Feedback modules, which appear ubiquitously in biological regulations, are often subject to disturbances from the input, leading to fluctuations in the output. Thus, the question becomes how a feedback system can produce a faithful response with a noisy input. We employed multiple time scale analysis, Fluctuation Dissipation Theorem, linear stability, and numerical simulations to investigate a module with one positive feedback loop driven by an external stimulus, and we obtained a critical quantity in noise attenuation, termed as “signed activation time”. We then studied the signed activation time for a system of two positive feedback loops, a system of one positive feedback loop and one negative feedback loop, and six other existing biological models consisting of multiple components along with positive and negative feedback loops. An inverse relationship is found between the noise amplification rate and the signed activation time, defined as the difference between the deactivation and activation time scales of the noise-free system, normalized by the frequency of noises presented in the input. Thus, the combination of fast activation and slow deactivation provides the best noise attenuation, and it can be attained in a single positive feedback loop system. An additional positive feedback loop often leads to a marked decrease in activation time, decrease or slight increase of deactivation time and allows larger kinetic rate variations for slow deactivation and fast activation. On the other hand, a negative feedback loop may increase the activation and deactivation times. The negative relationship between the noise amplification rate and the signed activation time also holds for the six other biological models with multiple components and feedback loops. This principle may be applicable to other feedback systems.
It has been identified that feedback loops play important roles in a variety of biological processes, such as calcium signaling [1], [2], p53 regulation [3], galactose regulation [4], cell cycle [5]–[8], and budding yeast polarization [9]–[13]. Although the detailed regulation of feedback loops may vary in different systems, the overall functions of feedback loop modules may be similar. For example, positive feedback loops are mainly used for promoting bi-stable switches and amplifying signals. One example is the cell cycle system [5]–[8] in which the mitotic regulator CDK1 activates Cdc25, which in turn activates CDK1, forming a positive feedback loop. Conversely, Wee1 and CDK1 inactivate each other, forming a double-negative feedback loop, equivalent to a positive feedback loop. The overall positive feedback regulation gives rise to a bi-stable switch that toggles between the inter-phase state and the mitotic-phase state. Another example is the system of yeast mating [9]–[15], in which multi-stage positive feedback loops enable the localization of signaling molecules at the plasma membrane by amplifying signals to initiate cell polarization and mating. While most studies of feedback loops have been concerned with their roles in signal amplification, switch (or switch-like) responses [16]–[20], and oscillations [21] (See [22], [23] for the latest review.), recently, another important aspect of feedback loops has drawn more and more attention: modulating (accelerating or delaying) timing of signal responses [22], [24], [25]. Intuitively, positive feedback could amplify signals inducing an expeditious activation, or delay an activation by setting a higher threshold such that the system is activated only when the response accumulates beyond that threshold [22], [25]. Because characteristics of noises (e. g. , the temporal frequency of a noise) in a biological process are closely related to timing of a signaling system, feedbacks clearly play a critical role in noise attenuation [26]–[29]. Thus, one of the central questions on noise analysis is how the architecture of a feedback circuit affects its noise property. Some studies suggested that positive feedbacks tended to amplify noise and negative feedbacks typically attenuated noise [30]–[32]; on the other hand, some other studies demonstrated that the positive feedbacks could attenuate noises and there were no strong correlations between the sign of feedbacks (negative or positive) and the noise attenuation properties [28], [33]. In their novel work [34], Brandman et al. linked the effect of positive feedback loops on noise attenuation to the time scales of the feedback loops. They studied a canonical feedback module consisting of three components, i. e. , an output and two positive feedback loops, and. The output is turned on by the two positive feedback loops and, which are stimulated by an external (or upstream) stimulus and are also facilitated by (Figure 1A). The output becomes active (or stays inactive) as the pulse stimulus is high (or low). Through numerical simulations, Brandman et al. [34] showed that, if one of the positive feedback loops (e. g. , loop) was slow and the other one was fast (termed as dual-time loops), the system could lead to distinct active output even in the presence of noise in the stimulus (at the high state). Following this work, Zhang et al. [35] studied dual-time loops in producing a bi-stable response with a constant input (unlike a pulse input in [34]). They concluded that dual-time loops were the most robust design among all combinations in producing bi-stable output for a slightly different system in which the stimulus could activate or without the participation of. Kim et al. [36] considered systems coupled with negative and positive feedback loops. By assuming all the positive feedback loops have the same time scale but different time delays, they obtained a system that was capable of performing fast activation, fast deactivation, and noise attenuation. What remains unclear are the sufficient and necessary conditions for a feedback system to achieve noise attenuation. Are two, or at least two, positive feedback loops (as used in [34]–[36]) required for controlling noise amplification in the input? Is a fast loop necessary for a positive feedback loop system to achieve noise attenuation? Are there any intrinsic quantities that connect the dynamic property of a system in absence of noises with the system' s capability of noise suppression? If such quantities exist, how do positive feedbacks or negative feedbacks affect them? In this work, we find that the capability of noise suppression in a system strongly depends on a quantity that measures the difference between the deactivation and activation times relative to the input noise frequency. Specifically, this quantity, termed as the “signed activation time”, has an inverse relationship with the noise amplification rate, with larger signed activation time leading to better noise attenuation. In addition, the signed activation time, representing one of the essential temporal characteristics of the system in absence of noises, may be controlled by either negative or positive feedbacks. We explore the properties of the quantity through both analytic approach (including linear stability analysis, multiple time scale analysis, and Fluctuation Dissipation Theorem) and numerical simulations. We first consider the same modules as in [34], and find that, for example, an additional positive feedback loop could drastically increase the signed activation time by speeding up the activation time while still keeping the deactivation time slow, as consistent with the previous observation [34] that dual-time-loop systems suppress noises better than single-loop systems. We next add a negative feedback loop to the positive-feedback-only system and show that a negative feedback loop usually slows down both activation and deactivation processes, leading to better or worse noise attenuation depending on which process (between activation and deactivation) is more significantly affected. Finally, we study the signed activation time and its relations to the noise amplification rate in different systems involving various feedbacks (e. g. , positive, negative, and feedforward), including a yeast cell polarization model [14], [37], a polymyxin B resistance model in enteric bacteria [38], and four connector-mediated models [39]. All simulations confirm that the capability of noise attenuation in those systems improves as the signed activation time increases.
A simple model with one positive feedback loop may have two components with one upstream stimulus (inside the red dashed box in Figure 1A). In this system, the output is activated by, and is triggered by a stimulus and regulated by. The stimulus drives the output of the system with a high (or low) stimulus that corresponds to an active (or inactive) state of. Many biological circuits have positive feedback regulations of this nature [1], [2], [19], [40]. For example, is a kinase to phosphorylate to, and once is activated, it catalyzes a conversion from an inactive form to an active form [5]. Neglecting the mechanistic details, while keeping the essential interactions, we model the dynamics of the above module by the following system of ordinary differential equations (Text S1): (1) where and represent normalized concentrations of and, respectively. The normalized stimulus, as a function of time, , usually varies (continuously) between two states, i. e. , an inactive state in which (or the “off” state) and an active state in which (or the “on” state). The parameters, , , and are kinetic constants, and indicates the time scale for loop. Once the output of the system reaches the “on” state driven by the stimulus, how does system (1) respond to temporal noises in the input signal? What are the strategies for effectively maintaining the system in the “on” state even with noises presented by the stimulus? We find that the time scales, denoted by and (Figure 2), for the system to switch from the “on” state to the “off” state and from “off” to “on” respectively in the absence of noises in the signal, play a critical role. Specifically, when is significantly larger than the time scale of the noise, i. e. , , where is the frequency of the noise, the output of the system remains in the “on” state (Figures 3A–3B). Intuitively, when the system in the stable “on” state receives a noisy signal with an instantaneous value possibly near, it needs time to react and detour to the “off” state. In the case of, before the system settles down to the “off” state, a noisy signal with an instantaneous value near shows up, forcing the system to synchronize with the new value of the input signal. If, the output recovers fast from the drift towards the inactive state, and is more likely to maintain around the “on” state. The above intuition suggests that the noise attenuation at the “on” state depends positively on and negatively on. Thus, the quantity, i. e. , the signed activation time, could be a good indicator of a system' s ability of attenuating noise. To investigate how noise level in the solution depends on the signed activation time, we study the noise amplification rate, defined as the relative ratio of the coefficients of variation of the output () and the noise () [28]: First, we perform numerical simulations on system (1) (Methods) to study the relationship between and by varying the activation and deactivation time scales while fixing. This is achieved by changing the kinetic parameteres, and individually in the system, and is found decreasing in (Figure 3C). Next, we hold constant, corresponding to no changes in all parameters, and vary the noise frequency. The trend of remains the same (Figure 3D). We also consider the dependence of on and individually (Figure S1). In the single loop case, it turns out that is always decreasing in (Figures S1A–S1C), but it might be increasing in (Figure S1D). Similar results are also obtained for positive-positive-loop systems (Figure S2). Both suggest that neither deactivation nor activation alone can fully characterize the noise amplification rate, and the noise amplification rate is more likely determined by the difference between the deactivation and activation time scales. Next, we further explore this system through the following two analytical approaches. In the previous section, we have demonstrated that the noise amplification rate depends negatively on the signed activation time. Thus, if a system is persistent to noise at the “on” state, it should have a large signed activation time. In this section, by studying the dynamics of the noise-free system, we show that a small is necessary for a slow deactivation, but not sufficient. With a fixed small, larger or could lead to slower deactivation and faster activation. In many biological processes, such as cell cycle [5], [6], often two positive feedback loops and activate the output simultaneously (Figure 1A). Similar to system (1), the corresponding equations take the form: (13) Through direct numerical simulations, we find that the noise amplification rate decreases in the signed activation time (Figures 4A–4B), following the same principle as in the single-positive-loop system (1). The activation time scale decreases in and, while the deactivation time scale increases in and (Figures 4C–4D, Table 1). We also find that an additional feedback loop can lead to a faster activation (red and black versus blue in Figures 4C–4D, bottom) and a slower (or similar) deactivation (red and black versus blue in Figures 4C–4D, top), compared to a single-positive-loop system, and a positive-positive-loop system can achieve similar activation and deactivation rates with larger ranges of kinetic parameters than a single-positive-loop system (Table 2). Consequently, noise attenuation can be better achieved in the positive-positive-loop system (Figures 4E–4F). Below are details of the mathematical analysis for the roles of the additional positive feedback. In this section, we study how an additional negative feedback loop affects noise attenuation in a system. One of the simplest ways to introduce negative feedback to the single-positive-loop system (1) is to let deactivate (Figure 1B) [21]. In this case, the model becomes (20) Our analytical results show that the additional negative feedback loop leads to slower deactivation and slower (or slightly faster) activation compared to its single-positive-loop counterpart (red and black versus blue in Figures 5C–5D). Moreover, the deactivation time scale increases in and, and the activation time scale decreases in and (Figures 5C–5D, Table 1), similar to the single-positive-loop (Figures 3E–3F) and positive-positive-loop systems (Figures 4C–4D). Numerical simulations reinforce these findings and demonstrate that the noise amplification rate of negative-positive-loop systems decreases in the signed activation time, following the same principle as their single-positive-loop counterparts (Figures 5A–5B). Below, we provide detailed analysis to show how the deactivation and activation time scales depend on various kinetic parameters, compared to the single-positive-loop case. In our analytical studies, we assume for simplicity. However, and are varied independently in numerical simulations. Unlike the simple models in the previous section, a yeast cell polarization signaling pathway model that we study next (Figure 6A) consists of more than three components and multiple feedback regulations [37], [48]. Polarization in yeast cells (a or cells) is activated by pheromone gradients [48]. The pheromone (L) binds to the receptor (R) and becomes activated (RL). The activated receptor facilitates the conversion of the heterotrimeric G-protein (G) into an activated -subunit (G) and a free G dimmer [49]. G is then deactivated to an inactive -subunit (Gd), which in turn binds to G and forms the heterotrimeric G-protein. The free G recruits cytoplasmic Cdc24 to the membrane, forming the membrane-bounded Cdc24 (Cdc24m), an activator of Cdc42. Accumulation of the activated Cdc42 (Cdc42a) at the projection site is a key feature of polarization, and thus is regarded as the output of the proposed system. The activated Cdc42 participates in other polarization processes, forming positive or negative feedback loops. For example, the activated Cdc42 sequesters the scaffold protein Bem1 to the membrane, which then recruits Cdc24 to the membrane [50]. This forms a positive feedback loop. Other functions of Cdc42 include the activation of Cla4 (Cla4a), an inhibitor of Cdc24, resulting in a negative feedback loop [51]. Following the model proposed in [37] but ignoring the spatial effect, we have the following system of equations: (25) Here, denotes the concentration of the corresponding protein; [L] is the input signal, and [Cdc42a] is the output; the concentrations of G, Gd, the inactive form of Cdc42, the cytoplasmic Cdc24, and the cytoplasmic Bem1 are derived through conservation relations: Here, is the volume of the cell; is the surface area of the cell; , and are the total numbers of molecules per cell of the corresponding proteins. The two Hill functions and are defined asThese two functions represent two different ways of bringing Cdc24 to the membrane. One is by the free G (function), and the other is through Bem1. The Bem1 recruitment is known to be facilitated by G' s binding to Ste20 [52], and the influence from G is modeled by the function. Kinetic parameters take the same values as in [37], and see also the caption of Figure 6. Starting from zero Cdc42a, giving high ([L]nM) or low ([L]nM) constant inputs, the output reaches active and inactive states, respectively, which are clearly distinguished (Figure 6B). Inputs with small amplitude (Figure 6C) can be detected by the system (Figure 6D). On the other hand, the output is robust to noise when it is around the active state (Figures 6E–6F). To study how the noise amplification rate depends on the relative time scales, we vary ten parameters systematically in their -fold ranges. All of them show the same decreasing trend of the noise amplification rate as a function of the signed activation time (Figure 6G). This suggests that the negative relation between the noise amplification rate and the signed activation time, derived from the simple models, could also apply to models of complex interactions and combinations of positive and negative feedback loops. Such negative relationship may be a generic principle on noise suppression for input-output systems with feedback loops.
Our theoretical and numerical studies have demonstrated that it is not the sign of the feedback that determines the degree of noise attenuation. In searching for a general framework for a relation between feedback and noise attenuation, we have identified a critical quantity, termed as the “signed activation time”. Its relation with the system' s ability of noise attenuation has been explored, and we have revealed that the noise amplification rate decreases in the signed activation time. These results are concluded through employing multiple time scale analysis, Fluctuation Dissipation Theorem, and linear stability analysis, combined with numerical simulations, in three feedback modules (Figure 1): single-positive-loop, positive-positive-loop, and positive-negative-loop systems. To test the generality of the conclusion, we have explored models (Figure 1A) with saturation effect, i. e. , modeling feedback loops by Hill functions (Text S1, Section 6, Figures S5, S6), a yeast cell polarization model consisting of multiple intermediate components (Figure 6), a polymyxin B resistance model in enteric bacteria (Figure 7), and four connector mediated models (Table 3, Figure 8). In all cases, the noise amplification rate has been confirmed to be a decreasing function in the signed activation time. To analyze the roles of multiple positive and negative feedback loops in our toy models, we have found that: 1) an additional positive feedback loop could drastically reduce the activation time scale, improving performance in noise attenuation; 2) the time scales in positive-positive-loop feedback systems are more robust to rate constant variations (e. g. due to variability of organisms or variation of environments); and 3) adding a negative feedback loop usually sustains both deactivation and activation processes, and thus its overall effect on the signed activation time could be either negative or positive. To obtain slow deactivation and fast activation, we have identified two key parameters, , the association constant of to, and, the association constant of to (Figure 1A), that tightly control the deactivation and the activation time scales (Tables 1 and 2). Interestingly, under appropriate conditions, even the simplest single positive feedback loop system could display slow deactivation and fast activation, which were not observed in previous works [34]–[36]. The idea of connecting noise attenuation with the time scales of signal responses was mentioned in other works, for example, [53], in which only the activation time scale was considered. However, we have shown that in our models neither the deactivation time scale nor activation time scale alone predict correctly the trend of the noise amplification rate (comparing Figure 3C to Figures S1C–S1D, for example) and the noise amplification rate is an interplay between the two time scales. Our proposed quantity, the signed activation time, provides a more consistent relation linking to the noise attenuation rate. Direct approaches for analyzing noise may be applied to feedback systems, such as the energy landscape method [4], [35], [54]–[56] and the methods used for noise attenuation or amplification in signaling cascades [28], [57]–[60] and covalent modification cycles [53]. To characterize signaling time scales, we have studied the magnitude of eigenvalues and their corresponding eigenvectors of the Jacobian matrices at each distinct state of the signal. Questions concerning how the magnitude of signal output and signal duration depend on properties of pathway components (e. g. , the effect of cascades) were explored from a system control point of view in other works [61]–[63]. Our study features a novel approach using multiple time scale asymptotic expansion [41]. Different from the one-time-scale expansion, this approach provides an explicit relation between the solutions and the two separated time scales, suggesting that the single-positive-loop system can function as a low-pass filter and explaining why the relative size of noise time scale and a system' s intrinsic time scales is important to noise attenuation. This approach may be applied to other biological systems with time scale separations. Our findings suggest that the negative relationship between the noise amplification rate and the signed activation time could be a general principle for many biological systems regardless of specific regulations or feedback loops. Notice that the deactivation and activation time scales are widely defined and could be measured without detailed knowledge of a system' s internal structure. Thus, the underline system could be treated as a black box and its ability of noise attenuation could be estimated based on the signed activation time. In general, if a system prefers to better attenuate noise at the “on” state, the system should have a large signed activation time. We would like to point out that the studies done here mainly focus on time scale changes within a fixed system, although comparisons across different systems are likely to be consistent with our result (e. g. the four connector-mediated models). However, we might not expect two drastically different systems with equal signed activation time to exhibit the same noise amplification rate, which is likely to depend on other factors in the system as well. We hope that the present work can shed some light on general principles of noise attenuation, in particular, their connections with timing of a system in the absence of noises.
All simulations are performed using Mathematica 6. 0. 0. To compute the noise amplification rate, we useto approximate, whereWe useto approximate, whereThe noise is generated by dividing the time interval into sub-intervals of length, and on each sub-interval the signal takes a random number from a uniform distribution in. See Figure 3A for a typical noisy signal. See Text S1.
Many biological systems use feedback loops to regulate dynamic interactions among different genes and proteins. Here, we ask how interlinked feedback loops control the timing of signal transductions and responses and, consequently, attenuate noise. Drawing on simple modeling along with both analytical insights and computational assessments, we have identified a key quantity, termed as the “signed activation time”, that dictates a system' s ability of attenuating noise. This quantity combining the speed of deactivation and activation in signal responses, relative to the input noise frequency, is determined by the property of feedback systems when noises are absent. In general, such quantity could be measured experimentally through the output response time of a signaling system driven by pulse stimulus. This principle for noise attenuation in feedback loops may also be applicable to other biological systems involving more complex regulations.
computational biology/signaling networks
mathematics
computational biology/systems biology
cell biology/cell signaling
A Critical Quantity for Noise Attenuation in Feedback Systems