Molecular mechanisms of detection and discrimination of dynamic signals

Many molecules decode not only the concentration of cellular signals, but also their temporal dynamics. However, little is known about the mechanisms that underlie the detection and discrimination of dynamic signals. We used computational modelling of the interaction of a ligand with multiple targets to investigate how kinetic and thermodynamic parameters regulate their capabilities to respond to dynamic signals. Our results demonstrated that the detection and discrimination of temporal features of signal inputs occur for reactions proceeding outside mass-action equilibrium. For these reactions, thermodynamic parameters such as affinity do not predict their outcomes. Additionally, we showed that, at non-equilibrium, the association rate constants determine the amount of product formed in reversible reactions. In contrast, the dissociation rate constants regulate the time interval required for reversible reactions to achieve equilibrium and, consequently, control their ability to detect and discriminate dynamic features of cellular signals.


Results
Mechanisms for the detection and discrimination of the durations of signals. Thermodynamic and kinetic parameters regulate chemical reactions, but their individual contributions vary 18,24 . For a reversible reaction of binding and unbinding (reaction 1) between a molecule M and a ligand L forming the complex LM: the dissociation constant (K D ) quantifies the binding affinity of the complex LM formed at equilibrium, which is mathematically defined by equation 1: D where the brackets indicate concentrations. According to equation 1, the K D of a reversible reaction specifies which species are more abundant at equilibrium (the reactants L and M or the product LM). The K D of a reversible reaction is related with its Gibbs free energy (ΔG), which designates the stability of the product LM relative to the reactants L and M ( Fig. 1A) 24,25 . As thermodynamic quantities, K D and ΔG define the relative concentrations of its components at equilibrium, but do not indicate whether the reversible reaction occurs in a feasible time 24 . It is the energy barrier (energy of activation, E A ) that must be overcome during a reaction that determines its velocity 24 . A low-energy barrier corresponds to a fast reaction and a high-energy barrier corresponds to a slow reaction (Fig. 1A). E A regulates the rate constant (k) of a reaction, but not whether it is thermodynamically favourable 24 . When reactions occur at equilibrium, they are under thermodynamic control and regulated by thermodynamic parameters such as K D 24,25 . When they proceed outside equilibrium, they are under kinetic control and their rate constants determine their outcomes 24,25 .
In biological systems, the concentrations of drugs and endogenous signals vary often with time scales faster than the rates of binding and unbinding from their cellular targets 16,18,20 . In consequence, many cellular reactions do not achieve equilibrium or steady-state 16,18,20 . We hypothesised that only reactions that proceed outside mass-action equilibrium detect and discriminate dynamic cellular signals. To test this hypothesis, we simulated the interactions of twelve different molecules (M1-M12) with a ligand L to form the corresponding complexes LM1-LM12 (Fig. 1B). We simulated the formation of three complexes (LM1-LM3) with high affinity at equilibrium (K D = 0.01 µmol.L −1 ), six with moderate affinity (LM4-LM6 with K D = 0.1 µmol.L −1 and LM7-LM9 with K D = 1 µmol.L −1 ) and three (LM10-LM12) with low affinity (K D = 10 µmol.L −1 ). For each K D , we implemented three different sets of rate constants of association (k f ) and dissociation (k b ) to simulate reactions with varied velocities (Fig. 1B). We then obtained the dose-response curves for the formations of LM1-LM12 as functions of free concentrations of L ([L] free ) at equilibrium (Fig. 1C) to ensure that the values of K D used in the model matched the concentration of free ligand ([L] free ) required to promote the half-maximum activation of each complex implemented, which we verified by fitting the equation (2): where A is the activity (i.e. normalized concentration) of the complexes LM1-LM12, A max corresponds to their maximum activity (=1), the term n hill is the Hill coefficient and K Dapp is the apparent K D . As expected, independently of the rate constants used for the reactions simulated, the K Dapp s of the dose-responses corresponded exactly to the K D s implemented (Fig. 1C). We set these K Dapp s as the control K D s of LM1-LM12 hereafter. All curves presented n Hill equal to 1. Next, we used square pulses of [L] free with different durations and peak concentrations to verify how the thermodynamic and kinetic parameters used regulate the detection and discrimination of dynamical signals, which we defined as the ability of molecules to respond and display different levels of activation to changes in the temporal properties of their signal activators. The durations and amplitudes of the pulses of [L] free were set in the simulations in a non-conservative manner. Thus, the concentrations of L used in the pulses were buffered. Consequently, all molecules M1-M12 were exposed to the same signals and there was no competition among them.
The association and dissociation of complexes that have identical affinities at equilibrium proceeded with different time courses when we used square pulses of [L] free as input signals (Fig. 2). Moreover, complexes that have the same K D at equilibrium displayed different levels of activation (Fig. 2). These differences were strongly pronounced for short pulses, which possess durations within the range of pivotal cellular signals (varying from milliseconds to few seconds) [26][27][28] , and gradually disappeared as we stimulated the model with pulses that were long enough to allow the reactions to achieve equilibrium. For instance, LM10, LM11 and LM12 presented very different levels of activation when stimulated by brief pulses of [L] free (10 ms) ( Fig. 2A,B), but equivalent activations for pulses of [L] free of 100 s (Fig. 2C,D To analyse these data, we used equation 2 to fit dose-responses curves of the peak concentrations of LM1-LM12 obtained as functions of the peak amplitudes of the pulses of [L] free with different durations, and estimated the values of their K Dapp and n Hill for comparisons with the results of the system at equilibrium. The dose-response curves of most complexes showed that the durations of pulses of [L] free modulated their formations (Fig. 3A) by changing the values of K Dapp in comparison to their control K D s in a duration-dependent manner. Figure 3B shows the K Dapp /K D ratios to facilitate their comparisons, we listed the exact values of K Dapp in Suppl. Table S1. As we increased the durations of pulses of [L] free , the values of K Dapp decreased until they matched the control K D s (K Dapp /K D = 1) indicating that the reversible reactions had reached equilibrium, which happened at different pulse durations for the molecules simulated (Fig. 3B). The dynamic changes of K Dapp s showed that the molecules detected the durations and the peak concentrations of the pulses of [L] free by temporally integrating these signals over time. Once the durations of pulses of [L] free were sufficiently long for the reactions to achieve mass-action equilibrium, they became insensitive to time and detected only the variations in the concentrations of L.
Our results demonstrated that the key point for the temporal discrimination of cellular signals relies on the different time scales in which each reversible reaction reach thermodynamic equilibrium. The longer it takes for a reaction to reach equilibrium, the larger is the range of durations of signals it can detect and discriminate by dynamically changing its K Dapp (Fig. 3B).  The results of Fig. 3B also revealed that the dissociation rate constants (k b s) used in our simulations played a pivotal role in determining the time required for each reversible reaction to reach equilibrium (Fig. 3B). In a reversible reaction, the slower is the k b the longer it takes for the activation of a given molecule to peak 29 . Our results indicated that, for the conditions that we simulated, the slower was the k b the longer was the time interval required for the reactions to reach equilibrium independently of the k f used. For instance, the reactions of formations of LM7 and LM10 occurred with very fast k b s in our simulations. Their formations proceeded at equilibrium for all pulse durations tested, consequently, they only detected the concentrations of L (Fig. 3B). However, the formation of LM1, which happened with the same k f used for the formations of LM7 and LM10 but a slower k b , required pulses of 500 ms to exhibit K Dapp compatible to its control K D . Reactions that have same k b s required identical durations of pulses of [L] free to reach equilibrium independently of their k f s ( Fig. 3B and Suppl. Table S1,  . Table S1). For instance, the values of K Dapp obtained for the formation of LM3 were much more similar to the K Dapp s of LM9 for most pulse durations tested than the K Dapp s of LM1 (Suppl. Table S1), even though LM1 and LM3 have identical K D s at equilibrium and the K D of LM9 is 100-fold weaker. The larger were their k f s, the lower were their K Dapp s observed at non-equilibrium. This result indicates that complexes with faster k f s activate preferentially outside mass-action equilibrium. However, the closer the reversible reactions got to reaching equilibrium, the lesser their outcomes depended on their k f s and the more they depended on their thermodynamic affinities as expected 30 .
In addition to the changes in K Dapp , we verified that durations of the pulses of [L] free promoted variations in the n Hill for the reactions that proceeded outside mass-action equilibrium, which displayed n Hill larger than 1 even though the components of our system have no allosteric cooperativity (Fig. 4). The parameter n Hill is commonly defined as an "interacting-coefficient" that reflects the cooperative binding of ligands to multiple sites of a molecule 31 . Nevertheless, it is important to note that, in addition to allosteric cooperativity, n Hill larger than 1 can indicate ultrasensitivity. Multiple mechanisms promote ultrasensitivity including feedback loops, small changes in reactions near saturating conditions, distributive phosphorylations, among others [32][33][34][35][36] . In a ultrasensitivity system, n Hill designates the degree of bistability 33,34,36 . In our results, we verified that the values of n Hill became larger than 1 only for reactions happening outside mass-action equilibrium. For these reactions, the values of n Hill increased as we reduced the durations of pulses of [L] free . The changes of n Hill resulted from ultrasensitivity promoted by the filtering of fast signals with low amplitudes as if they were noise. Previously, it was proposed that biology evolved to use non-equilibrium to efficiently discriminate signals from noise 16 , which is consistent with our results. We had observed similar changes of n Hill previously 18 . Detection and discrimination of frequencies and number of pulses of dynamic signals. Next, we investigated how the kinetic and thermodynamic parameters underlie the discrimination of interpulse intervals and number of pulses of trains of signals of L, a property displayed by several enzymes and signalling pathways 18,22,23,37 . We stimulated the formation of LM1-LM12 with trains of ten pulses of [L] free delivered at 1 Hz (1 s of interpulse interval), 10 Hz (100 ms of interpulse interval), or 100 Hz (10 ms of interpulse interval). Each pulse had duration of 50 ms (Suppl. Fig. S1A-C) or 100 ms (Suppl. Fig. S1D-F). Figure 5 and Suppl. Fig. S2 show examples of the time courses of LM1-LM12 observed. To verify whether the formation of LM1-LM12 detected the interpulse interval and the number of pulses of L simulated, we measured the peak amplitude of LM1-LM12 formed as functions of the peak of each pulse of [L] free within a train (Suppl. Fig. S3). We used these data to fit ten dose-response curves for each frequency tested using equation 2 (Suppl. Figs S4 and S5). Each curve corresponded to the formations of LM1-LM12 observed for a specific pulse number (Suppl. Figs S4 and S5). With these curves, we investigated whether the values of K Dapp and n Hill varied during each train and quantified their discrepancies from the control K D s (Fig. 1C). The K Dapp s obtained ( Fig. 6A and B) revealed different types of dynamic signal discriminations that relied heavily on the k b s used in the reactions of the different complexes. The K Dapp s for the formations of LM4 (k b = 100 s −1 ), LM7 (k b = 1000 s −1 ), LM10 (k b = 10000 s −1 ) and LM11 (k b = 100 s −1 ) corresponded to the control K D s to all situations tested and demonstrated that these complexes were insensitive to the number of pulses, the interpulse intervals and the durations of pulses used in the simulations. However, as we decreased the k b s, this scenario changed.
The K Dapp s of LM1 and LM8, which had k b = 10 s −1 , detected and discriminated mainly the interpulse interval between the signals of [L] free used. Moreover, we observed changes of their K Dapp s during the initial pulses of [L] free released at 10 Hz and 100 Hz. Thus, LM1 and LM8 also discriminated a limited number of pulses released at intermediary or high frequencies, because their interpulse intervals were shorter than the time required for the inactivation of both complexes. Consequently, there were summations of their activations for the initial pulses of trains released at 10 Hz and 100 Hz (Figs 5 and S2), which promoted alterations in their K Dapp s ( Fig. 6A and B). In contrast, pulses released at 1 Hz had a long interpulse interval (1 s) that prevented the accumulation of LM1 and LM8 from one pulse to another (Figs 5 and S2). The K Dapp s of LM1 and LM8 matched the control K D s when stimulated with 3 or more pulses of L with 100 ms of duration released at 100 Hz.
LM5 and LM12 (k b = 1 s −1 ) exhibited K Dapp s that changed as functions of both the interpulse interval and the number of pulses of [L] free released at 10 Hz and 100 Hz. For signals of [L] free released at 1 Hz, the K Dapp s of LM5 and LM12 changed only during the initial four pulses (Fig. 6A and B). Thus, LM5 and LM12 acted as good detectors and discriminators of the interpulse intervals for all frequencies tested and of the number of pulses of [L] free released at moderate to high frequencies, but poor detectors of the number of pulses released at a low frequency ( Fig. 6A and B).
The K Dapp s of LM2 and LM9 (k b = 0.1 s −1 ) detected and discriminated the number of pulses of [L] free for all frequencies tested. In addition, LM2 and LM9 discriminated the interpulse interval of pulses released at 1 Hz from pulses released at 10 Hz or 100 Hz. However, their K Dapp s did not discriminate the interpulse interval of pulses released at 10 Hz from pulses released at 100 Hz ( Fig. 6A and B). LM3 and LM6, the two complexes with the slowest k b s implemented (k b = 0.001 s −1 and 0.01 s −1 , respectively), presented K Dapp s that discriminated the number of pulses of [L] free , but were insensitive to their interpulse intervals ( Fig. 6A and B).
None of the K Dapp s of LM2, LM3, LM5, LM6, LM9 and LM12 matched their control K D s ( Fig. 6A and B, black dashed lines) evidencing that they did not reach mass-action equilibrium in the situations simulated. Moreover, the values of n Hill for all the complexes that did not present values of K Dapp compatible with the control K D s were larger than 1, which indicated that their activations had a bistability not observed at equilibrium (Suppl. Fig. S6). The k b s used in the simulations regulated the K Dapp /K D ratio, as observed in the previous session. Hence, molecules with identical k b s (LM1/LM8, LM2/LM9, LM5/LM12) exhibit K Dapp s that diverged from K D with equivalent magnitudes (Suppl. Fig. S7).
Detection and discrimination of dynamic signals in sequential reactions. Next, we explored the detection and discrimination of dynamic signals in sequential reactions. Firstly, we implemented the reactions of association/dissociation of LM1-LM12 with the targets T 1 -T 12 using one set of rate constants (k f = 10 µmol −1 .L.s −1 , k b = 0.1 s −1 , K D = 0.01 µmol.L −1 ). Specifically, LM1 reacted with T f1, LM2 with T f2, LM3 with T f3, and so on, which resulted in twelve ternary complexes LM1T 1 -LM12T 12 formed according to the sequential reactions: where n = 1, 2, 3, …, 12. The parameters k f and k b refer to the rate constants used for the association/dissociation of LM1-LM12 (Fig. 1B).
In a different set of simulations, we implemented the interactions of LM1-LM12 with the targets T′ 1 -T′ 12 to form LM1T′ 1 -LM12T′ 12 using a different set of rate constants (k f = 0.1 µmol −1 .L.s −1 , k b = 0.001 s −1 , K D = 0.01 µmol.L −1 ), but equivalent sequential reactions: Initially, we simulated the formations of LM1T 1 -LM12T 12 and LM1T′ 1 -LM12T′ 12 at steady-state as functions of different [L] free to obtain dose-response curves fitted with equation 2 and estimate the control K D s and n hill . Note that the K D used for the interactions of LM1-LM12 with T 1 -T 12 and with T′ 1 -T′ 12 are identical (K D = 0.01 µmol.L −1 ). Nevertheless, the control K D s for the formations of LM1T 1 -LM12T 12 and LM1T′ 1 -LM12T′ 12 as functions of [L] free varied according to the K D of their binary precursors. Thus, the ternary complexes formed by LM1, LM2 and LM3 (LM1T 1 , LM2T 2 , LM3T 3 , LM1T′ 1 , LM2T′ 2 and LM3T′ 3 ) exhibited control K D s as functions of [L] free of approximately 0.00001 µmol.L −1 (Fig. 7, Suppl. Table S2), which is 1000-fold lower than the control K D s of their binary precursors (Suppl. Table S1). The ternary complexes formed by LM4, LM5 and LM6 (LM4T 4 , LM5T 5 , LM6T 6 , LM4T′ 4 , LM5T′ 5 and LM6T′ 6 ) had control K D s as a function of [L] free of 0.0001 µmol.L −1 , which is higher than the K D s of the ternary complexes LM1T 1 -LM3T 3 and LM1T′ 1 -LM3T′ 3 , but is also 1000-lower than the K D s for the formations of their precursors LM4, LM5 and LM6 (Fig. 7, Suppl. Table S2). The same pattern was also observed for the other ternary complexes simulated. Consequently, all the ternary complexes exhibited control K D s for their activations as functions of [L] free at steady-state approximately 1000-fold lower than the K D s of their binary precursors, which demonstrated that the binding of each binary complex to a target affected its interaction with L. This type of alteration is commonly observed in biological systems 18,38 .
Next, we used square pulses of [L] free with different durations (100 ms, 500 ms, 1 s, and 5 s) and peak concentrations to investigate how they regulated the formations of the ternary complexes LM1T 1 -LM12T 12 and LM1T′ 1 -LM12T′ 12 . The results obtained were used to trace dose-responses curves of the peak concentrations of LM1T 1 -LM12T 12 and LM1T′ 1 -LM12T′ 12 as functions of the peak [L] free using equation 2. The curves were used to verify whether the ternary complexes simulated detected and discriminated the durations of the signals of [L] free by changing their values of K Dapp , n Hill and maximum activation (A max ) in comparisons to the values observed at steady-state (Fig. 7A,B).
In the previous sessions, we demonstrated that the rate constants used for the interactions of L with its targets M1-M12 modulated their K Dapp and n Hill . The results showed in Fig. 7A,B revealed that the rate constants used in the reactions of M1-M12 with L can also modulate the values of A max obtained for the dose-response curves of the formations of their respective ternary complexes. Thus, binary precursors that dissociated with fast k b s (≥1 s −1 ) from L impaired the A max observed for the activation of their corresponding ternary complexes. However, such impairment only occurred for the ternary complexes formed with slow k f (0.1 µmol −1 .L.s −1 ), which indicates that it is the combination of the k b of the precursor with the k f for its interaction with its target that regulates A max (Fig. 7B) and, in addition, also affected the values of K Dapp and n Hill (Fig. 7C,D and Suppl. Fig. S8).
Our results demonstrated that all ternary complexes simulated decoded the pulses duration tested and exhibited changes in their K Dapp values in comparisons to their control K D values observed at steady-state (Fig. 7C,D). Yet, all the ternary complexes that exhibited impairments of A max for the durations of pulses of [L] free also showed higher shifts in their values of K Dapp in comparison to the values observed at steady-state (Fig. 7C,D). Thus, the combination of short half-lives of fast dissociating binary precursors greatly impaired the formation of ternary complexes that associate with slow k f . For instance, the binary complexes LM4, LM5 and LM6, which share the same control K D , dissociated with k b of 100 s −1 , 1 s −1 and 0.01 s −1 , respectively. Due to the fast inactivation rate of LM4, the dose-response curves of activation of LM4T′ 4 as functions of pulses of [L] free with different durations exhibited strong modulations of A max , but a similar modulation was not observed for the activation of LM4T 4 , which reacted faster with its precursor. Moreover, LM4T′ 4  The curves for the activations of LM5T′ 5 showed a similar pattern observed for the curves of LM4T′ 4 , but with less pronounced modulations as its binary precursor had a slower k b . LM5T′ 5 also exhibited higher values of K Dapp than LM5T 5 , though both complexes have identical control K D values and interact with the same binary precursor. In contrast, the dose-response curves of LM6T′ 6 had no variation in their A max because its precursor, LM6, had a slow k b . In addition, the values of K Dapp verified for LM6T 6 were very similar to the values observed for LM6T′ 6 . Thus, the slow time course for the inactivation of LM6 allow it to act as a "molecular memory" and propagate the transient signals of L for longer periods in comparison to LM4 and LM5.
In Suppl. Fig. S8A and B we plotted the n Hill obtained for each dose-response curve showed in Fig. 7A and B. Our results demonstrated that, outside mass-action equilibrium, the formations of LM1T 1 -LM12T 12 showed higher values of n Hill in comparison to their binary precursors LM1-LM12, which indicated an increase in bistability along the sequential reactions simulated caused exclusively by kinetic factors (Suppl. Fig. S8A). Nevertheless, for the ternary complexes LM1T′ 1 -LM12T′ 12 , the curves that presented impairments of A max exhibited values of n Hill close to 1 and often lower than the values observed for their binary precursors (Suppl. Fig. S8B). Detection and discrimination of dynamic signals in competing systems. The last stage of our work consisted in investigating how competition among different molecules shapes their response. For this analysis, we used a simplified version of our system containing only the formation of LM4, LM5 and LM6, which exhibit very distinct patterns of activation even though they share the same control K D . Each one of this species were responsible for the activation of two targets simulated as described in the previous session. However, for this stage of the work, we simulated the two targets activated by each LM complex competing for their activators (Fig. 8A). We then used pulses of L with different durations (0.5 s and 5 s) to verify the consequences of competition in the results previously described (Fig. 8B-D).
The competition for a common activator had two consequences in our system. For the formation of fast-reacting ternary complexes (LM4T 4 , LM5T 5 , and LM6T 6 ), independently of the pulse duration tested, the competition with slow-reacting ternary complexes (LM4T′ 4 , LM5T′ 5 , and LM6T′ 6 ) did not affect the maximum amplitude of activation for each pulse tested, but accelerated the inactivation of LM6T 6 ( Fig. 8B-D). In contrast, for the slow-binding complexes (LM4T′ 4 , LM5T′ 5 , and LM6T′ 6 ), the presence of competition affected their maximum activation, which reduced slightly the A max of their dose-response curves of activation (8E-F, Suppl. Table S3). Moreover, for LM6T′ 6 competition promoted a delay in its activation curves (Fig. 8B-D) and shifted the K Dapp of its dose-response curves (8E-F). Taken together, these results indicated that the effects of competition vary depending on the combination of the half-life of the initial precursor (LM4, LM5, and LM6) with the rates of activation of the subsequent molecules.

Discussion
Time is an important variable in the biological environments. The temporal dynamics of cellular signals regulate many molecules and signalling networks 13,14,18,22,23,37 . However, the comprehension of the molecular mechanisms that underlie the temporal regulation of cellular processes remains a challenge because much of our understanding of signalling processes results from data obtained at equilibrium or steady-state conditions. In this work, we focused on the identification of the molecular mechanisms that underlie the detection and discrimination of the temporal features of signals. For that, we simulated reactions of association and dissociation between molecules and a ligand. Previously, we used realistic models of different biomolecules with intricate interactions with endogenous ligands to investigate their modes of activation 18 . However, the level of complexity of our previous work made the comparison among different molecules difficult. Thus, in this work we have opted to use a generic and simpler system.
Biological systems are open systems in constant change 16,18,19 . The concentrations and levels of activation of biomolecules fluctuate continually, which sets perfect conditions for several reactions to proceed at non-equilibrium 16,18,19 . Consequently, many reactions in biological systems detect and discriminate dynamic signals and use temporal properties to display differential patterns of activation 13,14,22,23,37 . Because these reactions are sensitive to the temporal features of their components, they are under kinetic control and thermodynamic parameters such as K D do not predict their outcomes, which is a conclusion fully supported by our results.
Several data have revealed that the association and dissociation rate constants (k f and k b , respectively) for the interaction between biomolecules or of a drug with its targets are often more important than the binding affinity of the resulting complexes 19,21,39,40 . However, though in some systems the values of k f are crucial 21 , especially for the interaction of drugs with their targets the k b appears to play a more fundamental role 19,40 . Our results indicated that both rate constants are important in the detection of dynamical signals because they play different roles. At non-equilibrium, the k f s used in our simulations played a predominate role in determining the levels of activation of the ligand/molecule complexes simulated. The faster were k f s, the lower were the K Dapp s obtained, which indicate that molecules with fast k f s activate better at non-equilibrium. Similar results were observed previously 21 . Our results showed that the affinities observed at equilibrium do not ensure which molecules will activate in larger amounts 18,19,21 . A high affinity complex with slow rate constants can display a K Dapp equivalent to the K Dapp of a weak affinity complex when their reactions occur outside mass-action equilibrium. Only when the reactions approach mass-action equilibrium their rate constants become less important and their outcomes gradually become defined by their control K D s 30 .
In our simulations, the k b s played a pivotal role determining the time required for each reaction to achieve equilibrium. The slower was the k b used, the larger was the range of signal durations that a reaction detected and discriminated and the longer was the time required for it to reach equilibrium. Slow k b s also promoted reactions sensitive to the frequencies and number of pulses of reacting signals. Nevertheless, the slower was the k b used in our simulations, the better the reactions detected the number of pulses despite of their interpulse interval, which indicates that reactions with slow k b s integrate the signals over time more efficiently. These results contribute to explain why some molecules are sensitive to the interpulse interval of their signals, while others count pulses of signals regardless of their frequencies 18,23 . In addition, this type of information is crucial for the design of artificial signalling systems and probes 41,42 . We also demonstrated that the k b s played a crucial role in the propagation of dynamic signals. Thus, ligand/molecule complexes that dissociate with fast k b s do not propagate efficiently fast signals for slow-interacting targets. Consequently, in this scenario, complexes that dissociate slowly propagate dynamic signals better. Several observations have demonstrated that often drugs that dissociate slowly from their endogenous targets are more efficient, though the reasons for this process are not totally understood 19,40 . In this work, we have not explored this process specifically. The time intervals of the dynamic signals that we investigated are more compatible with physiological signals. However, our observations are not restricted to endogenous molecules and might explain the role of rate constants on the efficacy of drugs as well.

Methods
We implemented the computational models using BioNetGen 43 , a rule-based software for modelling signaling networks and pathways. All simulations were solved deterministically.
To define the parameters of the model, we used K D s (0.01 µmol.L −1 , 0.1 µmol.L −1 , 1 µmol.L −1 and 10 µmol. L −1 ) commonly found for the interactions between biomolecules 10,18,38,44,45 . We defined the kinetic parameters of the model by setting a k f of 1000 µmol −1 .L.s −1 as our upper limit, which is consistent with the second order rate constant of a diffusion limited reaction in the cellular milieu. The other two k f s used in the model (10 µmol −1 .L.s −1 and 0.1 µmol −1 .L.s −1 ) were defined by dividing 1000 µmol −1 .L.s −1 by 100 and 10000, respectively, in order to simulate reactions that cover a large range of velocities. All the k f s used are within the range of values observed for the interactions of biomolecules, which typically vary from 0.001 µmol −1 .L.s −1 to 1000 µmol −1 .L.s −1 46,47 . For instance, calcium ions interact with many calcium-binding proteins with rates typically in the range of diffusion-limited reactions 48 . The complex calcium/calmodulin activates many targets with rate constants of binding around 1-10 µmol −1 .L.s −1 18,49 . In contrast, protein kinase A, a tetrameric enzyme involved in several signaling processes, has rate constants for the binding of its catalytic and regulatory subunits that vary around 0.5 to 0.05 µmol −1 .L.s −1 44 . We estimated the k b s for the reactions of the model using equation 3: The concentration of M1-M12 was set to 1 µmol.L −1 initially (Figs 1-6). In the simulations showed in Figs 7 and 8, we set the initial concentrations of M1-M12 (M4-M6 in Fig. 8) to 10 µmol.L −1 and the concentrations of T 1 -T 12 and T′ 1 -T′ 12 to 1 µmol.L −1 (T4-T6 and T4′-T6′ in Fig. 8). The interaction of LM1-LM12 with T 1 -T 12 and T′ 1 -T′ 12 were simulated separately.
To obtain the dose-response curves at steady-state (Figs 1 and 7A,B), we performed the simulations until the reactions had reached steady-state. Then, we annotated the final concentrations of the complexes investigated and the concentration of free L ([L] free ). To trace the dose-responses curves using square pulses of L, we simulated non-conservative signals of [L] free . The durations and peak concentrations of the pulses were set by the simulations and were not changed due to interactions with M1-M12, which prevented the competition among them. We defined the durations of the pulses of L setting 10 ms as our lower limit, which corresponds to fast calcium ion signals 28 . We then systematically increased the durations of the pulses until all complexes LM1-LM12 exhibited K Dapp s compatible with their control K D s. The data used in the dose-responses curves showed in Figs 3 and 6 corresponded to the peak activations of LM1-LM12 obtained as functions of the peak [L] free . In Figs 3 and 6, we varied the peak concentrations of the pulses of L from 0 µmol.L −1 to ~450 µmol.L −1 to achieve saturation of all complexes LM1-LM12. In Fig. 7, we varied the peak concentrations of the pulses of L from 0 µmol.L −1 to ~200 µmol.L −1 to saturate the complexes LM1T 1 -LM12T 12 and LM1T′ 1 -LM12T′ 12 . Nevertheless, we opted to plot the curves of Figs 3, 6 and 7 with a smaller range of concentrations of [L] free for better visualization. We fitted the dose-response curves showed in Figs 3, 7 and 8 and Suppl. Figs S4, S5 and S8 using Matlab curve fitting tool with 95% of confidence interval. The full description of the reactions and the parameters used in the models are listed in Suppl. Table S4.