Likelihood for transcriptions in a genetic regulatory system under asymmetric stable L\'evy noise

TL;DR

This paper investigates how asymmetric stable Lévy noise influences gene transcription likelihood, revealing phenomena like stochastic bifurcation and turning points, and offers methods to control transcription probability through noise parameter tuning.

Contribution

It introduces a novel analysis of asymmetric Lévy noise effects on gene regulation, identifying key phenomena and providing parameter strategies for gene transcription control.

Findings

Asymmetric noise skews increase transcription likelihood with positive skewness.

Stochastic bifurcation occurs at non-Gaussianity index α=1 in additive noise.

Turning points in likelihood are observed at β≈-0.5, depending on skewness.

Abstract

This work is devoted to investigating the evolution of concentration in a genetic regulation system, when the synthesis reaction rate is under additive and multiplicative asymmetric stable L\'evy fluctuations. By focusing on the impact of skewness (i.e., non-symmetry) in the probability distributions of noise, we find that via examining the mean first exit time (MFET) and the first escape probability (FEP), the asymmetric fluctuations, interacting with nonlinearity in the system, lead to peculiar likelihood for transcription. This includes, in the additive noise case, realizing higher likelihood of transcription for larger positive skewness (i.e., asymmetry) index $\beta$, causing a stochastic bifurcation at the non-Gaussianity index value $\alpha=1$ (i.e., it is a separating point or line for the likelihood for transcription), and achieving a turning point at the threshold value $\beta \approx -0.5$ (i.e., beyond which the likelihood for transcription suddenly reversed for $\alpha$ values). The stochastic bifurcation and turning point phenomena do not occur in the symmetric noise case ($\beta =0$). While in the multiplicative noise case, non-Gaussianity index value $\alpha=1$ is a separating point or line for both the mean first exit time (MFET) and the first escape probability (FEP). We also investigate the noise enhanced stability phenomenon. Additionally, we are able to specify the regions in the whole parameter space for the asymmetric noise, in which we attain desired likelihood for transcription. We have conducted a series of numerical experiments in `regulating' the likelihood of gene transcription by tuning asymmetric stable L\'evy noise indexes. This work offers insights for possible ways of achieving gene regulation in experimental research.

Equations

$$\dot{x}=\frac{k_f{x}^2}{x^2+K_d}-k_{d}x+R_{bas}.$$

$$U(x)=k_f\sqrt{K_d}\arctan \frac{x}{\sqrt{K_d}}+\frac{k_d}{2}{x}^2-(R_{bas}+k_f)x,$$

$$\dot{x}=f(x)=-U^{\prime }(x).$$

$$[-(\frac{k_f+R_{bas}}{3k_d}{)}^3+\frac{K_d(k_f+R_{bas})}{6k_d}-\frac{K_d{R}_{bas}}{2k_d}{]}^2+[\frac{K_d}{3}-(\frac{k_f+R_{bas}}{3k_d}{)}^2{]}^3<0.$$

$$x_{-}\approx 0.62685nM,x_{+}\approx 4.28343nM,$$

$$x_u\approx 1.48971nM.$$

$$\dot{X_t}=\frac{k_f{X}_t^2}{X_t^2+K_d}-k_d{X}_t+R_{bas}+{\dot{L}}_t^{\alpha ,\beta },X_0=x,$$

$$\dot{X_t}=\frac{k_f{X}_t^2}{X_t^2+K_d}-k_d{X}_t+R_{bas}+X_t{\dot{L}}_t^{\alpha ,\beta },X_0=x,$$

$${\int }_{{\mathbb{R}}^1\backslash \{0\}}(|y{|}^2\wedge 1)\nu (dy)<\infty .$$

$${\nu }_{\alpha ,\beta }(dy)=\frac{C_1{I}_{\{0<y<\infty \}}(y)+C_2{I}_{\{-\infty <y<0\}}(y)}{\mid y{\mid }^{1+\alpha }}dy,$$

$$Au(x)$$

$$u(x)$$

$$Au(x)=(f(x)+\epsilon M_{\alpha ,\beta })u^{\prime }(x)+\epsilon {\int }_{{\mathbb{R}}^1\backslash \{0\}}[u(x+y)-u(x)-I_{\{|y|<1\}}(y)y{u}^{\prime }(x)]{\nu }_{\alpha ,\beta }(dy),$$

$$M_{\alpha,\beta}=\left\{\begin{array}[]{ll}\frac{C_{1}-C_{2}}{1-\alpha},&\hbox{$\alpha\neq 1$,}\\ \ (\int_{1}^{\infty}{\frac{\sin(x)}{x^{2}}}dx+\int_{0}^{1}{\frac{\sin(x)-x}{x^{2}}}dx)(C_{2}-C_{1}),&\hbox{$\alpha=1$.}\end{array}\right.$$

$$(f(x)+\epsilon M_{\alpha ,\beta })u^{\prime }(x)$$

$$(\frac{2}{b-a})f(\frac{b-a}{2}s+\frac{b+a}{2}+\epsilon M_{\alpha ,\beta })u^{\prime }(s)+\epsilon (\frac{2}{b-a}{)}^{\alpha }{\int }_{{\mathbb{R}}^1\backslash \{0\}}[u(s+r)-u(s)-I_{\{|r|<1\}}(r)r{u}^{\prime }(s)]$$

$$\frac{[C_1{I}_{\{0<r<\infty \}}(r)+C_2{I}_{\{-\infty <r<0\}}(r)]}{\mid r{\mid }^{1+\alpha }}dr=-1.$$

$$Au(x)=(f(x)+\epsilon x{M}_{\alpha ,\beta })u^{\prime }(x)+\epsilon {\int }_{{\mathbb{R}}^1\backslash \{0\}}[u(x+yx)-u(x)-I_{\{|y|<1\}}yx{u}^{\prime }(x)]{\nu }_{\alpha ,\beta }(dy),$$

$$g(x)+\epsilon x^{\alpha }{\int }_{{\mathbb{R}}^1\backslash \{0\}}[u(x+z)-u(x)-I_{\{|y|<1\}}(z)z{u}^{\prime }(x)]{\nu }_{\alpha ,\beta }(dz),$$

$$g(x)=\left\{\begin{array}[]{ll}\varepsilon x^{\alpha}u^{\prime}(x)(C_{2}-C_{1})\ln x,&\hbox{$\alpha=1$,}\\ \ \varepsilon x^{\alpha}u^{\prime}(x)(C_{2}-C_{1})\frac{x^{(1-\alpha)}-1}{1-\alpha},&\hbox{$\alpha\neq 1$.}\end{array}\right.$$

$$c(x)u^{\prime }(x)$$

$$c(x)=\left\{\begin{array}[]{ll}f(x)+\varepsilon xM_{\alpha,\beta}+\varepsilon x^{\alpha}(C_{2}-C_{1})\ln x,&\hbox{$\alpha=1$,}\\ \ f(x)+\varepsilon xM_{\alpha,\beta}+\varepsilon x^{\alpha}(C_{2}-C_{1})\frac{x^{(1-\alpha)}-1}{1-\alpha},&\hbox{$\alpha\neq 1$.}\end{array}\right.$$

$$(\frac{2}{b-a})c(\frac{b-a}{2}s+\frac{b+a}{2})u^{\prime }(s)+\epsilon (\frac{2}{b-a}{)}^{\alpha }(\frac{b-a}{2}s+\frac{b+a}{2}{)}^{\alpha }{\int }_{{\mathbb{R}}^1\backslash \{0\}}[u(s+r)-u(s)-I_{\{|r|<1\}}(r)r{u}^{\prime }(s)]$$

$$\frac{[C_1{I}_{\{0<r<\infty \}}(r)+C_2{I}_{\{-\infty <r<0\}}(r)]}{\mid r{\mid }^{1+\alpha }}dr=-1.$$

$$\tau (\omega ,x)=\inf \{t\ge 0:X_t(\omega ,x)\notin D\},\omega \in \Omega ,$$

$$Au(x)$$

$$u(x)$$

$$p(x)=\mathbb{P}\{X_{\tau }(x)\in E\},$$

$$Ap(x)$$

$$p(x)$$

Full text

PACS:

87.18.Tt , 87.10.Mn , 87.18.Cf.

Keywords:

Asymmetric stable Lévy motions , non-Gaussian noise in gene regulation , likelihood for transcription, stochastic differential equations , stochastic bifurcation in transcription

**Noise plays a crucial role in gene regulation. It is a recent challenge to better understand how noise affects gene transcriptions and protein production. The bursty and intermittent transcription processes resemble the features of a stable Lévy motion. As a non-Gaussian stochastic process, a stable Lévy motion is characterized by its skewness (i.e., asymmetry) and non-Gaussianity indexes. In extension to former results, we study here the impact of asymmetric stable Lévy fluctuations on the likelihood of transcriptions in a prototypical gene regulatory system. We find that the interaction between the system’s nonlinearity and fluctuations induces various possibilities for transcriptions. We discover certain effects of the asymmetry index and other noise parameters on the likelihood for transcriptions. Hence we are able to select combinations of these parameters, in order to achieve the desired likelihood for transcription. **

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1 Introduction

Gene regulation is a crucial but noisy biological process [1, 2]. The significance of noise in genetic networks has been recognized and studied [3, 4, 5, 6, 7, 8, 9, 10]. It has been shown recently that noise is vital for regime transitions in gene regulatory systems [11, 12, 13, 14, 15]. In these works, noisy fluctuations are, however, taken to have Gaussian distributions only [6, 16, 17, 18, 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, 21, 22, 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, 25, 26, 27, 28, 29, 30]. This intermittent manner [31, 32, 33, 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, 38, 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 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 Section 2, we briefly describe a genetic regulation model with noisy fluctuations in the synthesis reaction rate. In Section 3, we recall basic facts about asymmetric stable Lévy motions. In Section 4, 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 Section 5, we study the multiplicative asymmetric Lévy motion case. Finally, we make some concluding remarks in Section 6. The Appendix contains the mathematical formulation for the first mean exit time and the first escape probability, in terms of deterministic integral-differential equations.

2 A stochastic genetic regulatory system

Smolen et al. [40] introduced the following model for the concentration ‘$x$’ of the transcription factor activator (TF-A)

[TABLE]

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, 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 TF-A dimer concentration to a maximal rate $k_{f}$, TF-A degrades with first-order kinetics with the rate $k_{d}$, and 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

[TABLE]

the model equation (1) becomes

[TABLE]

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

[TABLE]

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: $K_{d}=10$, $k_{d}=1min^{-1}$, $k_{f}=6min^{-1}$, and $R_{bas}=0.4min^{-1}$. Then the two stable states are

[TABLE]

and the unstable state (a saddle) is

[TABLE]

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 behaves like bursty perturbations as we discussed in the introduction. Therefore, we incorporate an asymmetric stable Lévy motion as a random perturbation of the synthesis rate $R_{bas}$. Thus the model (1) becomes the following stochastic gene regulation model:

[TABLE]

The effect of multiplicative noise has been investigated in the literature [41, 42, 43]. Here we also consider that the synthesis rate $R_{bas}$ is perturbed by a multiplicative asymmetric Lévy motion as follows:

[TABLE]

where $L_{t}^{\alpha,\beta}$ is an asymmetric stable Lévy motion with the jump measure $\nu_{\alpha,\beta}$, for the non-Gaussianity index $0<\alpha<2$ and skewness index $-1\leq\beta\leq 1$. This Lévy motion will be recalled in the next section. Here we assume that the generating triplet of asymmetric stable Lévy motion is $(0,0,\varepsilon\nu_{\alpha,\beta})$, where $\varepsilon$ is the noise intensity. The noise intensity plays an important role in the noise source [44, 45, 46, 47]. We will discuss the effects of noise intensity $\varepsilon$ in section 5. 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 the TF-A monomer concentration in this model. We focus on the likelihood for the TF-A monomer concentration transitions from the low concentration domain $D$ to the high concentration domain $E=[x_{u},+\infty)$ (the high concentration domain), via analyzing two deterministic quantities: the mean residence time (also called mean first exit time) in the 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 the high degree of activity. That is, high concentration indicates effective transcription and translation activities.

3 Asymmetric stable Lévy motion

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

[TABLE]

The asymmetric stable Lévy motion $L_{t}^{\alpha,\beta}$ is a stochastic process defined on a sample space $\Omega$ equipped with probability $\mathbb{P}$. It has independent and stationary increments, together with stochastically continuous sample paths: for each $s$, $L_{t}^{\alpha,\beta}\rightarrow$ $L_{s}^{\alpha,\beta}$ in probability. This means for every $\delta>0$, $\mathbb{P}$(\mid$$L_{t}^{\alpha,\beta}-L_{s}^{\alpha,\beta}$$\mid $>$ $\delta$)$\rightarrow 0$, as $t\rightarrow s$.

The jump measure, which describes jump intensity and size for sample paths, for the asymmetric Lévy motion $L_{t}^{\alpha,\beta}$ is [37, 38],

[TABLE]

with $C_{1}=\frac{H_{\alpha}(1+\beta)}{2}$, $C_{2}=\frac{H_{\alpha}(1-\beta)}{2}.$ When $\alpha=1$, $H_{\alpha}=\frac{2}{\pi}$; when $\alpha\neq 1$, $H_{\alpha}=\frac{\alpha(1-\alpha)}{\Gamma(2-\alpha)\cos(\frac{\pi\alpha}{2})}.$

Especially for $\beta=0$, this is the symmetric stable Lévy motion, which is usually denoted by $L_{t}^{\alpha}\triangleq L_{t}^{\alpha,0}$. The well-known Brownian motion $B_{t}$ may be regarded as a special case (i.e., Gaussian case) corresponding to $\alpha=2$ (and $\beta=0$); see [38].

We can see a clear difference to the symmetric case from Figure 3, which shows the probability density functions for $L_{t}^{\alpha,\beta}$ at $t=1$ for various $\alpha,\beta$.

For the stable Lévy motion with the jump measure in (4), the number of larger jumps for small $\alpha$ $(0<\alpha<1)$ are more than that for large $\alpha$ $(1<\alpha<2)$, while the number of smaller jumps for $0<\alpha<1$ are less than that for $1<\alpha<2$, as known in [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., (Appendix) and (Appendix), respectively, in Appendix at the end of this paper.

4 Gene regulation with synthesis rate under additive asymmetric Lévy fluctuations

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

4.1 Numerical algorithms

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

[TABLE]

Here $D^{c}$ is the complement set of $D$ in $\mathbb{R}^{1}$.

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

[TABLE]

with $\nu_{\alpha,\beta}(dy)=\frac{C_{1}I_{\{0<y<\infty\}}(y)+C_{2}I_{\{-\infty<y<0\}}(y)}{\mid y\mid^{1+\alpha}}dy,$ $C_{1}=\frac{H_{\alpha}(1+\beta)}{2}$ and $C_{2}=\frac{H_{\alpha}(1-\beta)}{2}.$ When $\alpha=1$, $H_{\alpha}=\frac{2}{\pi}$; when $\alpha\neq 1$, $H_{\alpha}=\frac{\alpha(1-\alpha)}{\Gamma(2-\alpha)\cos(\frac{\pi\alpha}{2})}.$ Additionally,

[TABLE]

The MFET $u$ satisfies the following equation:

[TABLE]

On an open interval $D=(a,b)$, we make a coordinate conversion $x=\frac{b-a}{2}s+\frac{b+a}{2}$ for $s\in[-1,1]$ and $y=\frac{b-a}{2}r$, to get finite difference discretization for $Au(x)=-1$ as in [48]:

[TABLE]

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

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

4.2 Numerical experiments

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 ($\beta=0$) in [36].

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

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

Figure 5 shows that when $\beta$ is fixed, the MFET values decrease with the increasing $\alpha$, i.e., the likelihood for gene transcription increases with increasing $\alpha$. In comparison, Figure 4(b) contains the case with Brownian noise (i.e., corresponding to $\alpha=2,\beta=0$) and the MFET values break this monotonicity and stay roughly between those for $\alpha=1.5$ and $\alpha=1.9$. Figure 4 and Figure5 indicate that if we start in the low concentration, then increasing $\alpha$ and $\beta$ values leads to the higher concentrations, corresponding to higher likelihood for transcription.

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

**When skewness $\beta\neq 0$: It makes a great difference on MFET for $\alpha<1$ and $\alpha>1$. ** Figure 7 exhibits that, when $\beta\neq 0$, MFET has a bifurcation or discontinuity point at $\alpha=1$ when $\beta\neq 0$. We can see that the MFET has a ‘phase transition’ or bifurcation at the critical non-Gasussian index value $\alpha=1$. This result is consistent with a theoretical analysis in [49]. When the asymmetry index $\beta\neq 0$, in the low concentration region, MFET decreases with the increasing $\alpha$ for $0<\alpha<1$, while for $1<\alpha<2$, MFET increases firstly but then decreases with the increasing $\alpha$. In the symmetric Lévy nose case ($\beta=0$), MFET is decreasing for all $\alpha$ (no bifurcation). Hence in the asymmetric Lévy noise case ($\beta\neq 0$): We gain higher likelihood for transcription by increasing non-Gaussian index $\alpha\in(0,1)$, while for $\alpha\in(1,2)$ there is a specific $\alpha_{s}$ leading to the minimum likelihood for transcription.

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

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

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

As shown in Figure 10, we find that, when $\beta<0$, FEP deceases with the increasing $\alpha$ for initial concentration $x<x_{-}$, then increases with the increasing $\alpha$ for $x_{-}<x<x_{u}$. This leads to the conclusion that larger initial concentrations are more likely leading to the transcription. If we consider FEP at the low concentration $x=0.62685$, we see that when $\beta<0$, FEP increases with the increasing $\alpha$, while when $\beta\geq 0$, FEP decreases with the increasing $\alpha$. A small $\alpha$ (and $\beta>0$) or a large $\alpha$ (and $\beta<0$) contributes to large FEP (i.e., more likely for transcription).

**FEP has ‘turning points’ with respect to $\alpha,\beta$. ** Figure 11 (a) exhibits that FEP increases with the increasing $\beta$, i.e., the likelihood for transcription improves with increasing $\beta$, when the system starts in low concentrations. When starting system at low stable concentration $x=0.62685$, we find that the evolution of FEP has ‘turning points’ for $\beta=\beta_{turning}\approx-0.5$ (this threshold value varies slightly with various $\alpha$). As shown in Figure 11 (b), before and after a turning point $\beta_{turning}$, FEP presents a reverse relationship: Higher FEP for larger $\alpha$ suddenly switches to higher FEP for smaller $\alpha$. That is, the higher likelihood for transcription is attained for larger non-Gaussianity index $\alpha$ before the turning point $\beta_{turning}$, 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 $\{(\alpha,\beta)\}=(0,2)\times[-1,1]$. ** Figure 12 displays the combined effects of both non-Gaussianity index $\alpha$ and skewness index $\beta$ on MFET $u$ and FEP $p$, at the initial concentration $x=x_{-}\approx 0.62685$ (i.e., the lower stable state).

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

In Figure 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 $\alpha$ and big skewness index $\beta$ (i.e., $\alpha\in(0,0.7)$ and $\beta\in(0.5,1)$) lead to 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 $\alpha$ and skewness index $\beta$ appropriately.

5 Gene regulation with synthesis rate under multiplicative asymmetric Lévy fluctuations

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

5.1 Numerical algorithms

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

[TABLE]

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

[TABLE]

where

[TABLE]

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

[TABLE]

where,

[TABLE]

For an open interval $D=(a,b)$ (in our computations below, we will take $D=(0,x_{u})$), we make a coordinate transformation $x=\frac{b-a}{2}s+\frac{b+a}{2}$ for $s\in[-1,1]$ and $z=\frac{b-a}{2}r$ to get finite difference discretization for $Au(x)=-1$ as in [48]:

[TABLE]

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

5.2 Numerical experiments

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 the previous section.

Figure 13 plots the evolution of MFET under multiplicative asymmetric Lévy motion. From Figure 13(a), we observe that for $\alpha=1.5$, MFET decreases with the increase of the initial concentration $x$ in the low concentration domain. We also find that the MFET becomes short if we tune the skewness index $\beta$ 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. MFET increases with the increase of non-Gaussianity index $\alpha$.

Figure 14 indicates the effects of $\alpha$ and $\beta$ on MFET for various initial concentrations $x$. Figure 14(a) shows the dependency in the low concentration on the skewness index $\beta$. As is shown for the case of $\alpha=0.5$, MFET decreases with the increase of $\beta$, and the higher initial concentrations correspond to 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 $\alpha=1$ like in the additive asymmetric Lévy case. For $0<\alpha<1$, MFET increases firstly then decreases when near to $\alpha=1$, however, for $1<\alpha<2$, MFET increases all the way very quickly. The maximum value is reached at $\alpha$ close to $2$.

As shown in Figure 15(a), when $\alpha=0.5$, FEP increases with the increase of $\beta$. So the large positive $\beta$ can induce larger FEP. From Figure 15(b), we see that FEP increases in the low concentration domain with the increase of $x$, and FEP with different $\alpha$ has intersections. Especially, the parts of the FEP for $\alpha=0.5$ overlaps with that for $\alpha=1.0$. 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 $\alpha$ and $\beta$ on FEP for various initial concentrations $x$. Figure 16(a) shows the FEP with respect to $\beta$ for various initial concentrations $x$. FEP increases as $\beta$ increases. Figure 16(b) shows the FEP with respect to $\alpha$ for various initial concentrations $x$. It presents that $\alpha=1$ is also a bifurcation point for FEP. This is totally different from that in additive asymmetric Lévy case (see Figure 10). We can see that for $0<\alpha<1$, FEP decreases with small initial concentrations $x$, while increases firstly then decreases with large initial concentrations $x$. For $1<\alpha<2$, FEP decreases with $x$ in the low concentration domain with the increase of $\alpha$.

Figure 17 demonstrates that the effects of noise intensity $\varepsilon$ on MFET and FEP. Figure 17(a) shows us that the larger $\varepsilon$, the smaller MFET. This indicates that large noise intensity helps to exit from the low concentration domain. Figure 17(b) shows that the FEP with various noise intensity $\varepsilon$. 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, FEP presents a reverse relationship with the initial concentration $x$. After the crossing point, FEP increases with the increase of $\varepsilon$.

We plot the MFET as a function of noise intensity $\varepsilon$ in Figure 18. Inspired by the literatures [50, 51, 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 is $[0,2]$. We could see that the MFET has a monotonic behavior with the noise intensity $\varepsilon$. The inflection point value of MFET decreases with the increase of the initial value. When MFET passes the inflection point, it changes more modestly. Comparing Figure 18 and Figure 18, we could observe clearly that the smaller $\alpha$ makes the MFET shorter. The results are shown as a character of noise enhanced stability effect [50, 51, 52].

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

6 Discussion

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 noise are considered in this paper. We have examined possible switches or transitions from the low concentration states to the high concentration ones (i.e., likelihood for transcriptions), excited by 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) $\beta$ and a small non-Gaussianity index $\alpha$. In contrast to the symmetric case, we have observed a bifurcation for the likelihood of transcription at the critical value $\alpha=1$ under asymmetric stable Lévy noise ($\beta\neq 0$), as shown in Figure 7, Figure 14 and Figure 16. Comparing Figure 12 and Figure 19, we see a striking difference between additive and multiplicative noises. There is also a turning point in the skewness index $\beta$ for the likelihood of transcription, as seen in Figure 11 (b). The bifurcation and turning point phenomena do not occur in the symmetric noise case ($\beta=0$).

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

Acknowledgements:

We would like to thank Dr. Xiao Wang for helpful discussions.

Appendix

**1. Mean first exit time **

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

[TABLE]

where $X_{t}(\omega,x)$ is the solution orbit of the stochastic differential equation (2), starting with the initial TF-A concentration $x$. Then the MFET is denoted as $u(x)=\mathbb{E}\tau(\omega,x)$. Here the mean $\mathbb{E}$ is taken with respect to the probability $\mathbb{P}$. The MFET $u(x)$ of the solution orbit $X_{t}(\omega,x)$, 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 $Au=\lim_{t\rightarrow 0}\frac{P_{t}u-u}{t}$, where $P_{t}u(x)=\mathbb{E}u(X_{t})$. The generator $A$ for the gene regulatory system (2) is explicitly given in (6), Section 4.1. Then the mean exit time $u$ satisfies the following nonlocal equation [38] with an exterior boundary condition

[TABLE]

Here $D^{c}$ is the complement set of $D$ in $\mathbb{R}^{1}$.

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.

**2. First escape probability **

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,

[TABLE]

where $\tau$ is the exit time from $D$, as in (A.1). This first escape probability $p$ satisfies the following nonlocal equation [38] with a special, exterior boundary condition:

[TABLE]

where $A$ is the generator for the stochastic differential equation (2), as in (6), Section 4.1.