Nonlinear Research Institute, Baoji University of Arts and Sciences, Baoji 721016, China

Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, China

National Center for Mathematics and Interdisciplinary Sciences, Chinese Academy of Sciences, Beijing 100190, China

Department of Physics and Institute of Biophysics, Huazhong Normal University, Wuhan 430071, China

Abstract

Background

Noise, nonlinear interactions, positive and negative feedbacks within signaling pathways, time delays, protein oligomerization, and crosstalk between different pathways are main characters in the regulatory of gene expression. However, only a single noise source or only delay time in the deterministic model is considered in the gene transcriptional regulatory system in previous researches. The combined effects of correlated noise and time delays on the gene regulatory model still remain not to be fully understood.

Results

The roles of time delay on gene switch and stochastic resonance are systematically explored based on a famous gene transcriptional regulatory model subject to correlated noise. Two cases, including linear time delay appearing in the degradation process (case I) and nonlinear time delay appearing in the synthesis process (case II) are considered, respectively. For case I: Our theoretical results show that time delay can induce gene switch, i.e., the TF-A monomer concentration shifts from the high concentration state to the low concentration state ("on"

Conclusions

The stochastic delay dynamic approach can identify key physiological control parameters to which the behavior of special genetic regulatory systems is particularly sensitive. Such parameters might provide targets for pharmacological intervention. Thus, it would be highly interesting to investigate if similar experimental techniques could be used to bring out the delay-induced switch and stochastic resonance in the stochastic gene transcriptional regulatory process.

Background

In recent years, a plenty of researches show that noises play a positive role in many fields. Many novel phenomena are found, such as noise induced transition

Regulation of gene expression by signals outside and inside the cell plays important roles in many biological processes. As the basic principles of genetic regulation have been characterized, it has become increasingly evident that nonlinear interactions, positive and negative feedback within signaling pathways, time delays, protein oligomerization, and crosstalk between different pathways need to be considered for understanding genetic regulation _{d }_{bas}

Stochastic resonance (SR), which was originally discovered by Benzi and Nicolis

We would like to emphasize that the combined effects of correlated noise and time delay on dynamical behaviors of gene regulatory network are rarely investigated. In this article, the statistical properties of gene switch and stochastic resonance induced by time delay in two different cases (i.e., linear time delay case and nonlinear time delay case) are explored. Our investigation is a significant try forward understanding the basic mechanisms of the delay induced gene switch and stochastic resonance in realistic yet complex organisms from a view of theory, and will motivate the further experimental research for gene network.

Model

Deterministic gene transcriptional regulatory model

To examine the capability of genetic regulatory systems for complex dynamic activity, Smolen _{f}_{bas}_{d}

Model of genetic regulation with a positive autoregulatory feedback loop

**Model of genetic regulation with a positive autoregulatory feedback loop**. The transcription factor activator (TF-A) activates transcription with a maximal rate _{f }_{d }_{bas}

where _{d }

The potential function corresponding to Eq.(1) is

Two stable steady states are presented as _{d }_{bas }+ k_{f}_{d}^{2 }/3, _{d}_{f }- 2_{bas}_{d}_{bas }_{f})/(_{d}^{3 }and

An interesting aspect of the model is that, based on the different initial conditions, the concentration of TF-A can be one of the two stable steady states. It is a bistable system for certain values of _{f }_{f }<_{f }_{d }_{d }_{bas }_{+ }≈_{u }≈

Bifurcation plot for the steady state of TF-A on the control parameter of transcription rate _{f}

**Bifurcation plot for the steady state of TF-A on the control parameter of transcription rate k**. The system in the region (i.e., 5.45

The bistable potential of Eq.(3)

**The bistable potential of Eq.(3)**. The parameter values are _{f }_{d }_{d }_{bas }

Stochastic model with correlated noise and time delay

Cells are intrinsically noisy biochemical reactors: low reactant numbers can lead to significant statistical fluctuations in molecule numbers and reaction rates _{bas }_{d }_{bas }_{d}. Namely _{bas }→ R_{bas }+ η_{d }_{d }+ ξ(

where ξ(

Where

In order to more exactly predict the dynamics of the genetic regulation model, it is necessary to consider macromolecular transport in these biochemical reactions. Transport can be diffusive or active, and in some cases a time delay may suffice to model active transport. Smolen _{f }_{d }

Case I: Linear time delay appearing in the degradation process

First, we consider the local time delay due to the degradation of TF-A in the nucleus. The simplest kinetic model of genetic regulation with the local time delay is described by **Case I **in Figure _{1 }appearing in the TF-A degradation process can affect the TF-A monomer concentration _{d }+ ξ_{d }+ ξ_{1}), and Eq. (4) is further rewritten:

Model of genetic regulation with a positive autoregulatory feedback loop and time delays

**Model of genetic regulation with a positive autoregulatory feedback loop and time delays**. **Case I: **time delay _{1 }in the degradation process of TF-A; **Case**** II****:** time delay _{2 }=

where the _{1 }(time delay) previous to the time when _{d}x

Case II: Nonlinear time delay appearing in the synthesis process

Second, the rate constant _{f }_{2 }=

Then,

where the first term on the right side is evaluated at a time _{2 }(delay time) previous to the time when

Below, the statistics properties of our theoretical model subjected to correlated noise and time delay are explored in the two different cases (i.e., linear time delay case and nonlinear time delay case). Considering the difficulties in theoretical analysis, we will investigate the two different time delays in the gene model, respectively.

Methods and results

Results for case I

Steady-state probability distribution

The small time delay approximation of the probability density approach is employed _{1}) in Eq.(9), we obtain

where

In Eq.(11)-(12),

where

with

In the steady-state regime (given by Eq.(2)) and under the constraint

where

The stationary probability distribution (SPD) corresponding to Eq. (19) is obtained

where

where

In the bistable region, the time course of TF-A monomer concentration _{1}. If we regard the low concentration state as the "off" state and the high concentration state as the "on" state, the above result indicates that a switch process can be induced by the delay time. Figure

Sample paths and probability distribution of _{1}

**Sample paths and probability distribution of x(t) for different delay time τ**. From top to bottom

Mean value

In order to quantitatively investigate the stationary properties of the system, we introduce the moments of the variable

The mean of the state variable

The theoretical and the numerical simulation results of 〈_{st }as a function of _{1 }is plotted in Figure _{st }is decreased with increasing _{1}. When _{1 }is small, the TF-A monomer concentrates on the high concentration state. When _{1 }is increased, the TF-A monomer concentrates on the low concentration state. Namely, for large _{1}, it is more easy to be at the "off" state (the low concentration state). The effect is similar to the effect of _{1 }on SPD shown in Figure

_{st }_{1 }with

** < x >**. The other parameter values are the same as those in Figure

Mean first passage time

For the delay time-induced switch, we will quantify the effects of delay time on the switch between the two stable steady states. When the system is stochastically bistable, a quantity of interest is the time from one state to the other state, which is often referred to as the first passage time. We consider the mean first passage time (MFPT). Here the MFPT of the process _{+ }

When the intensities of noises terms _{u}

Here, the potential _{0}(

By virtue of Eq.(28), the effects of _{1 }on the MFPT can be analyzed. MFPT as a function of _{1 }is plotted in Figure _{1 }increases. From the view point of physics, it means that the delay time can speed up the transition between the two steady states (low concentration state and high concentration state). Namely, the delay time can accelerate the transition of gene switch from "on" state to "off" state.

Effects of time delay on stochastic resonance

In the gene transcriptional regulatory process, the external environmental factors, such as the electromagnetic field on the earth, the solar terms and seasonal variation, are the common features. This means that the transcript of gene should have a periodic form. For simplicity, a cosinoidal form

Model of genetic regulation with a positive autoregulatory feedback loop, delay time and an additive signal

**Model of genetic regulation with a positive autoregulatory feedback loop, delay time and an additive signal Acos(Ω t)**.

where _{1 }is the delay time.

Signal to noise ratio

Making use of the small delay time approximation of the probability density approach and the stochastic equivalence method, the approximated delay Fokker-Planck equation of this model is given by

Under the constraint _{qst}

where _{n}

where _{0},_{1},_{2},_{3 }and _{4 }are given by Eq.(24). And

Since the frequency _{± }_{+}_{-},

in which _{+}_{-}, _{u }_{n}

For the general asymmetric nonlinear dynamical system, the SR phenomenon has been found, and the related theory has been developed

The system is subjected to a time dependent signal

where the constants _{1, 2 }and _{1, 2 }depend on the detailed structure of the system under study. For the asymmetric case, _{1 }≠ _{2 }and _{1 }= _{2}.

For the general asymmetric case we defined _{S N R}

where

According to the expression of SNR in Eq.(37), the effects of the additive noise intensity _{1 }on the SNR are analyzed. These results are plotted in Figures

_{S N R }

_{S N R }_{1 }= 0.1, _{1 }= 0.1,

The SNR as a function of multiplicative noise intensity _{1 }= 0.1, 0.3, 0.4 is plotted in Figure _{SNR }_{1 }increases, and the position of the peak shifts from the large _{SNR }_{1}. It must be pointed out that the observed SR is obvious when the additive noise intensity

The SNR as a function of the multiplicative noise intensity

The SNR as a function of the multiplicative noise intensity

Why these different control parameters exhibit various regulatory properties on the SR? One possible reason is that the potential function of the bistable gene model is adjusted differently. The symmetry of potential wells and the height of potential barrier have different dependences on these parameters. The quantitative analysis about the underlying mechanisms of time delay

In order to check the valid of our theoretical approximate method, the numerical simulation is performed by directly integrating the Eq.(28) with Eqs.(5)-(8). Using the Euler method, the numerical data of time series are calculated using a fast Fourier transform. To reduce the variance of the result, the 1024 ensembles of power spectra are averaged. The output signal-to-noise ratio is defined as _{p}_{s}_{n}_{s}_{p}_{s}_{s }_{n}_{s}_{s}. The parameters are chosen as the same value in the theoretical analysis. The results are plotted in Figure

Results for case II

When the time delay appears in the Hill function, Eq.(10) becomes a nonlinear time delay stochastic equation. It is difficult to deal with the small time delay approximate method from the aspect of the theory. Hence the following results are given by direct simulation for the stochastic delay differential equation, i.e., Eq.(10), which can be formally integrated by using a simple forward Eular algorithm with a small time step for time delay.

The forward Euler algorithm with a small time step

Where ^{6 }realizations of _{st}_{st }can be obtained and shown in Figures _{2 }is unlimited. But in the case I the time delay _{1 }is very small since the theoretical approximate method is only valid for small time delay _{1}.

The numerical simulations of the probability distribution _{st}_{2 }with _{2 }= 0.1, 0.5,1.0 and 2.0. The other parameter values are the same as those in Figure

**The numerical simulations of the probability distribution P**

The numerical simulations of (a) _{st }_{2}.

**The numerical simulations of (a) < x >**

Steady-state probability distribution

Figure _{2 }can be also used as a control parameter for the switch process in the genetic regulatory system. However, compared with case I, the time delay _{1 }induces the transition of gene switch from "on" to "off".

Mean value

The numerical results of the mean value of _{2 }are plotted in Figure _{2 }increasing. In summary, when the model incorporated a nonlinear time delay _{2 }and _{1 }play the opposite roles in our genetic regulatory process.

Mean first passage time

Similar, making use of the MFPT of the process _{+}_{-}, we can investigate the transition time from "on" state to "off" state. According to the definition of MFPT given by Hu _{2 }is shown in Figure _{2 }increases. Physically, it means that the delay time _{2 }can speed up the transition between the two steady states (low concentration state and high concentration state). Namely, the delay time can accelerate the transition of gene switch from "on" state to "off" state. The roles of _{1 }and _{2 }here are similar.

Stochastic resonance

Similar, we consider the gene transcriptional regulatory process subjected to a periodic signal _{2 }=

Model of genetic regulation with a positive autoregulatory feedback loop, an additive signal _{2 }=

**Model of genetic regulation with a positive autoregulatory feedback loop, an additive signal Acos(Ωt), and time delay τ**

where _{2 }is the delay time.

Applying the numerical simulation method of calculating signal to noise ratio given by Ref. _{2 }on the SR. The SNR is defined as the ratio of the peak height of the power spectral intensity to the height of the noisy background at the same frequency. Figure _{2 }= 0.1, 0.3, 0.5, when the other parameters are fixed. It is found that there is a single peak in _{SNR }_{2 }increases, and the position of the peak shifts from the small _{SNR }_{2}. It should be noted that _{2 }can restrain the SR to occur. Comparing Figure _{1 }and _{2 }on the SR is different. _{1 }can enhance the SR, but _{2 }can weaken the SR.

Numerical simulation results of _{S N R }_{2 }= 0.1, 0.3 and 0.5 with

**Numerical simulation results of R**

Conclusions

In this article, the regulatory properties of time delay on gene switch and stochastic resonance are systematically studied based on a bistable gene transcriptional regulatory model. This gene model is driven by the correlated noise and time delay simultaneously. Two cases, including linear time delay appearing in the degradation process (case I) and nonlinear time delay appearing in the synthesis process (case II) are considered, respectively. We mainly focus our research on two aspects, i.e., the dynamical switch characters (including the steady probability distribution, the mean value and the mean first passage time) and the stochastic resonance phenomenon.

For case I: Our theoretical results show that (i) the delay time _{1 }resulting from the degradation process can induce the gene switch process, i.e., the TF-A monomer concentration _{1 }can further speed up the transition from "on" to "off" state. (ii) The stochastic resonance can be enhanced by the time delay _{1 }and the correlated noise intensity λ. However, the additive noise original from the synthesis rate _{bas }

For case II: Our numerical simulation results show that time delay _{2 }can also induce the gene switch, while different from case I, the TF-A monomer concentration shifts from the low concentration state to the high concentration state ("off"→ "on"). The time delays in two cases play the opposite roles. With increasing the time delay _{2}, the transition from "on" to "off" state can be further accelerated, which is similar to case I. Moreover, it is found that the stochastic resonance can be weaken by the time delay _{2}. These insights on the combined effects of noises and time delay would be beneficial to understanding the basic mechanism of how living systems optimally facilitate to function under real environments.

The main result of our works is the time delays in both case I and case II induce gene switch, and the switch process can be further accelerated with increasing time delay. In order to demonstrate this theoretical result, an example is provided by using a biological system, i.e., the inducible lac genetic switch for Escherichia coli cells

To test our predictions quantitatively, one would ideally like to perform an experiment on this gene transcriptional regulatory model with tunable time delay and noise intensity, in which all the parameters concentrations of components and rate constants are the same as our theoretical model. To our knowledge, this clearly seems a very difficult experiment to perform, what we do is to give a primary picture of the integrated effects of time delay and noise. Recently, with the development of synthetic biology, some artificial gene networks are designed by genetic engineer. Moreover, it is increasingly being recognized that some biological parameters, including time delay and feedback strength, can be controlled by using micro-fluidic devices in gene regulatory network. So we wish that the time delay-accelerated transition of gene switch and time delay-enhanced or suppressed stochastic resonance could be examined in future.

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

CJ Wang and M Yi conceived and designed the structure of this work. CJ Wang, KL Yang and LJ Yang performed the numerical experiments and analyzed the data. CJ Wang and M Yi wrote the paper. All authors have read and approved the final manuscript.

Acknowledgements

This project was supported by the Natural Science Foundation of China (Grant No.11047146 (CJ Wang), Grant No.10905089 (M Yi) and Grant No.11047107 and No.11105058 (LJ Yang)), the Natural Science Foundation of Shaanxi province of China (Grant No.2010JQ1014 (CJ Wang)) and the Science Foundation of Baoji University of Science and Arts of China (Grant No.ZK11053 (CJ Wang)).

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