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Edited by: Francisco Monroy, Complutense University of Madrid, Spain

Reviewed by: Tianshou Zhou, Sun Yat-sen University, China; Juan Belmonte, Universidad de Castilla-La Mancha (UCLM), Spain

*Correspondence: Ming Yi

This article was submitted to Biophysics, a section of the journal Frontiers in Physiology

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

Coherent feed-forward loops exist extensively in realistic biological regulatory systems, and are common signaling motifs. Here, we study the characteristics and the propagation mechanism of the output noise in a coherent feed-forward transcriptional regulatory loop that can be divided into a main road and branch. Using the linear noise approximation, we derive analytical formulae for the total noise of the full loop, the noise of the branch, and the noise of the main road, which are verified by the Gillespie algorithm. Importantly, we find that (i) compared with the branch motif or the main road motif, the full motif can effectively attenuate the output noise level; (ii) there is a transition point of system state such that the noise of the main road is dominated when the underlying system is below this point, whereas the noise of the branch is dominated when the system is beyond the point. The entire analysis reveals the mechanism of how the noise is generated and propagated in a simple yet representative signaling module.

The biological world is filled with interaction of deterministic laws and randomness (Monod,

A prominent feature of gene transcription regulatory networks is the presence of a large number of motifs, i.e., patterns of interconnection. These motifs include the auto-regulation loop, feedback loop, feed-forward loop and so on. An important task in the post-genome era is to understand how these different regulation mechanisms of expression noise affect the functioning of cells and how they contribute to cell-to-cell variability. Our ultimate purpose is to understand how the characteristics of noise in the complex networks can be derived from the properties of modules that are used to compose these networks.

It has been recognized that feedback loops play significant roles in a variety of biological processes, such as calcium signaling (Berridge,

It is well-known that the feed-forward loop is also a typical of biological motif. The feed-forward loop, a pattern of three genes, consists of two input transcription factors, one of which regulates the other, conjointly regulating a target gene. Each of the interactions of three genes in the feed-forward loop can be activation or inhibition so that the feed-forward loop has eight possible structural types. Among them, the coherent feed-forward loop appears with the highest frequency in the organism (Mangan and Alon,

Few studies, however, focused attentions on the effect of feed-forward on expression noise in biochemical systems. The current study has ever studied the mechanisms of feed-forward regulation in cell fate decisions in budding yeast (Li et al.,

To address the above questions, fluctuation and noise propagation in the coherent feed-forward transcriptional regulatory loop are investigated in this paper. Our motivation is to clarify the potential relationships between network structure, noise characteristics, and biological function. The main contribution of our study is that we decompose the expression noise of each element in the coherent feed-forward loop into different noise sources (denoted as fine structure here). We believe that our study presents a possible understanding for why has the biological system evolved into a coherent feed-forward regulatory mechanism. This paper is organized as follows. In Section Mathematical modeling and analytical noise, the mathematical models of coherent feed-forward loop, including its main road and branch subsystems, are presented first. Then the related theoretical methods are introduced to calculate variances and normalized variations of each expression production in these motifs. Further, in Section Results, we analyse the fluctuation and noise propagation in the coherent feed-forward loop and its subsystems. Finally, the conclusions and discussions are given in Section Conclusion.

A coherent feed-forward loop is composed of three components: Two transcription factors X and Y, where the former regulates the latter, and a target gene Z, where X and Y both bind the regulatory region of Z and jointly modulate the transcription rate (Figure

Based on the biochemical reaction rules, the mathematical models of these corresponding gene regulation motifs are built by virtue of ordinary differential equations. Two conditions are considered, respectively.

Treating the level of X as an adjustable parameter.

First, the expression process of X is neglected and the concentration of X is regarded as the control parameter. Hence, the model of coherent feed-forward loop has only two variables. The deterministic dynamics can be described by the following equations:

where _{y} and α_{z} are the maximum level of activated protein production for Y and Z, which are set to ten, respectively. _{y} and δ_{z}, which are set to unity, respectively. The model of the branch is defined by ω_{1} = 0, ω_{2} = 1, the main road by ω_{1} = 1 and ω_{2} = 0, and the coherent feed-forward loop by ω_{1} = 1 and ω_{2} = 1.

For equations (1) and (2), there exists an attracting fixed point

Treating the level of X as a variable.

Then, we consider the birth and death processes of X. Therefore, the model of coherent feed-forward loop has three variables whose dynamics can be described by the following equations:

Similarly, the branch is defined by ω_{1} = 0, ω_{2} = 1, the main road by ω_{1} = 1 and ω_{2} = 0, and the coherent feed-forward loop by ω_{1} = 1and ω_{2} = 1. We will consider α_{x} as a control parameter under this condition. α_{y} = α_{z} = 10, δ_{x} = δ_{y} = δ_{z} = 1.

Equations (3–5) have an attracting fixed point _{x} is five. The three-dimensional phase diagram (Figure

Treating the level of X as an adjustable parameter.

Under small molecular numbers, the stochastic noise can be described exactly with the master equations. A theoretical derivation about the noise expression is given below, starting from the master equations. The numbers of the expression productions can be expressed as _{i}, _{y} = Ω_{z} = Ω

The joint probability distribution _{y}, _{z},

where the symbol ^{1/2} in the expansion of Equation (6) reproduces the concentration form of the macroscopic rate equation and the terms of Ω^{0} get a linear Fokker-Planck equation,

Thus,

The linear noise approximation is summarized by (Elf and Ehrenberg,

where matrix _{yy}, _{zz},which characterizes the fluctuation in Y and Z. Substituting Equations (8) and (9) into Equation (10), we obtain

To quantify the noise propagation around the steady state, Equation (10) is normalized as

To measure how the balance between production and elimination of _{i} is affected by _{k} (Paulsson, _{i} and _{k} represent the numbers of the expression productions, respectively. _{ik} represents a common method of the sensitivity of a response to changes in parameter _{k}, also known as logarithmic gain (Savageau,

Under the steady state (i.e., _{i} is determined by the total rate of elimination,

Then its normalized formation

In order to study noise propagation, we substitute Equation (15) into Equation (16), and then solve Equation (14) for the normalized variations _{yy}, and _{zz}. Hence we have

It is seen that the upstream X has no contribution to the total noise in downstream Z, while the intermediate Y transmits a part of expression noise to Z, as shown in Figure

Equations (1) and (2) can be translated into the following set of birth-death processes:

where Equations (19) and (20) describe the production of Y and Z,respectively. In Equations (19),

Treating the level of X as a variable.

The joint probability distribution _{x}, _{y}, _{z},

Thus,

The linear noise approximation is summarized by ^{T} + Ω

Taking into account the constants of both the self-proliferation rates and the death rates, the logarithmic gain _{ik} can be obtained

Under the steady state, the average lifetime τ_{i} is determined by the total rate of elimination, _{xx}, _{yy}, and _{zz} here, and then we have

We can see that, accompanied by signal transduction in the network, the noise is also transmitted along these pathways. The upstream X has contributions to the total noise in Y and Z via one-step (branch road) and two-step (main road) propagation, respectively (see Figure _{two−steps propagation noise from X} on the right describes the two-step propagation noise from X to Z.

Based on the theoretical formulas of variances and normalized variations obtained above, we can make a further analysis about how noise is transmitted in the coherent feed-forward loop. Some interesting results are observed.

In order to illustrate noise characteristic of different nodes in these small gene networks, as observed in Equations (18)and(35), we investigate the normalized variations of each element in a variety of situations, which can present a global feature.

When the concentration of X is constant and regarded as a control parameter, the normalized variations _{yy} and _{zz} under different conditions are given in Figures

When we treat the production rate α_{x} as a control parameter, both X and Y have contributions to the total noise of Z. The noise curves of X are calculated and supplied in Figures _{x} is less than certain threshold for all cases. When α_{x} increases to a certain value, the concentration of X increases, hence the intrinsic noise of X is reduced. It is illustrated that, in the region below a certain critical point, the total noise level of downstream component is the smallest compared with other upstream components. Basically, if the stochastic birth-death process of X is involved, the noise levels are enhanced, but the critical points are only modified slightly.

Because the target gene Z is a downstream gene and can be considered as the system's output, we focus on the noise characteristics of gene Z. A comparison between the normalized variations _{zz} for different motifs is plotted in Figure _{x} is small. Our theoretical findings reveal that the multi-step process is beneficial for signal transmission from X to Z, i.e., the coherent feed-forward loop can attenuate effectively the noise level of downstream gene. Therefore, living organisms could utilize feed forward for better survival in fluctuating environments.

For the one-step cascade, signal X can be propagated directly to Z, however, the noise level is always larger than that of the other motifs. In addition, the two noise curves of one-step cascade and two-step cascade intersect at a certain point.

As is mentioned above, the total noise can be decomposed into a series of noise terms, and each noise term represents different sources. As observed in Equation (18), when X is considered as an adjustable constant, the total noise in Z consists of two noise components, in which the first one is pure intrinsic noise in Z, the second one is the fluctuation propagated from the neighbor Y (i.e., one-step noise propagation). However, as seen in Equation (35), if we treat X as a variable, then the total noise in Z contains more noise sources, including pure intrinsic noise of Z, one-step propagation noise from X (X regulates Z directly via feed forward loop), one-step propagation noise from neighboring Y, and two-step propagation noise from X (X modulates Z indirectly through Y). In order to study noise propagation in the coherent feed-forward loop, the total noise and different-step noise sources are theoretically discussed in Equations (18) and (35).

Case (i): The concentration of X is an adjustable constant. The inset in Figure

Case (ii): The concentration of X is a variable. The inset in Figure

In order to study the propagation feature of the intrinsic expression noise along the direction of signaling transduction, we add the one-step and two-step noise terms of X together. Then, the contribution of noise propagation from X to Z is compared with the contribution from Y to Z, as shown in Figure

We further consider how the expression noise of upstream component X is transferred to downstream Z via the main road and branch. The curves of noises propagated through main road and branch are plotted in Figure

A population of genetically identical cells exposing to the same extracellular environment may exhibit considerable noise in the mRNA or protein level. This cell-to-cell noise is generated largely due to the limited number of reacting molecules such as gene copies, mRNA, or proteins. Understanding the dynamics of noise propagation in gene regulation systems is an important question on noise analysis in biophysics and system biology.

In this paper, we studied how the expression noise is propagated through a coherent feed-forward loop. First we established a toy yet representative model of gene regulation with feed-forward. Then the theoretical formulas for noise propagation were derived by using the linear noise approximation of master equation and logarithmic gain. We have analytically shown that the total noise is a simple sum of different noise sources including the intrinsic noise and the transmitted noise from other elements in the loop. Therefore, signal transmission is accompanied by noise propagation. In principle, the sub-processes in signal transduction may also contribute to the total noise. By analyzing this decomposition of expression noise, we have further obtained some interesting results about noise characteristics and propagation.

Compared with other upstream components, the noise level of downstream component is smaller in the coherent feed-forward loop due to the addition of branch. The multi-channel process in the coherent feed-forward loop is advantageous to the propagation of signals, which means that the expression noise level of downstream gene can be reduced due to feed forward loop. Our finding may present a clue to understand why the fate decision system in budding yeast would evolve into a coherent feed forward structure (Li et al.,

The main noise source in the total noise for downstream component is intrinsic noise. The noise propagated from upstream factor is weaker than the noise transmitted from intermediate component when the system is below this critical point. By comparing the different noise contributions of upstream factors, a transition point in the coherent feed-forward loop is observed. When the system is below this transition point, the noise of the main road is relatively higher while the noise of branch is higher when the system is beyond this point. The two-phase mechanism could be advantageous to the signal propagation.

Phosphotransferase system (PTS) is a signaling network in bacteria, responsible for sensing and using a certain nutrient. In the PTS, enzyme I (EI) is first autophosphorylated and then transfers the phosphoryl group to enzyme IIA-Glucose (EIIA^{Glc}) via enzyme histidine phosphorcarrier (HPr). As an alternative pathway, we found that, EIIA^{Glc} can be phosphorylated directly by EI in the absence of HPr. Therefore, the PTS is similar to the above feed-forward system. Recently, using the paramagnetic NMR spectroscopy and mathematical model, we have investigated the mechanism of phosphoryl transfer between the EI and EIIA^{Glc} and demonstrated the physical basis for their ultraweak interaction (Xing et al.,

Our research has clarified the potential relationships between feed-forward structure, noise characteristics, and signal transduction. By comparing the properties of three different motifs, we easily find that the feed-forward motif is a best design for signal transduction because it excels in the noise-reduction function. So far, it is the first time to clarify noise characteristics and propagation mechanism in a coherent feed-forward loop by analytical method. The ability to dissect theoretically noise propagation through complex biological networks enables the researchers to understand the role of noise in function and evolution. Our work provides a preliminary result for noise decomposition in gene regulation circuits. However, there still exist some deficiencies in our theoretical work. (i) Because the coherent feed-forward loop has the highest abundance in nature, we have only analyzed this form of coherent feed-forward loop. It is worthy to study the noise transmission in other feed-forward loops. (ii) Due to lack of experimental data, it is difficult to build a quantitative coherent feed-forward loop model. Only a coarse model of coherent feed-forward loop is used to explore the qualitative behaviors. For example, in the reaction process, the Hill coefficient is equal to 1 without taking into account the case of N greater than 1. (iii) We only select one set of parameter values to make a simple analysis. We can see the fine structure of total noise for each component, but the noise level is very small, especially the first-step propagation noise via branch.

Conceived and designed the experiments: RG, YJ, and MY. Performed the experiments: RG, QL, YY, HD, and CM. Analyzed the data: RG, QL, YY, HD, CM, YJ, and MY. Contributed reagents/materials/analysis tools: RG, QL, YY, HD, and CM. Wrote the paper: RG, YJ, and MY.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

This work was supported by the Fundamental Research Funds for the Central Universities (Grant No. 2662015QC041 and 2662014BQ069), Huazhong Agricultural University Scientific & Technological Self-innovation Foundation (Program No. 2015RC021), and the National Natural Science Foundation of China (Grant No. 11675060, No.11547244, No.91330113, No.11275259, No.11175068, No.11474117).