binding is very low, it will require a relatively small amount, perhaps as small as produced in a few average-sized bursts, of protein molecules to turn on the autoregulation loop. A low value of the dissociation constant will therefore have a similar effect like a high value of the basal rate. If the dissociation constant is very large, perhaps as large as the maximum production rate (a₀+a₁)×b, it will be almost impossible for the protein to turn its production on, thus becoming trapped in the low-expression state for good.

While a sound physical understanding of the model can elucidate the basic logic of the parameter space, mathematical analysis needs to be put to use in order to paint a finer picture of it, which is interesting especially for the intermediate values of the basal rate a₀ and the dissociation constant k_p.

In the absence of binding sites, cf. Fig. 2, the response of the linear-scale density to an increase in promoter affinity, i.e. to a decrease in the dissociation constant k_p, is binary if the basal rate a₀ is very low (parameter sets 1–3 in Fig. 2). For high values of k_p, much of the probability mass is concentrated at the low-expression singular peak (parameter set 1). As k_p decreases down to a critical level, indicated by the solid black curve in the figure, the linear-scale density acquires an inflection point (cf. parameter set 2), which then develops into a regular peak of high expression with further improvement of promoter affinity (parameter set 3). We note that unless a₀ is very low and k_p is close to the critical level, the low-expression model will contain only a very small fraction of the entire probability mass (a₀ = 0.1 for parameter set 3), with the rest of it residing around the regular high-expression peak.

The solid black curve of Fig. 2, on which the transition from unimodality to bimodality occurs, has been determined by solving the system of algebraic Equations (20)–(21) numerically in unknowns a₀ and k_p for a range of values of x_f. The curve encloses the region of bimodality from above; the dashed line a₀ = 1 forms the border on the right-hand side. In order to understand the nature of the transition from bimodality to unimodality at a₀ = 1, we note that if a₀ ≥ 1, then the linear-scale density (11) is bounded from above, while it has a singularity at x = 0 if a₀ < 1. As a₀ converges to one from left, the singularity becomes ever thinner, until it peters out at a₀ = 1 (see parameter set 6 of Fig. 2 as compared to parameter set 3), where the linear-scale density p(x) has a finite limit as x tends to zero; if a₀ exceeds one, the density vanishes at x = 0 (see parameter sets 7, 8 or 9 of Fig. 2).

The logarithmic-scale density is always unimodal in the absence of binding sites, as is evidenced by the examples of Fig. 2; indeed, Equation (24) for local extrema of the logarithmic-scale density can easily be shown to possess one solution at most if y = 0. For low rates of basal transcription, the logarithmic density is highly skewed and possesses a fat left tail.

The parameter-space cross-section in which the number of decoy binding sites is set to y = 100 is visibly different, see Fig. 3, from the previous one. Binding sites are able to sequester the produce of basal transcription, thus stabilising the low-expression mode. The response of the linear-scale, as well as logarithmic scale, densities to an increase in binding affinity to the promoter (i.e. to decreasing k_p) is binary for a relatively wide range of basal rates (see the parameter sets selected in Fig. 3). As k_p descends to a critical level, indicated by the upper branch of the solid black curve, the linear-scale density acquires an inflection point which turns into a peak of high expression upon further decrease in k_p (parameter sets 2, 5 and 8 of Fig. 3). On the lower branch of the solid black curve, it is the low-expression peak that turns into an inflection point (not shown; however, parameter set 9 of Fig. 3 is close to such a scenario).

The two branches of the solid black curve enclose the parameter region in which the linear-scale density is bimodal. The region of bimodality for the logarithmic-scale density is obtained, as explained above, by shifting these two branches by one unit to the left. The shifted curve is shown in grey; the parameter sets 1 and 4 that lie on the grey curve are seen to exhibit a transition, via an inflection point, between unimodality and bimodality on the logarithmic scale.

Thus, in comparison with the situation that arises in the absence of decoy binding sites, in their presence the probability mass is evenly distributed between the modes of low and high expression for a wide ranges of a₀ and k_p. Bimodality is observed not only on the linear scale, but also on the logarithmic one.

Deterministic model

We multiply the master Equation (7) by x and integrate over all positive values of x, finding that the mean protein concentration satisfies (25) Equation (25) is not closed in the mean since the transcription and degradation rates are, in general, nonlinear functions of x.

Neglecting any fluctuations in (25), we obtain a deterministic formulation of the model, (26) which is an autonomous nonlinear first-order differential equation in protein concentration x.

At steady state, we have ba(x) = c(x), i.e. (27) having expressed the transcription and degradation rates in terms of the free protein concentration, as given by Equations (6) and (8).

Equation (26) is bistable if there are multiple solutions x_f to (27). Notably, if there are no decoy binding sites (y = 0), or if proteins bound to decoy binding sites are not degraded (γ_b = 0), then (27) possesses a single solution, and bistability cannot occur.

In the linear degradation case, i.e. for γ_b = 1, the steady-state Equation (27) is equivalent to the Equation (24) for the extrema of the logarithmic-scale density of the stochastic model. Consequently, the deterministic model exhibits bistability if the stochastic model is bimodal on the logarithmic scale, i.e. for parameter values from the region which is enclosed by the grey curve in Fig. 3.

Decoy sites and protein degradation

In regards to the impact of binding to decoy sites on protein stability, it is enlightening to juxtapose the following two contrasting cases, cf. [18]: (i) protein molecules are degraded with the same rate regardless of whether they are bound or not; (ii) they are protected from decay while bound. Having previously explored the former in detail, we now discuss the latter possibility; more details can be found in [23].

If bound proteins are stable, then sequestering of protein molecules at decoy binding sites decreases the effective protein degradation rate constant, thus increasing the mean steady-state protein level. Unlike in the previous case, adding decoy binding sites does not stabilise the low-expression mode: the effective increase in the autoregulation loop threshold, to which we attributed the stabilisation of the low-expression mode previously, will be balanced in its effect by the increase in the mean protein level.

Fig. 4 shows the cross-section of the parameter space of our model in which the basal transcription rate a₀ and the dissociation constant k_p for promoter binding are varied, while the remaining parameter values are fixed at constant values. The fixed values are the same as those used in Fig. 3, except for the degradation rate constant γ_b for bound protein, which is set to zero.

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Fig 4. Bimodality and unimodality for the linear-scale and logarithmic-scale densities of total protein concentration in the presence of decoy binding sites, which protect the protein from degradation.

The basal transcription rate a₀ and the dissociation constant for promoter binding k_p are varied, the regulable transcription rate is set to a₁ = 25 bursts per protein lifetime, the number of binding sites is y = 100 with which protein molecules interact at dissociation constant of k_b = 1 molecule; the mean burst size is b = 50; protein molecules are degraded with unit rate constant γ_f = 1 if they are free, but no degradation occurs if they are bound to decoys (γ_b = 0).

https://doi.org/10.1371/journal.pone.0120555.g004

The critical values of a₀ and k_p, for which a transition between unimodality and bimodality in the linear-scale density occurs, form the black curve in Fig. 4. The grey curve separates the bimodal and unimodal regions of the logarithmic-scale density. The curves were determined using a modification of the method used to classify the parameter space in the case of γ_b = 1.

Comparing Figs. 2 and 4, we note that if γ_b = 0, then adding decoy binding sites does not lead to bimodal expression on the linear scale, unless bimodality is already observed in their absence. In the right panel of Fig. 4, we show the linear and logarithmic scale densities for a₀ = 0.1 and three specific choices of k_p. The densities have been calculated by taking the limit, as γ_b tends to zero, of (11). The response of the density, on either scale, to an increase in promoter binding affinity (i.e. to a decrease in k_p) is binary. In contrast with the response observed at similarly low levels of a₀ in the absence of binding sites (parameter sets 1–3 in Fig. 2), the lower peak is regular and can also be identified on the logarithmic scale.

Thus, adding decoy binding sites that protect the protein from degradation increases the mean protein expression level and decreases the heterogeneity in protein expression. On the other hand, if decoy binding sites are added that do not interfere with protein decay, protein expression decreases in the mean and can become more heterogeneous.

Modelling assumptions

The modelling framework

The evolution in time, of the cell-to-cell distribution for the total protein level, is described by master Equation (7). It is an integro-differential equation, in which protein degradation is manifested by the convective term, while burst-like synthesis manifests itself in the integral term. The total degradation rate in the convective term is given by (6) as a sum of the degradation rate of free protein and that of bound protein (the rate constants for the two being not necessarily equal). The burst rate in the integral term increases with the number of free protein molecules, the functional dependence being of the Michaelis-Menten form (8), corresponding to non-cooperative positive autoregulation: examples of our interest, HIV-1 Tat protein and the synthetic yeast system, exhibit this kind of autoregulation [15, 24].

The exact stationary distribution (10) of the total protein level is found by solving, at steady state, master Equation (7), along the lines of [10]. The analytic solution is given by a closed-form expression, up to a normalisation constant which has to be determined by numerical integration. The free protein and logarithmic-scale (total or free protein) distributions can be obtained from the total protein distribution using the transformation rule.

Connection with previous work

The cell-to-cell distribution of protein copy number for a self-regulating gene, which is being expressed in bursts, has been characterised in a number of previous studies using mathematical models based on the chemical master equation, to a varying degree of detail [10, 32]. In these models, however, decoy binding sites have not been included.

A fine-grained mathematical model for a self-regulating gene in the presence of decoy binding sites, accounting for individual promoter transitions, DNA binding and dissociation, etc., has been investigated in [18]. The master equation associated with the fine-grained model was solved numerically [18]. Some analytic results have been obtained using a Langevin approximation to the master equation [19].

Our paper extends the model of [10] by including the interaction between the protein and decoy binding sites. Our results also contribute, by finding an analytically tractable stationary protein distribution, to the previous approaches in modelling the dynamics of a self-regulating protein in the presence of decoys due to [18, 19] and [15].

Conditions for the protein copy number distribution to possess two modes are derived in a similar manner as was done in [26] for the model of [10] without decoy binding sites. In comparison to [26], bimodality is investigated not only on the linear, but also on the logarithmic scale. Our model also represents a biologically important example to which the results of [22] on the existence, uniqueness, and long-time behaviour of solutions to a master equation such as (7), are applicable.

New results

Fig. 1 illustrates some basic properties of the stationary distribution (for the total protein number, on a logarithmic scale). Most parameter values are adopted from [15], in which a deterministic version of our model was studied and compared to experimental measurements on a synthetic yeast system. Specifically, the interaction between the transcription factor and decoy binding sites is assumed not to affect the propensity for the protein to be degraded; secondly, decoy binding sites are assumed to attract proteins with much greater affinity than the autoregulatory binding site.

Adding decoy binding sites is shown to lead to bimodality in the protein distribution (Fig. 1, top). Biologically, decoy binding sites sequester much of the produce of basal gene expression, at the expense of autoregulation, leading to the stabilisation of the basal expression mode, and to a bimodal profile of the distribution.

The middle and bottom panels of Fig. 1 illustrate the response of the distribution to changing doses of an upstream repressor (modelled, for simplicity, by varying the parameter corresponding to the maximal achievable transcription rate). In the absence of decoys (Fig. 1, middle), the response is graded, while in their presence, the response is binary (Fig. 1, bottom), a distinction which is in qualitative agreement with empirical distributions for a fluorescent reporter protein in the synthetic yeast system [15].

The tenacity of the basal expression mode is sensitive to changes in the basal transcription rate a₀ and the affinity of the transcription factor for its autoregulatory site (this being equal to the reciprocal of the dissociation constant k_p). As a₀ and k_p vary, the total protein distribution can transition between unimodal and bimodal behaviour. In the absence of decoys (cf. Fig. 2), bimodality is observed on the linear scale only if basal transcription occurs at a rate lower than one burst per protein lifetime. If observed, the basal expression mode is always singular. On the logarithmic scale, the distribution is unimodal for all combinations of parameters, being highly skewed in the regions of parameter space where the linear-scale distribution is bimodal. In the presence of decoys, bimodality is observed for a broader region of parameter values, both on the linear and also on the logarithmic scale (cf. Fig. 3). On the linear scale, the low expression mode can be either singular or regular.

Bimodality of stationary distributions in stochastic models is often thought of as a counterpart of bistability in deterministic systems. We have shown that the conditions for bistability for a deterministic reduction of our model coincide with the conditions for bimodality on the logarithmic scale.

The above results have been concerned with the case of equal degradation for free and bound protein. A radically different scenario follows if decoy binding sites protect the protein from being degraded. If such decoys are added, bimodal behaviour is observed for a narrower region of parameter values than in their absence (cf. Fig. 4). The effect of adding decoy binding sites is one of an increase in the protein level, due to a reduction in the overall degradation rate, rather than a stabilisation of the low expression mode. In the presence of protective decoys, the basal expression mode becomes regular and is maintained on the logarithmic scale.

If free protein molecules are degraded with a different rate constant than the bound ones, conditions for bimodality no longer coincide with those for bistability of the deterministic model; indeed, if bound proteins are completely protected from degradation, bistability cannot occur, while bimodality on the logarithmic scale can.

In summary, we propose a minimalistic model for burst-like gene expression for a transcription factor that can regulate itself as well as bind to a large number of targets on the DNA. Our main results are the derivation of an exact stationary distribution for the protein level, and its classification according to the number of modes.