Regulatory Effects of Cooperativity and Signal Profile on Adaptive Dynamics in Incoherent Feedforward Loop Networks

Abstract

Cellular adaptation to external signals is essential for biological functions, and it is an important field of interest in systems biology. This study examines the impact of cooperativity on the adaptation response of the Incoherent Feedforward Loop (IFFL) network motif to various signal profiles. Through comprehensive simulations, we studied how the IFFL motif responds to constant and pulse-type signals under varying levels of cooperativity. The results of our study demonstrate that positive cooperativity generally enhances the system’s ability to adapt to different signal profiles. Nevertheless, given specific signal profiles, higher levels of cooperativity may decrease the system’s adaptability. On the other hand, the adaptive response breaks down for negative cooperativity. For constant signals, increased positive cooperativity leads to a response with higher amplitude, and it accelerates the response time but delays the return time required to settle back down to the pre-stimulus state. Upon signal cessation, high positive cooperativity not only slows the system’s response and return times but, in some cases, can lead to a complete temporary halt in response. For the pulse-like signal, cooperativity increases the maximum amplitude of the oscillatory response. These insights highlight the delicate balance between cooperativity and signal profile in cellular adaptation mechanisms involving the IFFL network motif.

Introduction

Cells are subjected to many external stimuli and have developed intricate adaption mechanisms to maintain a state of equilibrium. This adaptive capacity is more than a mere cellular response; it is essential for the survival and efficient functioning of cells. It enables cells to precisely react to the continuously shifting external stimuli (Bleris et al., 2011; Ma et al., 2009). This complex regulatory relationship between stimulus and response is partly governed by small, evolutionarily conserved interaction networks, referred to as network motifs. The network motifs that are capable of displaying adaptive dynamics include negative feedback and incoherent feedforward loops (IFFLs). These motifs, characterized by unique structural designs, have a key role in the complex dynamics of adaptation. They determine how cells perceive and respond to different environmental signals (Alon, 2007; Mangan & Alon, 2003).

The IFFL motif, particularly its Type I form, is crucial in gene regulatory networks for processing environmental signals through two opposing regulatory pathways on a specific protein. It initially triggers an increase in the abundance of a response protein via an activatory pathway. Then, an inhibitory pathway counteracts this effect, leading to a precise regulation of the protein’s level. This results in an initial rapid increase in the protein levels, followed by a gradual return to its basal level, a process known as perfect adaptation. It has been shown that an IFFL motif with adaptive dynamics must have at least three proteins (Ma et al., 2009).

The proteins regulated by IFFL motif are involved in a wide range of biological functions, including specific cellular tasks and interacting with other transcription factors (Mangan et al., 2006; Mangan & Alon, 2003; Novák & Tyson, 2008). The IFFL adaptive dynamics are observed in biological processes, such as light detection in plants, homeostatic adaptation, neuronal regeneration, and eukaryotic chemotaxis, highlighting the importance of the motif’s role in different cellular contexts (Frick et al., 2017; Lee et al., 2014; MacGillavry et al., 2009; Shi & Iglesias, 2013; Takeda et al., 2012; Tendler et al., 2018). Besides its widespread occurrence in many biological systems, the IFFL motif is a major focus in synthetic biology research. Biologists have constructed the IFFLs in many systems to carry out a variety of tasks, including mitigation of resource competition, and signal filtering (Bleris et al., 2011; Frei & Khammash, 2021; Jones et al., 2019). To effectively utilize this network in the regulation of biological systems, a better understanding of its mechanics is crucial.

Cooperativity among proteins, such as transcription factors, is vital in controlling biological processes. Cooperative molecule binding is identified as a key mechanism for ultra-sensitive responses in biological systems (Goldbeter & Koshland Jr, 1984). Within transcriptional networks, cooperativity can either enhance or diminish the reaction to external stimuli, highlighting its dual role in biological regulation (Berg et al., 2004; Forsén & Linse, 1995). Furthermore, cooperativity tends to increase in cascading systems, which are prevalent in developmental networks (Blake et al., 2003). Such cascading effects contribute significantly to the complex behavior of these networks (Chickarmane et al., 2006; Demarez et al., 2018; Pimanda & Göttgens, 2010; Yoshida et al., 2020). The incoherent feedforward loop can still exist with cascading proteins (Ferrell, 2016). Specifically, in IFFLs, cooperativity significantly impacts these motifs’ signal-processing abilities, leading to significant changes in the system’s adaptability and response characteristics (Benary et al., 2013; Sun et al., 2022).

Positive cooperativity is commonly observed in biological systems, where the binding of one molecule increases the likelihood of subsequent molecule’s binding. However, negative cooperativity also plays a significant role, where the binding of one molecule decreases the affinity for the next molecule to bind (Koshland Jr, 1996). Specifically, negative cooperativity is instrumental in the application of Weber’s Law within transcription systems (Ferrell, 2009). Weber’s Law describes how the perception of a change in a stimulus (e.g., light intensity) is proportional to the magnitude of the stimulus itself, suggesting that organisms are sensitive to relative changes rather than absolute ones. This principle is crucial for fold-change detection in biological systems, a process that requires perfect adaptation to accurately measure relative changes in stimulus intensity (Shoval et al., 2010).

Our study investigates the role of cooperativity within the Type-I IFFL motif and focuses on how cooperativity influences the motif’s capacity for cellular dynamics and adaptation. To unravel the impact of varying levels of cooperativity on IFFL dynamics, we employ computational models and analyze these effects across different signal profiles. We utilize adaptation metrics to quantify response dynamics such as response level, response time, return time, and adaptation error to comprehensively study and gain a better understanding of the adaptive dynamics and its regulation in the IFFL motif under different cooperativity conditions (Frei & Khammash, 2021; Shi et al., 2017; Yildirim, 2024). The results of this study are expected to provide a novel understanding regarding the dynamic control of proteins, a critical component in cellular adaptation and function.

The Mathematical Modeling and Analysis of the IFFL Network

This section presents the mathematical model for the Type-I IFFL motif and derives the necessary and sufficient conditions for the adaptive dynamics. An IFFL network with fewer than three proteins is unable to generate adaptive dynamics (Ma et al., 2009; Shi et al., 2017). A cartoon showing the interactions of three proteins, A, B, and C, in the Type-I IFFL network is displayed in Figure 1(a). Here we name protein A as “a signal protein” on which environmental input signal S increases its abundance. The signal protein A positively regulates two proteins downstream, which are “intermediate protein” B and “response protein” C. Protein B also negatively regulates the response protein C.

Figure 1.

(a): A cartoon showing IFFL network involving three proteins, which is capable of producing adaptive dynamics. In this cartoon, environmental signal S increases the abundance of the signal protein A, which positively interacts with both intermediate protein B and response protein C, while the intermediate protein B negatively regulates the abundance of the response protein C. (b): A plot illustrating the response metrics ${\tau }_{rs}^{On}$ , ${\tau }_{rt}^{On}$ , and $C_{max}$ when the signal S is turned on. The y-axis represents the % of increase in the protein product C, and the x-axis represents dimensionless time $\tau$ . (c): A plot depicting the response metrics ${\tau }_{rs}^{Off}$ , ${\tau }_{rt}^{Off}$ , and $C_{min}$ when the signal is turned off. Again, the y-axis represents the % of decrease in the response protein C, and the x-axis represents the dimensionless time $\tau$ .

The model equation governing IFFL motif is given by Equations (1) to (3) (Shi et al., 2017). Equation (1) describes the dynamics of the signal protein A, where $v_{A}f(S)$ represents the gain rate, which is assumed to be a first order reaction, and the second term models the degradation of this protein with a parameter ${\beta }_A$ determining the half-life of this protein.

$$\frac{dA}{dt}=v_{A}f(S)-\frac{A}{{\beta }_A}$$

(1)

$$\frac{dB}{dt}=v_B\frac{A^{{}^{n_{AB}}}}{A^{{}^{n_{AB}}}+K_{{}_{AB}}^{{}^{n_{AB}}}}-\frac{B}{{\beta }_B}$$

(2)

$$\frac{dC}{dt}=v_C\frac{A^{{}^{n_{AC}}}}{A^{{}^{n_{{}_{AC}}}}+K_{AC}^{{}^{n_{{}_{AC}}}}}\frac{K_{{}_{BC}}^{{}^{n_{BC}}}}{B^{{}^{n_{BC}}}+K_{BC}^{{}^{n_{BC}}}}-\frac{C}{{\beta }_C}$$

(3)Equation (2) models the dynamics of the intermediate protein B, in which the first term represents the gain rate of B due to A, which is assumed to be Hill type with the parameters of $n_{AB}$ , $v_B$ and $K_{AB}$ , which are respectively Hill coefficient, maximum gain rate, and the dissociation constant. The dissociation constant $K_{AB}$ determines the amount of A required for half of the maximum gain rate for B. Similar to the first equation, the second term is for the degradation of B, with the parameter ${\beta }_B$ determining this rate. Lastly, the dynamics of C is governed by Equation (3). Here, the gain rate for C is the product of two functions. The first term $\frac{A^{{}^{n_{AC}}}}{A^{{}^{n_{AC}}}+K_{AC}^{{}^{n_{AC}}}}$ describes the positive regulation of C by A, which is again taken to be Hill type with parameters of $n_{AC}$ , $v_C$ and $K_{AC}$ . $n_{AC}$ is the Hill coefficient, $v_C$ is the maximal gain rate of C, and $K_{AC}$ is dissociation constant. The second term in this product $\left(\frac{K_{{}_{BC}}^{{}^{n_{BC}}}}{B^{{}^{n_{BC}}}+K_{{}_{BC}}^{{}^{n_{BC}}}}\right)$ models the negative regulation of C by B, and $K_{BC}$ is the dissociation constant and $n_{BC}$ is the Hill coefficient. Finally, the last term describes the degradation of this protein, which is controlled by a parameter ${\beta }_C$ .

Steady State Analysis and Adaptive Dynamics

To carry out analytical work and study model dynamics, the model is written in dimensionless form to reduce the number of parameters while retaining its dynamic properties. By defining new variables as $\tau =\frac{t}{{\beta }_C},a=\frac{A}{K_{AC}},$ $b=\frac{B}{K_{BC}},c=\frac{C}{v_C{\beta }_C}$ , the model equations in Equations (1) to (3) simplifies to the dimensionless form,

$$\begin{array}{rl}\frac{da}{d\tau }& ={\alpha }_0-{\beta }_{0}a\\ \frac{db}{d\tau }& =\frac{{\alpha }_1{a}^{n_{AB}}}{a^{n_{AB}}+{\alpha }_2^{n_{AB}}}-{\beta }_{1}b\\ \frac{dc}{d\tau }& =\frac{a^{n_{AC}}}{a^{n_{AC}}+1}\frac{1}{b^{n_{BC}}+1}-c\end{array}$$

where ${\alpha }_0=\frac{{\beta }_C{v}_{A}f(S)}{K_{AC}},{\beta }_0=\frac{{\beta }_C}{{\beta }_A}$ , ${\alpha }_1=\frac{{\beta }_C{v}_B}{K_{BC}},{\alpha }_2=\frac{K_{AB}}{K_{AC}},{\beta }_1=\frac{{\beta }_C}{{\beta }_B}$ . Here $\tau$ is the dimensionless time, and $a$ , $b$ and $c$ represent dimensionless variables for the signal protein $A$ , intermediate protein $B$ and the response protein $C$ . The degradation coefficients directly affect the model dynamics. After exposure to a signal, a longer time is needed for a stable protein to reach its steady state compared to an unstable protein. To make a fair comparison of dynamics, we assumed equal degradation rates for all three proteins by setting ${\beta }_A={\beta }_B={\beta }_C$ , which leads to ${\beta }_0={\beta }_1=1$ (Adler & Alon, 2018).

The adaptive dynamic response requires the steady states of the response protein $c$ be independent of the signal protein $a$ and its parameters. Under the conditions (i) $a^{*}\ll {\alpha }_2$ , (ii) $a^{*}\ll 1$ , (iii) $b^{*}\gg 1$ and (iv) $n_{AC}=n_{AB}{n}_{BC}$ , the dimensionless model further simplifies to following the adaptive model,

$$\frac{da}{d\tau }={\alpha }_0-a$$

(4)

$$\frac{db}{d\tau }=\frac{{\alpha }_1{a}^{n_{AB}}}{{\alpha }_2^{n_{AB}}}-b$$

(5)

$$\frac{dc}{d\tau }=\frac{a^{n_{AB}{n}_{BC}}}{b^{n_{BC}}}-c$$

(6)Biologically, the first adaptation condition requires the steady state of dimensionless A to be much smaller than the parameter ${\alpha }_2$ . Since ${\alpha }_2=\frac{K_{AB}}{K_{AC}}$ is the ratio of the dissociation constants for proteins A and B and the dissociation constants for proteins A and C, this can also be interpreted as higher affinity of A to protein C compared to affinity of A to B. The second adaptation condition, $a^{*}\ll 1$ , indicates a low level of steady state of A which results in a non-saturating activation of C by A. The third adaptation condition, $b^{*}\gg 1$ , suggests the high steady state of B and which results in an unsaturated inhibition of C by B. The fourth condition, $n_{AC}=n_{AB}{n}_{BC}$ , represents the delicate balance between the level of cooperatively to achieve adaptive response.

When $\frac{da}{d\tau }=\frac{db}{d\tau }=\frac{dc}{d\tau }=0$ , the steady state $(a^{*},b^{*},c^{*})$ of the adaptive model becomes,

$$(a^{*},b^{*},c^{*})=({\alpha }_0,{\alpha }_1{\alpha }_0^{n_{AB}}{\alpha }_2^{-n_{AB}},{\alpha }_2^{n_{AB}{n}_{BC}}{\alpha }_1^{-n_{BC}})$$

(7)Notice that the steady state $c^{*}$ is independent of the signal protein $a*$ level and it is solely determined by the parameters of the intermediate protein $b$ under the four conditions listed above.

These state levels could further be simplified. For some large $\delta \gg 1$ , the adaptation conditions simplify to (i) $a^{*}={\alpha }_0={\delta }^{-1}$ , (ii) ${\alpha }_2=1$ , (iii) ${\alpha }_1={\delta }^{1+n_{AB}}$ while (iv) $n_{AC}=n_{AB}{n}_{BC}$ . Therefore, the steady state becomes

$$(a^{*},b^{*},c^{*})=({\delta }^{-1},\delta ,{\delta }^{-n_{BC}(1+n_{AB})})$$

(8)Notice that, for the Type I IFFL model to exhibit adaptive dynamics, $c^{*}\ll a^{*}\ll b^{*}$ must hold. Here ${\delta }^{-1}$ represents the basal level of $a*$ in the absence of the input signal, this parameter is increased up to 200 fold to mimic the external signal in our simulations.

Local Stability Analysis of the Adaptive Model

To study the local dynamics of the IFFL model in response to a small increase in the input signal S while the system is at resting state, the model equations are linearized around the steady state to obtain:

$$\begin{array}{rl}\frac{d\mathrm{\Delta }a}{d\tau }& =-\mathrm{\Delta }a+\gamma \\ \frac{d\mathrm{\Delta }b}{d\tau }& ={\delta }^2{n}_{AB}\mathrm{\Delta }a-b\\ \frac{d\mathrm{\Delta }c}{d\tau }& ={\delta }^{1-(1+n_{AB})n_{BC}}{n}_{BC}{n}_{AB}\mathrm{\Delta }a-{\delta }^{-1-(1+n_{AB})n_{BC}}{n}_{BC}\mathrm{\Delta }b-\mathrm{\Delta }c\end{array}$$

where $\mathrm{\Delta }a=a-a^{*},\mathrm{\Delta }b=b-b^{*}$ , $\mathrm{\Delta }c=c-c^{*}$ , and $\gamma$ is a new input parameter representing an activatory input signal applied when the system is at the resting state. The analytical solution of this system of differential equations with the initial condition $\mathrm{\Delta }a(0)=\mathrm{\Delta }b(0)=\mathrm{\Delta }c(0)=0$ in terms of the parameters $\gamma$ , $\delta$ and the Hill coefficients $n_{AB}$ , $n_{BC}$ , $n_{AC}$ becomes

$$\mathrm{\Delta }a(\tau )=\gamma (1-e^{-\tau })$$

(9)

$$\mathrm{\Delta }b(\tau )=\gamma n_{AB}{\delta }^2(1-(1+\tau )e^{-\tau })$$

(10)

$$\mathrm{\Delta }c(\tau )=\frac{1}{2}\gamma n_{AB}{n}_{BC}{\delta }^{1-(1+n_{AB})n_{BC}}{\tau }^2{e}^{-\tau }$$

(11)By differentiating Equation (11) with respect to $\tau$ and setting it equal to zero, one can compute the maximal level of $\mathrm{\Delta }c$ as $\mathrm{\Delta }{c}_{max}=2e^{-2}\gamma n_{AB}{n}_{BC}{\delta }^{1-(1+n_{AB})n_{BC}}$ , which is attained at $\tau =2$ . Notice that the percentage maximal increase in $\mathrm{\Delta }c$ relative to its basal level prior to the signal given in Equation (8) becomes

$$100\times \frac{\mathrm{\Delta }{c}_{max}}{c^{*}}=200e^{-2}\gamma \delta n_{AB}{n}_{BC}$$

Notice that the time required to reach maximum responses is independent of the model parameter values while the maximum response in $c$ is directly proportional to the input signal $\gamma$ , $n_{AB}$ and $n_{BC}$ . For example, increasing the basal level of signal protein $a$ by $p$ percent by setting $\gamma =\frac{p}{100}\times {\delta }^{-1}$ results in an increase of

$$100\times \frac{\mathrm{\Delta }{c}_{max}}{c^{*}}=0.27\times p\times n_{AC}$$

in response protein $c$ , where $n_{AC}=n_{AB}{n}_{BC}$ . Furthermore, changing the cooperativity coefficients $n_{AB}$ and $n_{BC}$ while keeping their product the same does neither increase nor decrease the maximal response.

Numerical Simulations

We conducted a series of numerical simulations to further study sensitivity of the adaptive dynamics of the Type I IFFL model to the changes in the cooperativity parameters $n_{AB}$ and $n_{BC}$ , and the signal parameter $S$ . In all of the simulations, the Hill coefficients are varied in the range of 1/3 to 3, and the signal intensity parameter is changed from 1 to 200. We defined adaptation as less than $5\mathrm{\%}$ adaptation error. Adaptation error is calculated as the absolute percentage difference between the steady-state levels before ( $c_1*$ ) and after ( $c_2*$ ) the stimulus, using the formula $|\frac{c_1*-c_2*}{c_1*}|\times 100$ .

Different metrics were used to quantify the IFFL’s adaptation response to different signal profiles. For analyzing the response to a signal when it is activated, three specific metrics were employed: (i) response time ${\tau }_{rs}^{on}$ which quantifies the duration required for the system to attain the maximal level of $c$ after the signal, (ii) maximum response $c_{max}$ which measures the percentage maximal response in $c$ relative to its basal level, calculated via $c_{max}=|\frac{c({\tau }_{rs}^{on})-c_1*}{c_1*}|\times 100$ , where $c({\tau }_{rs}^{on})$ is the abundance of $c$ at the response time ${\tau }_{rs}^{on}$ and $c_1*$ is the pre-stimulus steady state, (iii) return time ${\tau }_{rt}^{on}$ which computes the time for the system to revert back within 5% of pre-stimulus steady state $c_1*$ , defined by $|\frac{c_{5\mathrm{\%}}-c_1*}{c_1*}|\times 100\ge 5$ . Figure 1(b) depicts these metrics with the steady state set as the zero baseline. Here $c_{5\mathrm{\%}}$ represents the abundance of $c$ 5% above the steady state.

To explore the adaptive dynamics after a signal is turned off, we modify these metrics slightly: (i) response time ${\tau }_{rs}^{off}$ which calculates the duration needed for the system to attain the minimum level of $c$ after the signal is turned off, (ii) minimum response $c_{min}$ which quantifies the maximal percentage decrease of $c$ from its basal level, which is calculated by the formula $c_{min}=|\frac{c({\tau }_{rs}^{off})-c_1*}{c_1*}|\times 100$ . Here $c({\tau }_{rs}^{off})$ denotes the abundance at the response time ${\tau }_{rs}^{off}$ , and $c_1*$ is the steady state level prior to the stimulus, (iii) return time ${\tau }_{rt}^{off}$ which computes the duration for the response protein level to settle back within 5% of the pre-stimulus steady state, calculated by $|\frac{c_{5\mathrm{\%}}-c_1*}{c_1*}|\times 100\ge 5$ , where $c_{5\mathrm{\%}}$ is the level that is 5% away from $c_1*$ . Figure 1(c) depicts these metrics with the steady state set as the zero baseline.

We also utilized the amplitude metric to analyze pulse-type signals. Amplitude is defined as the difference between the maximum and minimum responses, expressed as $c_{max}-c_{min}$ .

Section “Adaptation Dynamics Under Varying Cooperativity Levels and Constant Signal Inputs” examines how the Type I IFFL network adapts under conditions of both positive and negative cooperativity using the adaptation error metric. Subsequent sections then concentrate solely on the effects of positive cooperativity. Section “System Response to Constant Signal Inputs” focuses on the system’s response to constant weak and strong signals and looks at the effect of different levels of cooperativity on the adaptation metrics. Section “System Response to Varying Signal Inputs” presents a similar analysis of the system to the signals of changing strengths as the cooperativity levels vary. Section “Adaptation Dynamics for Post-signal Deactivation Under Constant Signal Inputs” examines the system dynamics after the signal is turned off, investigating how the cooperativity regulates the adaptive system’s behavior once the input signal is ceased. Section “System Response to Varying Signal Inputs Post Signal Deactivation” extends this analysis to the influence of signals of differing strengths after the signal has been turned off as signal cooperativity levels vary. Finally, Section “Adaptation Dynamics Under Repetitive Pulse Type Signals” analyzes this adaptive system’s response to pulse-type signals. We assess how the system processes varying pulse strengths and profiles under different cooperativity levels.

Adaptation Dynamics Under Varying Cooperativity Levels and Constant Signal Inputs

To elucidate the model’s response dynamics to constant external stimuli under varying cooperativity levels, we conducted a detailed examination of system behavior under three distinct signal strengths: (i) a weak signal of $S=2$ , (ii) a medium signal of $S=20$ , and (iii) a strong signal of $S=200$ . The weak signal corresponds to a 2-fold increase in the abundance of the signal protein $a$ , while medium and strong signals mean 20- and 200-fold increases in the abundance of the signal protein $a$ , respectively. This analysis was performed by systematic tuning of cooperativity coefficients $n_{AB}$ and $n_{BC}$ ranging from 1/3 to 3, which falls in a biologically reasonable interval (Charoenwattanasatien et al., 2020; Houslay et al., 1994; Schwartz & Skafar, 1994; Thukral et al., 1991; Zhang et al., 2013). Here, $n_{AB}$ represents the activator Hill coefficient, dictating the steepness of the activation curve for protein B, while $n_{BC}$ , the inhibitor Hill coefficient, determines the binding efficacy of protein B and its inhibitory influence on C. An additional cooperative interaction is quantified by the Hill coefficient $n_{AC}=n_{AB}\times n_{BC}$ , which is one of the adaptation conditions.

As illustrated in Figure 2, our findings show a discernible difference in the response dynamics to the weak (Figure 2(a)), medium (Figure 2(b)), and strong (Figure 2(c)) signals. The system demonstrates more refined adaptability to the weaker signal compared to the stronger signal. Furthermore, it displays a better adaptive response under positive cooperativity as opposed to negative cooperativity. Our results show that negative cooperativity eliminates the system’s ability to adapt, especially as the signal becomes stronger (Figure 2(a) to (d)). Due to negative cooperativity at large hindering the system’s ability to adapt, we decided to base our further analysis on positive cooperativity. The time series plot in ((Figure 2(d))) shows three representative dynamics when $S=200$ as $n_{AB}=0.5$ and $n_{BC}$ changes from $0.67$ to $1.5$ . As seen in this plot, only the curve with $n_{BC}=1$ and $n_{AB}=0.5$ exhibits adaptive dynamics, not the other two. It is interesting to see that when $n_{BC}=1$ , the system always displays adaptive dynamics.

Figure 2.

Heat maps illustrating the system’s dynamics and adaptation for different cooperativity levels of $n_{AB}$ and $n_{BC}$ ranging from 1/3 to 3 under weak ( $\mathrm{a};S=2$ ), medium ( $\mathrm{b};S=20$ ) and strong ( $\mathrm{c};S=200$ ) signals. Black means no adaptation, and blue and red cells indicate adaptation at different levels. A representative time series plot (d) shows three cases. The horizontal red solid line shows the pre-steady state. Only the curve for $n_{BC}=1$ and $n_{AB}=0.5$ exhibits adaptive dynamics.

System Response to Constant Signal Inputs

To elucidate the model’s response dynamics to constant external input signals, we conducted a detailed examination of system dynamics under two distinct signal strengths (i) a weak signal, which corresponds to a 2-fold increase in the basal level of the signal protein $a$ by setting $S=2$ and (ii) a strong signal, which is equivalent of a 200-fold increase in the basal level of the signal protein $a$ by taking $S=200$ . This analysis was performed by systematic modulation of cooperativity coefficients $n_{AB}$ and $n_{BC}$ ranging from $1$ to $3$ and only covers the positive cooperativity range. Here, $n_{AB}$ represents the activator Hill coefficient, dictating the activation curve’s steepness for protein $B$ . Conversely, $n_{BC}$ , the inhibitor Hill coefficient, determines the binding efficacy of protein $B$ and its inhibitory influence on the protein $C$ .

As illustrated in Figure 3, our findings show a noticeable difference in the system’s response dynamics to the weak and strong input signals. The system demonstrates better adaptability to the weaker signal than the stronger signal. For example, adaptation fails when $n_{AB}=1$ and $n_{BC}\ge 1.5$ for the strong signal, but not the weak signal. As predicted, stronger signals invoke a more substantial (Figure 3(a) and (b)) and accelerated response (smaller $t_{rs}^{on}$ ) profile (Figure 3(c) and (d)), yet this is coupled with an extended return time $t_{rt}^{on}$ (Figure 3(e) and (f)) compared to the weak signal.

Figure 3.

Heat maps illustrating the system’s response to different cooperativity levels ( $n_{AB}$ and $n_{BC}$ ) under weak ( $S=2$ ) and strong ( $S=200$ ) signals. Panels (a, c, e) pertain to a weak signal, while panels (b, d, f) correspond to a strong signal. Displayed are the maximum response $c_{max}$ (first row), response time $t_{rs}^{on}$ (second row), and return time $t_{rt}^{on}$ (third row). Black regions indicate where the system does not meet the desired adaptation level of $5\mathrm{\%}$ .

The adaptation metrics exhibit sensitivity to the variations in the Hill coefficients. Increases in either $n_{AB}$ or $n_{BC}$ are associated with an increase in the maximum response (Figure 3(a) and (b)), a reduction in response time ${\tau }_{rs}^{on}$ (Figure 3(c) and (d)), and elongation of return time ${\tau }_{rt}^{on}$ (Figure 3(e) and (f)). The impact of $n_{BC}$ remains more pronounced across both signal intensities. For strong signal cases, the differential effects induced by alterations in $n_{AB}$ and $n_{BC}$ become more clear. For example, under weak signal conditions, an increase in the activator coefficient $n_{AB}$ or the inhibitory counterpart $n_{BC}$ increases the return time $t_{rt}^{on}$ comparably. In the strong signal case, the increase in $n_{BC}$ results in a longer return time $t_{rt}^{on}$ compared to an increase in $n_{AB}$ .

System Response to Varying Signal Inputs

To study how the adaptation dynamics change under varying signal levels in our system, we have simulated the model under a broad spectrum of signal strengths ranging from $1$ to $200$ as the positive cooperativity coefficient $n_{AB}$ is systematically tuned within the bounds of 1 to 3 when $n_{BC}=1,2$ or $3$ . The results of these simulations are depicted in Figures 4 and S1, presented as heat maps that show the changes in the three response metrics as the signal strength and the cooperativity coefficients change.

Figure 4.

Heat maps showing changes in $c_{max}$ (first row), ${\tau }_{rs}^{on}$ (second row), and ${\tau }_{rt}^{on}$ (third row) as the cooperativity levels ( $n_{AB}$ ) vary in $[1,3]$ and the input signal $S$ varies in $[1,200]$ for $3$ different values of $n_{BC}=1,2,3$ (columns 1–3). Heat maps for the $c_{max}$ metric (a, b, c); heat maps for the ${\tau }_{rs}^{on}$ metric (d, e, f); heat maps for the ${\tau }_{rt}^{on}$ metric (g, h, i). The black color represents where the system does not reach the desired adaptation level.

Our simulations unveiled a critical relationship between the model parameters $S$ , $n_{AB}$ , and $n_{BC}$ . The system’s adaptability deteriorates when faced with $n_{BC}\ge 2$ levels, strong signal, and low $n_{AB}$ levels. Echoing the trends identified in the previous analysis (Figure 3), we observed that increased cooperativity -both inhibitory and activatory- is associated with both an increase in maximum response (Figures 4(a) to (c) and S1 (a) to (c)) and a decrease in response time ${\tau }_{rs}^{on}$ (Figures 4(d) to (f) and S1 (d) to (f)). Furthermore, increased signal strength leads to elongated return time $t_{rt}^{on}$ (Figures 4(g) to (i) and S1 (g) to (i)). This simulation reinforces our previous findings of the interdependence of signal strength and cooperativity on the dynamics of system adaptation.

Adaptation Dynamics for Post-signal Deactivation Under Constant Signal Inputs

To study adaptation dynamics for post-signal deactivation, we sustained the signal for an extended period, $\tau =80$ , before its cessation and ran the simulation until $\tau =200$ . The model is solved for a weak signal ( $S=2$ ) and a strong signal ( $S=200$ ) when the cooperativity coefficients, $n_{AB}$ , and $n_{BC}$ , vary from 1 to 3. For each cooperativity level and signal profile, minimum response ( $c_{min}$ ), response time ${\tau }_{rs}^{off}$ , and return time ${\tau }_{rt}^{off}$ metrics are calculated.

Similar to prior analyses, we observed signal-dependent changes in response metrics (Figure 5). Specifically, the system exhibited a reduced minimum response $c_{min}$ (Figure 5(a) and (b)), and elongated response ( $t_{rs}^{off}$ ) (Figure 5(c) and (d)) and return ( $t_{rt}^{off}$ ) times (Figure 5(e) and (f)) in response to the stronger signal. These metrics also varied with different cooperativity levels under a constant signal condition. Across all simulations, as cooperativity coefficients get larger, the minimum response gets smaller while both response ( $t_{rs}^{off}$ ) and return ( $t_{rt}^{off}$ ) times get longer. Notably, the response time $t_{rs}^{off}$ exhibited an opposite relationship with cooperativity levels in post-signal deactivation in comparison to signal activation, marking a significant finding in the behavior of IFFL systems. The impact of $n_{AB}$ remains more pronounced across both signal intensities. For the strong signal, the differential effects induced by alterations in $n_{AB}$ and $n_{BC}$ become more observable. For example, under weak signal conditions, an increase in the activator coefficient $n_{AB}$ or the inhibitory counterpart $n_{BC}$ increases the return time $t_{rt}^{off}$ comparably. In the strong signal, the increase in $n_{AB}$ results in a longer return time $t_{rt}^{off}$ compared to an increase in $n_{BC}$ .

Figure 5.

Heat maps illustrating the system’s response to different cooperativity levels $n_{AB}$ and $n_{BC}$ under weak ( $S=2$ , left column) and strong ( $S=200$ , right column) signals post-signal deactivation. Panels (a, c, e) pertain to a weak signal, while panels (b, d, f) correspond to a strong signal. Displayed are the minimum response ( $c_{min}$ ), response time $t_{rs}^{off}$ and return time $t_{rt}^{off}$ . The first row is for $c_{min}$ , the middle row is for $t_{rs}^{off}$ , and the last row is for $t_{rt}^{off}$ .

System Response to Varying Signal Inputs Post Signal Deactivation

We extended our analysis to examine the system’s adaptive dynamics after signal termination, spanning a comprehensive range of signal strengths and cooperativity levels. Similar to the previous section, we sustained the signal for an extended period, $\tau =80$ , before its cessation and ran the simulation until $\tau =200$ . We varied the signal from $S=1$ to $S=200$ and the cooperativity coefficient $n_{AB}$ from $1$ to $3$ when $n_{BC}=1,2,3$ .

Our results delineated a discernible pattern: as signal strength and cooperativity coefficients increase, there is an observable decrease in the minimum response $c_{min}$ and a clear increase in both the response time ${\tau }_{rs}^{off}$ and the return time ${\tau }_{rt}^{off}$ , as depicted in Figures 6 and S2. Specifically, the lower $c_{min}$ with stronger signals underscores an increased reactivity of the system to more potent stimuli. Concurrently, the elongated ${\tau }_{rs}^{off}$ and ${\tau }_{rt}^{off}$ observed under conditions of higher $n_{AB}$ and $n_{BC}$ imply a delay in achieving both the minimum response and subsequent return to the baseline of post-stimulation.

Figure 6.

Heat maps showing changes in $c_{min}$ (first row), ${\tau }_{rs}^{off}$ (second row), and ${\tau }_{rt}^{off}$ (third row) as the cooperativity levels $n_{AB}$ vary in $[1,3]$ and the input signal $S$ varies in $[1,200]$ for $3$ different values of $n_{BC}=1,2,3$ (columns 1–3). Heat maps for the $c_{min}$ metric (a, b, c); heat maps for the ${\tau }_{rs}^{off}$ metric (d, e, f); heat maps for the ${\tau }_{rt}^{off}$ metric (g, h, i).

Adaptation Dynamics Under Repetitive Pulse Type Signals

Here, we concentrated on evaluating the amplitude of the response to discrete signal pulses across a gradient of cooperative interaction strengths for weak ( $S=2$ ) and strong ( $S=200$ ) signals. We quantified the response amplitude as the difference between the maximum response when the signal is on, $c_{max}$ , and the minimum response when the signal is off, $c_{min}$ . Specifically, we varied the signal duration (on time) and the time between pulses (off time). The heat maps depicted in Figure 7 show the response amplitudes to these signal inputs. We observed a direct correlation between the system’s response amplitude and the cooperativity coefficients $n_{AB}$ and $n_{BC}$ . As these coefficients increase, so does the amplitude of the response. Furthermore, the duration for which the system remains inactive (off time) also influences the amplitude, with a longer inactivity duration allowing the system to reach higher amplitudes upon reactivation.

Figure 7.

Heat maps showing response amplitude changes to pulse type signals as the cooperativity levels ( $n_{AB}$ and $n_{BC}$ ) vary in $[1,3]$ for weak ( $S=2$ , left column) and strong ( $S=200$ , right column) signals. Each row represents a different signal profile with active (On) and inactive (Off) time periods. In the top row, the signal is alternated between on and off states every 2 units of time until 80 units of time. The second row is for 5 units on and 2 units off, the third row is for 2 units on and 5 units off, and the last row is 5 units on and 5 units off.

It was observed that longer periods of signal activation led to significantly higher system responses compared to shorter periods of signal activation. This trend was consistent across both weak and strong signal strengths. Interestingly, the response dynamics varied with signal strength. The most substantial responses were elicited for weaker signals when both the activation and deactivation periods were prolonged. Conversely, stronger signals achieved the response with maximum amplitude when the activation period was brief, followed by an extended deactivation phase. This phenomenon underscores the intricate relationship between signal duration, system adaptability, and cooperative interactions.

Discussion

In this study, we have investigated the role of cooperativity and signal profiles in the adaptation dynamics of transcriptional type I IFFLs. The adaptation dynamics are assessed through metrics such as maximum/minimum response level, response, and return times as cooperativity level and signal profile are varied. Our analysis reveals that the system’s adaptability and response dynamics strongly depend on these parameters. Specifically, the Type I IFFL motif exhibits better adaptation under conditions of positive cooperativity compared to negative cooperativity. Our analysis of positive cooperativity shows that increased cooperativity and signal strength led to higher maximum response levels, shorter response time $t_{rs}^{on}$ , and longer return time $t_{rt}^{on}$ when the signal is on. We demonstrate that these findings highlight the nuanced role of cooperativity and signal characteristics in modulating system behavior, contributing new insights into the field. Our post-signal deactivation dynamic analysis suggests that increased cooperativity and signal strength prolong the response ( $t_{rs}^{off}$ ) and return ( $t_{rt}^{off}$ ) times and decrease the minimum response. Furthermore, we observed that the system’s response to varying signal profiles indicated that both the signal strength and signal on and off times are key regulatory factors for the system’s dynamics. While previous studies have outlined the influence of signal strength, our results show that cooperativity and signal-off times lead to increased response amplitudes, with this effect being more pronounced for weak signals with shorter signal-on and longer signal-off periods. Overall, our analysis emphasizes the complex interactions between cooperativity, signal characteristics, and adaptation dynamics for this network motif, clarifying previously ambiguous relationships and highlighting the nuanced ways these factors influence system behavior.

It has been shown that positive cooperativity could prevent the system’s adaptability in IFFLs for enzymatic reactions (Ma et al., 2009). In another study, it was suggested that adaptation in transcriptional IFFLs can be attained under specific cooperativity conditions (Shi et al., 2017). However, we extend these findings by showing that altering both signal and cooperativity levels simultaneously can disrupt the IFFL’s adaptive capability beyond known parameter regimes. By simultaneously altering signal and cooperativity levels beyond the parameter regimes considered in Shi et al. (2017), we observed disruptions in the IFFL’s adaptive capability. This discrepancy underscores the nuanced influence of cooperativity on network dynamics, particularly in challenging the previously established prerequisite of $n_{AC}=n_{AB}{n}_{BC}$ for adaptation. Our results show that for strong signals and negative cooperativity, the condition $n_{BC}>n_{AB}$ disrupts adaptation, providing a deeper understanding of the IFFL’s dynamic constraints. This highlights the need for further investigation of how signal strength and cooperative interactions within IFFLs influence the overall adaptability of the network, which has broader implications for interpreting the network’s behavior across varying environmental contexts.

Building upon the foundational work of Mangan and Alon (2003), our study extends the understanding of type I IFFL response dynamics. This study suggests that the response time $t_{rs}^{on}$ speeds up as the signal is activated, and response time $t_{rs}^{off}$ does not change upon signal deactivation in this motif in comparison to a simple transcription network. We extended this by showing that increased cooperativity can slow down the response time $t_{rs}^{off}$ after signal removal. This observation introduces a novel aspect of cooperativity’s dual role in both enhancing and impeding signal processing within the network. Additionally, their study showed that this motif is capable of producing a weak response to the input signals. Our results contribute to this understanding by demonstrating that cooperativity can substantially amplify signal responses, especially in specific signal profiles.

We have shown that cooperativity alters the function of the IFFL. One of the other major functions of the IFFL is fold change detection in which the behavior of the function depends on the relative change in the signal rather than the overall level of the signal (Goentoro et al., 2009). Perfect adaptation is a key attribute of fold change detection in the IFFL (Shoval et al., 2010). In our simulations, we have shown that with varying cooperativity, the IFFL may not always adapt, especially with stronger signals. Thus, our results suggest that high levels of negative and positive cooperativity can impair the fold change detection capability of the IFFL motif.

Another function of the IFFL motif is to act as a high-pass filter, allowing high-frequency signals to pass through (De Ronde et al., 2012). However, our findings reveal that the response level of the IFFL is influenced not only by the signal’s frequency but also by its duration and the level of cooperativity. Specifically, short pulses elicited the weakest response for weak signals and a slightly more robust response for strong signals. Conversely, longer-duration signals at lower frequencies resulted in the strongest response for weak signals and a comparably high response for strong signals, indicating that signal strength remains robust despite less frequent pulses. This suggests a new perspective on the high-pass filtering capability of the IFFL, showing that signal duration plays a critical role in the system’s response, beyond just signal frequency. This nuanced view of IFFL signal processing capabilities, particularly in handling short pulses and longer signals, adds depth to our understanding of network motif functionalities.

IFFL systems are shown to be widely unaffected by retroactivity (Wang et al., 2020). Retroactivity refers to the phenomenon where downstream binding sites alter the dynamics of an upstream system by sequestering transcription factors, making them unavailable for other molecular activities such as degradation, protein-protein interactions, or regulation of other genes (Wang et al., 2020). Wang and Belta (2019) analyzed the IFFL in the context of a pulse generator and response accelerator and also performed an analysis of cooperativity. They fluctuated the value of $n_{AC}$ independent of $n_{AB}$ and $n_{BC}$ so the adaptation parameter was not met. Their study suggests that retroactivity can strengthen the response for the IFFL, but more for negative cooperativity than positive cooperativity. Our study builds upon this by showing that high levels of positive cooperativity can also enhance adaptation. It could be interesting to see how retroactivity affects the adaptation capabilities of the IFFL. While another study did perform an analysis on the IFFL’s ability to adapt with retroactivity (Wang and Belta 2019), the study largely stemmed around randomizing parameters instead of focusing on a specific model. We propose further investigation into how retroactivity interacts with cooperativity to modulate network adaptability, providing a more targeted approach than random parameterization.

In synthetic biology, it has been of interest to make a more precise adaptive device (Tyson et al., 2003). There is an interest in fine-tuning the IFFL’s adaptive ability. One study performed by Hong et al. determined that adjusting the binding affinity of the genes would alter the system’s ability to adapt (Hong et al., 2018), and another study by Reeves suggested that incorporating negative feedback to the IFFL would help facilitate adaptation (Reeves, 2019). Our results provide a complementary perspective by showing that cooperativity and signal characteristics can be crucial in fine-tuning the IFFL’s adaptive responses, suggesting a potential avenue for engineering more robust adaptive motifs. This comprehensive understanding enables the fine-tuning of networks for enhanced functionality in biological systems. Even though our study involves type I IFFLs, cooperative effects and signal properties could also have an impact on other networks, like enzymatic IFFL networks and multi-IFFLs. Future studies could extend this research to explore cooperativity’s influence in broader contexts, such as enzymatic IFFLs and multi-IFFL systems, further enhancing our understanding of network behavior. These insights pave the way for future research to explore the effects of cooperativity and signal characteristics across a broader range of networks, aiming to deepen our understanding of their dynamic interactions. In essence, this study adds to the knowledge of network motifs’ complex behaviors, underscoring cooperativity and signal characteristics’ vital roles and setting the stage for further exploration into the mechanisms underlying cellular information processing.