Likelihood for transcriptions in a genetic regulatory system under asymmetric stable Lévy noise

Gene regulation is a crucial but noisy biological process.^1,2 The significance of noise in genetic networks has been recognized and studied.^3–10 It has been shown recently that noise is vital for regime transitions in gene regulatory systems.^11–15 In these works, noisy fluctuations are, however, taken to have Gaussian distributions only^6,16–19 and are expressed in terms of Brownian motion.

But when the fluctuations are present in certain events, such as bursty transition events, the Gaussianity assumption is not proper. In this case, it is more appropriate to model the random fluctuations by a non-Gaussian Lévy motion with heavy tails and bursting sample paths.^20–23 Especially, during the regulation of gene expression, transcriptions of DNA from genes and translations into proteins occur in a bursty, intermittent way.^9,10,24–30 This intermittent manner^31–34 resembles the features of a stable Lévy motion, which is a non-Gaussian process with jumps.

Recent studies^35,36 have recognized that symmetric stable Lévy motion can induce switches between different gene expression states. Note that symmetry (zero skewness) in stable Lévy motions is a special, idealized situation.^37–39 The asymmetric Lévy noise is more general and more representative.

In this present paper, we examine the likelihood for transitions from low to high concentrations (i.e., likelihood for transcriptions) in a genetic regulatory system under an asymmetric (i.e., non-symmetric) stable Lévy noise, highlighting the dynamical differences with the case of symmetric noise. To this end, we compute two deterministic quantities, the mean first exit time (MFET) and the first escape probability (FEP). The MFET is the mean time scale for the system to exit the low concentration state (i.e., the longer the exit time, the less likely for transcription), while the FEP is the switch probability from low concentration states to high concentration states (i.e., it is the likelihood for transcriptions).

Having a better understanding of the likelihood for transcriptions in the genetic regulatory networks, we could shed light on the mechanisms of diseases which are caused by the dysregulation of gene expressions.

This paper is organized as follows. In Sec. II, we briefly describe a genetic regulation model with noisy fluctuations in the synthesis reaction rate. In Sec. III, we recall basic facts about asymmetric stable Lévy motions. In Sec. IV, we investigate the transition phenomena under additive asymmetric Lévy motion by numerically computing two deterministic quantities, highlighting the differences with the symmetric stable Lévy noise case. In Sec. V, we study the multiplicative asymmetric Lévy motion case. Finally, we make some concluding remarks in Sec. VI. The  Appendix contains the mathematical formulation for the first mean exit time and the first escape probability, in terms of deterministic integral-differential equations.

Smolen et al.⁴⁰ introduced the following model for the concentration “x” of the transcription factor activator (TF-A)

This is a relatively basic model of positive and negative autoregulations of transcription factors (Fig. 1). A transcription factor activator, denoted by (TF-A), is considered as part of a pathway mediating a cellular response to a stimulus. The transcription factor forms a homodimer, which can bind to specific responsive elements (TF-REs). The TF-A gene includes a TF-RE, and when homodimers bind to this element, the TF-A transcription is increased. Only phosphorylated dimers can activate transcription. The regulatory activity of transcription factors is often modulated by phosphorylation. It is assumed that the transcription rate saturates with the TF-A dimer concentration to a maximal rate k_f, TF-A degrades with first-order kinetics with the rate k_d, and the TF-A dimer dissociates from TF-REs with the constant K_d. The basal rate of the synthesis of the activator is R_bas.

With the potential

the model equation (1) becomes

This system has two stable and one unstable equilibrium states, i.e., a double-well structure, when the parameters satisfy the following condition:

Under this “double-well” condition and as in Smolen et al.,⁴⁰ we choose proper parameters in this genetic regulatory system on the basis of genetic significance and also for convenience: K_d = 10, $k d = 1 min − 1 , k f = 6 min − 1$⁠, and $R b a s = 0.4 min − 1$⁠. Then, the two stable states are

and the unstable state (a saddle) is

That is, the deterministic dynamical system (1) has two stable states: $x −$ and $x +$ as well as one unstable state x_u (see Fig. 2).

However, the basal synthesis rate R_bas is unavoidably influenced by many factors,⁴⁰ such as the mutations, the biochemical reactions inside the cell, and the concentration of other proteins. These fluctuations in the genetic regulatory system behave like bursty perturbations as we discussed in Sec. I. Therefore, we incorporate an asymmetric stable Lévy motion as a random perturbation of the synthesis rate R_bas. Thus, model (1) becomes the following stochastic gene regulation model:

The effect of multiplicative noise has been investigated in the literature.^41–43 Here, we also consider that the synthesis rate R_bas is perturbed by a multiplicative asymmetric Lévy motion as follows:

where $L t α , β$ is an asymmetric stable Lévy motion with the jump measure $ν α , β$⁠, for the non-Gaussianity index $0 < α < 2$ and skewness index $− 1 ≤ β ≤ 1$⁠. This Lévy motion will be recalled in Sec. III. Here, we assume that the generating triplet of asymmetric stable Lévy motion is $( 0 , 0 , ε ν α , β )$⁠, where ε is the noise intensity. The noise intensity plays an important role in the noise source.^44–47 We will discuss the effects of noise intensity ε in Sec. V. In stochastic dynamics, it is customary to denote a state variable in a capital letter, with time dependence as subscript. The “x” here and hereafter denotes the initial concentration for the transcription activator factor or TF-A monomer in this gene regulatory system.

Under the interaction of the potential field U and these fluctuations, the concentration of the TF-A monomer may exit from the domain $D = ( 0 , x u )$ (the low concentration domain). Our goal is to quantify the effects of asymmetric Lévy noise on the dynamical behaviors of TF-A monomer concentration in this model. We focus on the likelihood for TF-A monomer concentration transitions from the low concentration domain D to the high concentration domain $E = [ x u , + ∞ )$⁠, via analyzing two deterministic quantities: the mean residence time (also called mean first exit time) in domain D before first exit, and the likelihood of first escape from D through the right side (i.e., becoming high concentration). It is desirable to focus mainly on the high TF-A monomer concentration, since that corresponds to a high degree of activity. That is, high concentration indicates effective transcription and translation activities.

The aforementioned asymmetric stable Lévy motion $L t α , β$ is an appropriate model for non-Gaussian fluctuations with bursts or jumps. The parameter α is the non-Gaussianity index (⁠ $0 < α < 2$⁠) and β is the skewness index (⁠ $− 1 ≤ β ≤ 1$⁠). A scalar Lévy motion has jumps that are characterized by a Borel measure ν, defined on the real line $ℝ 1$ and concentrated on $ℝ 1 \ { 0 }$⁠. The jump measure ν satisfies the following condition:

The asymmetric stable Lévy motion $L t α , β$ is a stochastic process defined on a sample space Ω equipped with probability $P$⁠. It has independent and stationary increments, together with stochastically continuous sample paths: for each s, $L t α , β →$  $L s α , β$ in probability. This means for every $δ > 0 , P ( | L t α , β − L s α , β | > δ ) → 0$⁠, as $t → s$⁠.

The jump measure, which describes jump intensity and size for sample paths, for the asymmetric Lévy motion $L t α , β$ is^37,38

with $C 1 = H α ( 1 + β ) 2 , C 2 = H α ( 1 − β ) 2 .$ When α = 1, $H α = 2 π$⁠; when $α ≠ 1 , H α = α ( 1 − α ) Γ ( 2 − α ) cos ( π α 2 ) .$

Especially for β = 0, this is the symmetric stable Lévy motion, which is usually denoted by $L t α ≜ L t α , 0$⁠. The well-known Brownian motion B_t may be regarded as a special case (i.e., Gaussian case) corresponding to α = 2 (and β = 0); see Ref. 38.

We can see a clear difference from the symmetric case from Fig. 3, which shows the probability density functions for $L t α , β$ at t = 1 for various α, β.

For the stable Lévy motion with the jump measure in (4), the number of larger jumps for small α  $( 0 < α < 1 )$ is more than that for large α  $( 1 < α < 2 )$⁠, while the number of smaller jumps for $0 < α < 1$ is less than that for $1 < α < 2$⁠, as known in Ref. 39.

To quantify the likelihood for transcription for the stochastic genetic regulatory system (2) under asymmetric (i.e., non-symmetric) stable Lévy noise, we will compute two deterministic quantities, the mean first exit time (MFET) and the first escape probability (FEP). They are solutions of nonlocal integral-differential equations, i.e., (A2) and (A4), respectively, in the  Appendix.

In this section, we first present the numerical schemes for solving the mean exit time u and escape probability p, and then conduct numerical simulations to gain insights about the likelihood for transcriptions modeled by (2).

For the stochastic differential equation (2) of the genetic regulation system with synthesis rate under an asymmetric Lévy noise, we present a numerical scheme to solve the following deterministic nonlocal integral-differential equation, (A2) in the  Appendix, in order to get the mean first exit time u

Here, D^c is the complement set of D in $ℝ 1$⁠.

The generator A for the stochastic differential equation (2) with asymmetric stable Lévy motion is^38,39

with $ν α , β ( d y ) = C 1 I { 0 < y < ∞ } ( y ) + C 2 I { − ∞ < y < 0 } ( y ) | y | 1 + α d y ,$ $C 1 = H α ( 1 + β ) 2$ and $C 2 = H α ( 1 − β ) 2 .$ When α = 1, $H α = 2 π$⁠; when $α ≠ 1 , H α = α ( 1 − α ) Γ ( 2 − α ) cos ( π α 2 ) .$ Additionally,

The MFET u satisfies the following equation:

On an open interval $D = ( a , b )$⁠, we make a coordinate conversion $x = b − a 2 s + b + a 2$ for $s ∈ [ − 1 , 1 ]$ and $y = b − a 2 r$⁠, to get the finite difference discretization for $A u ( x ) = − 1$ as in Ref. 48 

With the numerical simulation via (8), we obtain the MFET u for the stochastic gene regulation model (2).

A similar scheme is used for the numerical simulation for (A4) in the  Appendix, to get the first escape probability p.

We summarize major numerical simulation results below, and indicate their relevance to the likelihood for gene transcriptions. We highlight the peculiar dynamical differences with the case of symmetric stable Lévy noise (β = 0) in Ref. 36.

As we take domain $D = ( 0 , x u )$ to be in the low concentration region, a smaller MFET indicates a higher likelihood for gene transcription (and vice versa), and a larger FEP means a higher likelihood for gene transcription (and vice versa). Both MFET u and FEP p reflect the interactions between the nonlinear vector field f and the asymmetric stable Lévy noise $L t α , β$⁠.

Figure 4 shows the impact of the skewness index β on MFET, for $α = 0.5$ and $α = 1.5$⁠. When $− 1 < β < 0$⁠, the MFET increases first and then decreases, but for $0 < β < 1$⁠, the MFET decreases in the whole interval. This indicates that the asymmetry of the noise (characterized by β) plays an important role in the dynamical system: Increasing positive asymmetry leads to a higher likelihood for gene transcription, while for negative asymmetry, there is a minimum likelihood for transcription (⁠ $α = 0.5$⁠). But for $α = 1.5$⁠, the MFET increases to the maximum and then decreases to 0, i.e., there is a minimum likelihood for transcription for all asymmetry index β. Meanwhile, we observe that for $β < 0$⁠, the MFET decreases earlier than that for $β > 0$⁠. We also observe a peculiar feature. With $α < 1$⁠, the MFET reaches the maxima value (i.e., the least likelihood for transcription) near the exit boundary $x u = 1.48971$ for negative β; while with $α > 1$⁠, the MFET reaches the maxima value near (i.e., the least likelihood for transcription) the exit boundary $x u = 1.48971$ for positive β. This indicates that the skewness index β may function as a tuning parameter for transcription.

Figure 5 shows that when β is fixed, the MFET values decrease with an increase in α, i.e., the likelihood for gene transcription increases with an increase in α. In comparison, Fig. 4(b) contains the case with Brownian noise (i.e., corresponding to $α = 2 , β = 0$⁠) and the MFET values break this monotonicity and stay roughly between those for $α = 1.5$ and $α = 1.9$⁠. Figures 4 and 5 indicate that if we start in the low concentration, then increasing α and β values leads to higher concentrations, corresponding to a higher likelihood for transcription.

Figure 6 plots the dependency of MFET in the low concentration on the asymmetry index β. Since the transcription behavior is particularly sensitive to initial conditions,⁴⁰ we investigate the noise effect on different initial concentrations. In the case of $α = 0.5$⁠, the MFET increases at first and then decreases. Different initial concentrations x correspond to different maximum MFET values: By tuning the asymmetry index β (depending on the initial concentration), we can find the least likelihood for transcription. If we fix x = 0.62685 (low stable concentration), the MFET increases and then decreases, especially for $α = 0.5$ or 1.5: By increasing the non-Gaussian index α, we can achieve a higher likelihood for transcription.

When skewness  $β ≠ 0$: It makes a great difference on the MFET for  $α < 1$ and  $α > 1$⁠. Figure 7 exhibits that when $β ≠ 0$⁠, the MFET has a bifurcation or discontinuity point at α = 1 when $β ≠ 0$⁠. We can see that the MFET has a “phase transition” or bifurcation at the critical non-Gasussian index value α = 1. This result is consistent with a theoretical analysis in Ref. 49. When the asymmetry index $β ≠ 0$⁠, in the low concentration region, the MFET decreases with the increasing α for $0 < α < 1$⁠, while for $1 < α < 2$⁠, the MFET increases first but then decreases with the increasing α. In the symmetric Lévy nose case (β = 0), the MFET is decreasing for all α (no bifurcation). Hence, in the asymmetric Lévy noise case (⁠ $β ≠ 0$⁠), we gain a higher likelihood for transcription by increasing the non-Gaussian index $α ∈ ( 0 , 1 )$⁠, while for $α ∈ ( 1 , 2 )$ there is a specific α_s leading to the minimum likelihood for transcription.

We thus observe a smaller MFET for a larger non-Gaussianity index α and a larger skewness index β. We can always achieve a minimum MFET by tuning the non-Gaussianity index α and skewness index β. A smaller MFET means a high level of TF-A, corresponding to a higher likelihood for gene transcription.

Figure 8 demonstrates that the FEP increases with the increasing β, and the FEP for positive β is larger than that for negative β. Comparing (a) with (b), we find that the FEP for $α = 1.5$ increases more rapidly than that for $α = 0.5$⁠.

From Fig. 9, we observe that when $β = − 0.5$⁠, the FEP corresponding to different α has intersection or crossover points. Before and after the intersection point, there exists an opposite relationship. When $β = 0.5$⁠, the FEP decreases with the increasing α. So in order to get a high likelihood of gene transcription, we can tune the asymmetric index β larger and α smaller. In comparison, for the Brownian noise case in Fig. 9(b), the FEP is approximately linearly increasing in the initial concentration x.

As shown in Fig. 10, we find that, when $β < 0$⁠, the FEP deceases with the increasing α for an initial concentration $x < x −$⁠, then increases with the increasing α for $x − < x < x u$⁠. This leads to the conclusion that larger initial concentrations are more likely leading to the transcription. If we consider the FEP at the low concentration x = 0.62685, we see that when $β < 0$⁠, the FEP increases with the increasing α, while when $β ≥ 0$⁠, the FEP decreases with the increasing α. A small α (and $β > 0$⁠) or a large α (and $β < 0$⁠) contributes to a large FEP (i.e., more likely for transcription).

The FEP has “turning points” with respect to α, β. Figure 11(a) exhibits that FEP increases with the increasing β, i.e., the likelihood for transcription improves with increasing β, when the system starts in low concentrations. When starting system at a low stable concentration x = 0.62685, we find that the evolution of FEP has “turning points” for $β = β turning ≈ − 0.5$ (this threshold value varies slightly with various α). As shown in Fig. 11(b), before and after a turning point β_turning, the FEP presents a reverse relationship: A higher FEP for larger α suddenly switches to a higher FEP for smaller α. That is, the higher likelihood for transcription is attained for larger non-Gaussianity index α before the turning point β_turning, while the opposite is true after the turning point. This phenomenon does not occur when the system is under symmetric Lévy fluctuations.

Combined effects: Exploring the whole parameter space  ${ ( α , β ) } = ( 0 , 2 ) × [ − 1 , 1 ]$⁠. Figure 12 displays the combined effects of both non-Gaussianity index α and skewness index β on MFET u and FEP p, at the initial concentration $x = x − ≈ 0.62685$ (i.e., the lower stable state).

In Fig. 12(a), the blue region indicates a smaller MFET (corresponding to the higher likelihood for transcription), while the red region means a larger MFET. Small MFET values occur when $( α , β )$ is in the two blue “sectors,” with (1, 0) as the common vertex and with α = 1 as the separation or bifurcation line. Note that α = 1 is not a separation point or bifurcation point in the symmetric case β = 0. Thus, we could achieve a minimum MFET or higher likelihood for transcription by tuning the non-Gaussianity index α and the skewness index β appropriately.

In Fig. 12(b), we observe that the FEP is larger in the red region, but smaller in the blue region. The combined small non-Gaussianity index α and big skewness index β (i.e., $α ∈ ( 0 , 0.7 )$ and $β ∈ ( 0.5 , 1 )$⁠) leads to a bigger FEP, i.e., higher likelihood for transcriptions. Therefore, we could achieve the maximum FEP or higher likelihood for transcription by tuning the non-Gaussianity index α and skewness index β appropriately.

Now, we present numerical experiments for understanding the likelihood of transcriptions modeled by (3).

The generator A for stochastic differential equation model (3) is

Let $z = y g ( x )$⁠. Then, the integral term in Eq. (9) is transformed to

where

Then, we obtain that the MFET u satisfies the following equation:

where

For an open interval $D = ( a , b )$ (in our computations below, we will take $D = ( 0 , x u )$⁠), we make a coordinate transformation $x = b − a 2 s + b + a 2$ for $s ∈ [ − 1 , 1 ]$ and $z = b − a 2 r$ to get the finite difference discretization for $A u ( x ) = − 1$ as in Ref. 48 

With this discretization, we obtain the numerical solution for nonlocal (A2) and thus MFET u for stochastic model Eq. (3). A similar method is applied to the first escape probability p.

We summarize our major numerical simulation results below. In this section, we highlight the dynamical differences with the case of additive asymmetric Lévy noise in Sec. IV B.

Figure 13 plots the evolution of MFET under multiplicative asymmetric Lévy motion. From Fig. 13(a), we observe that for $α = 1.5$⁠, the MFET decreases with the increase in the initial concentration x in the low concentration domain. We also find that the MFET becomes short if we tune the skewness index β small. Figure 13(b) includes the case of multiplicative Brownian noise; in this case, the MFET is bigger than the case of multiplicative Lévy noise clearly. The MFET increases with the increase in the non-Gaussianity index α.

Figure 14 indicates the effects of α and β on MFET for various initial concentrations x. Figure 14(a) shows the dependency in the low concentration on the skewness index β. As is shown for the case of $α = 0.5$⁠, the MFET decreases with the increase of β, and the higher initial concentrations correspond to a smaller MFET, i.e., higher initial concentrations x benefit for the transition. Figure 14(b) exhibits that in the multiplicative asymmetric Lévy case, MFET has a bifurcation or discontinuity point at α = 1 as in the additive asymmetric Lévy case. For $0 < α < 1$⁠, the MFET increases first and then decreases when near to α = 1; however, for $1 < α < 2$⁠, the MFET increases all the way very quickly. The maximum value is reached at α close to 2.

As shown in Fig. 15(a), when $α = 0.5$⁠, the FEP increases with the increase in β. So, a large positive β can induce a larger FEP. From Fig. 15(b), we see that FEP increases in the low concentration domain with the increase of x, and FEP with different α has intersections. Especially, the parts of the FEP for $α = 0.5$ overlap with that for $α = 1.0$⁠. But in the multiplicative Brownian noise case, the value of FEP is 1, i.e., the low concentration states will get to high concentration surely, which is quite different from the additive case.

Figure 16 plots the effects of α and β on FEP for various initial concentrations x. Figure 16(a) shows the FEP with respect to β for various initial concentrations x. FEP increases as β increases. Figure 16(b) shows the FEP with respect to α for various initial concentrations x. It presents that α = 1 is also a bifurcation point for FEP. This is totally different from that in additive asymmetric Lévy case [see Fig. 10(a)]. We can see that for $0 < α < 1$⁠, the FEP decreases with small initial concentrations x, while it increases first and then decreases with large initial concentrations x. For $1 < α < 2$⁠, the FEP decreases with x in the low concentration domain with the increase of α.

Figure 17 demonstrates the effects of noise intensity ε on MFET and FEP. Figure 17(a) shows us that larger the ε, the smaller the MFET. This indicates that large noise intensity helps to exit from the low concentration domain. Figure 17(b) shows the FEP with various noise intensity ε. The curves are crossing around x = 0.4. Note that this phenomenon is a little complicated near the crossing point. Before and after the crossing point, the FEP presents a reverse relationship with the initial concentration x. After the crossing point, the FEP increases with the increase of ε.

We plot the MFET as a function of noise intensity ε in Fig. 18. Inspired by the literatures,^50–52 we are interested in computing the MFET of the stochastic genetic model (3) starting from an unstable initial position, so we here set the exit domain as $[ 0 , 2 ]$⁠. We could see that the MFET has a monotonic behavior with the noise intensity ε. The inflection point value of MFET decreases with the increase in the initial value. When MFET passes the inflection point, it changes more modestly. Comparing Figs. 18(a) and 18(b), we could observe clearly that the smaller α makes the MFET shorter. The results are shown as a character of noise enhanced stability effect.^50–52 

Figure 19 presents the combined effects of non-Gaussianity index α and skewness index β on MFET u and FEP p, at an initial concentration $x = x − ≈ 0.62685$ under multiplicative asymmetric Lévy noise. From Fig. 19(a), we see that smaller MFET mostly appears at $α < 1$⁠. In the domain $α ∈ ( 1 , 1.1 )$ and $β ∈ ( 0.9 , 1 )$⁠, the MFET is rather long; we could see that α = 1 is the separation line clearly here. In Fig. 19(b), the red region presents a larger FEP domain, while the blue region represents the smaller parts. We observe that α = 1 is the bifurcation line apparently. Note that this phenomenon does not occur in the additive asymmetric Lévy case [see Fig. 12(b)]. The larger FEP values occur when $( α , β )$ is in the two red domains. We could tune the non-Gaussianity index α and skewness index β to the corresponding domains to achieve large FEP, i.e., having higher likelihood for transcriptions.

We have studied the effects of asymmetric stable Lévy noise on a kinetic concentration model for a genetic regulatory system. Both additive and multiplicative asymmetric Lévy noises are considered in this paper. We have examined the possible switches or transitions from the low concentration states to the high concentration ones (i.e., likelihood for transcriptions), excited by the asymmetric non-Gaussian Lévy noise. Our results suggest that asymmetric stable Lévy noise can be used as a possible “regulator” for gene transcriptions, for example, in the additive case, to attain a higher likelihood of transcription by selecting a larger positive skewness index (asymmetry index) β and a small non-Gaussianity index α. In contrast to the symmetric case, we have observed a bifurcation for the likelihood of transcription at the critical value α = 1 under an asymmetric stable Lévy noise (⁠ $β ≠ 0$⁠), as shown in Figs. 7, 14, and 16. Comparing Figs. 12(b) and 19(b), we see a striking difference between additive and multiplicative noises. There is also a turning point in the skewness index β for the likelihood of transcription, as seen in Fig. 11(b). The bifurcation and turning point phenomena do not occur in the symmetric noise case (β = 0).

Our results offer a possible guide to achieving certain genetic regulatory behaviors by tuning the noise index,⁵³ and may also provide helpful insights to further experimental research.

We would like to thank Dr. Xiao Wang for helpful discussions. This work was partly supported by the National Science Foundation Grant No. 1620449, and the National Natural Science Foundation of China Grant Nos. 11531006 and 11771449.

The mean first exit time (MFET) quantifies how long the system resides in the domain D before the first exit. The first exit time is defined as follows:³⁸ 

where $X t ( ω , x )$ is the solution orbit of the stochastic differential equation (2), starting with the initial TF-A concentration x. Then, the MFET is denoted as $u ( x ) = E τ ( ω , x )$⁠. Here, the mean $E$ is taken with respect to the probability $P$⁠. The MFET u(x) of the solution orbit $X t ( ω , x )$⁠, starting with the initial TF-A concentration x, is the mean time to stay in the low concentration domain D.

Denote the generator of the stochastic differential equation (2) by A. It is defined as $A u = lim t → 0 P t u − u t$⁠, where $P t u ( x ) = E u ( X t )$⁠. The generator A for the gene regulatory system (2) is explicitly given in (6), Sec. IV A. Then the mean exit time u satisfies the following nonlocal equation³⁸ with an exterior boundary condition

Here, D^c is the complement set of D in $ℝ 1$⁠.

When we take the domain $D = ( 0 , x u )$⁠, containing the low concentration stable state “ $x −$⁠,” the MFET is the mean time scale for the system to exit the low concentration state. The longer the mean exit time is, the less likely the system is in transcription.

The first escape probability (FEP), denoted by p(x), is the likelihood that the TF-A monomer, with initial concentration x, first escapes from the low concentration domain D and lands in the high concentration domain E. That is,

where τ is the exit time from D, as in (A1). This first escape probability p satisfies the following nonlocal equation³⁸ with a special, exterior boundary condition:

where A is the generator for the stochastic differential equation (2), as in (6), Sec. IV A.