Dichotomous Nature of Bistability Generated by Negative

May 27, 2019 - Positive cooperativity in receptor–ligand binding plays an important role in cell signaling as it generates an ultrasensitive respons...
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Cite This: ACS Synth. Biol. 2019, 8, 1294−1302

Dichotomous Nature of Bistability Generated by Negative Cooperativity in Receptor−Ligand Binding Anupam Dey and Debashis Barik* School of Chemistry, University of Hyderabad, Central University P.O., Hyderabad, 500046 Telangana, India

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ABSTRACT: Positive cooperativity in receptor−ligand binding plays an important role in cell signaling as it generates an ultrasensitive response, a requirement for nonlinear phenomena such as bistability and oscillations in feedback regulated reaction networks. On the other hand, negative cooperativity typically produces a hyperbolic response and is thus less explored. However, recently negative cooperativity was shown to generate an ultrasensitive response under the condition of strong ligand affinity. In this work, we have used mathematical modeling to investigate the effect of negative cooperativity in receptor− ligand interaction on the bistability in a positive feedback regulatory motif. We systematically investigated the effect of negative cooperativity, modifying the two equilibrium constants of the receptor−ligand binding, on the robustness and tunability of bistability. We show that in the regime where negative cooperativity exhibits robust bistability, positive cooperativity results in poor bistability and vice versa. Further we find that the robustness and tunability of bistability depend crucially on the stability of singly and doubly engaged receptors. Our modeling highlights the ability of negative cooperativity to produce complex phenomena with potential applications in designing synthetic devices or in explaining experimental observations in cell biology. KEYWORDS: negative and positive cooperativity, positive feedback, bistability, bifurcation analysis ΔG in the subsequent binding steps becomes more negative, whereas in the negative cooperativity it becomes less negative in the subsequent steps. In the case of noncooperative binding, ΔG does not exhibit any change over subsequent binding steps.15 From a typical sigmoidal ligand binding curve, the use of the Hill function allows one to extract information about the cooperative nature of binding kinetics.16,17 The value of the Hill coefficient (nH) characterizes the cooperativity where nH > 1, nH < 1, and nH = 1 represent positive, negative, and noncooperative binding, respectively. Making conclusions about cooperativity from fitting the signal response curve with the Hill equation can sometimes be challenging as it requires an accurate estimation of concentrations of the free ligand and receptor.18,19 Positive cooperativity generates a stiff sigmoidal response with a large dynamic range, and it also creates a threshold in the amount of ligand needed for full engagement of the receptor.20 Thus, positive cooperativity exhibits an ultrasensitive switch-like behavior. Owing to its ultrasensitive nature, positive cooperativity carries a great importance in generating system level phenomenon such as bistability and oscillations in feedback regulated networks.21−23 On the contrary to positive cooperativity, negative cooperativity generally leads to a hyperbolic response with a

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large number of cellular responses against external cues are controlled by the binding of signaling molecules or ligands to the cell surface receptors. Receptor−ligand dynamics is central to many cellular functions as the engagement of ligands to cell surface receptors triggers intracellular signaling pathways allowing the cell to perform various physiological functions. For example, the class I and class II cytokine superfamily,1−5 TGF-β receptor,6 receptor tyrosine kinases,7,8 and G-protein coupled receptors9,10 mediated signaling control cell proliferation, apoptosis, differentiation, immune regulations, metabolism, and fertility and longevity at the organism level. The engagement of ligands to oligomeric receptors often leads to cooperative receptor−ligand binding dynamics. In cooperative binding, the engagement of one ligand to an oligomeric receptor influences the engagement of subsequent ligands to the same receptor, and allosteric interaction among the binding sites is often recognized as the mechanism for cooperativity.11,12 In positive cooperativity, the binding of a ligand to a receptor increases the receptor’s affinity toward the binding of subsequent ligands, whereas in negative cooperativity, the engagement of one ligand dampens the receptor’s affinity to bind subsequent ligands.13,14 Cooperativity in receptor−ligand engagement can be best described by the Gibbs free energy change (ΔG) in the subsequent binding of ligands to a multimeric receptor that is capable of engaging multiple copies of the same ligand. In positive cooperativity © 2019 American Chemical Society

Received: December 12, 2018 Published: May 27, 2019 1294

DOI: 10.1021/acssynbio.8b00517 ACS Synth. Biol. 2019, 8, 1294−1302

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ACS Synthetic Biology

active receptor also upregulates production of the receptor. This module is essentially a fusion of two positive feedback loops in the model. Before going into the details of both models, we describe the binding mechanism of the ligand (L) with the dimeric receptor (R2). In both models the receptor−ligand binding mechanism is the same. The binding of the ligand to the receptor is assumed to be distributive in which each ligand−receptor encounter leads to attachment of a single ligand molecule to the receptor. Therefore, the full engagement of the receptor is a two-step sequential process: first, one ligand binds to the unoccupied receptor (R2) and then another ligand binds to the singly occupied receptor (R2L1) to produce the fully engaged receptor (R2L2). Similarly the dissociation of the ligand from the engaged receptor is also a two-step process. A schematic representation of the ligand−receptor binding is given in Figure 1. We have considered both the ordered and disordered

small dynamic range and may not exhibit any signaling threshold.13,24 Negative cooperativity has been found in several receptor−ligand interactions such as in the epidermal growth factor receptor,25−27 insulin receptor,28−30 and glycoprotein hormone receptor.31 Although negative cooperativity is as common as positive cooperativity in biological systems, the former was never really explored much. In the case of the dimeric insulin receptor, Kiselyov et al.32 developed a detailed mathematical model based on the concept of the harmonic oscillator to account for several key experimental observations including negative cooperativity. Recently Ha et al.33 showed that in dimeric receptor−ligand binding kinetics, negative cooperativity can generate a stiff ultrasensitive response with a threshold if the affinity of the ligand toward the receptor is strong enough to cause sufficient depletion of the ligand from the reaction mixture. Therefore, negative cooperativity can also be a source of nonlinearity that is required for bistability and oscillations. In a bistable signal response, the regulatory system exhibits two stable steady states separated by an unstable steady state. Owing to its “all” or “none” kind of response, bistability has been found to be associated with various decision making processes in living cell, for example, proliferation,34−37 apoptosis,38 differentiation,39−41 and memory.42−44 Further, complex nonlinear behavior in gene regulatory networks have been achieved using synthetic biology techniques.45−47 Synthetic genetic oscillators48−53 and bistable switches in gene regulatory networks54−56 rely on nonlinearity in the binding of the transcription factor on the promoter site of the DNA. In light of its potential to generate an ultrasensitive response, we investigated the role of negative cooperativity in a positive feedback loop that may exhibit bistable signal response. In our model we considered a dimeric cell surface receptor that sequentially binds to two molecules of extracellular ligands. We incorporated this receptor−ligand module in a positive feedback loop for which the fully ligated receptor upregulates synthesis of the ligand. We systematically investigated the dependence of the robustness of bistability on the two equilibrium constants of sequential receptor−ligand engagement to study the effect of negative and positive cooperativity on bistability. We used one- and two-parameter bifurcation analyses of the model developed based on mass-action rate laws of chemical reactions. We found that besides positive cooperativity, negative cooperativity also generates robust bistable signal responses. However, the dependence of negative cooperativity on the robustness was dichotomic in naturethe robustness was crucially dependent on how negative cooperativity was achieved by adjusting the two equilibrium constants. Further the effect of negative cooperativity on the bistability can be very different depending on the nature of the input signal in the network.

Figure 1. Schematic representation of the model with a positive feedback loop in ligand production. The dimeric receptor (R2) sequentially binds to the ligand (L). The doubly engaged receptor (R2L2) is the active receptor that upregulates the production of the ligand creating a positive feedback loop. The solid and dashed lines represent chemical reactions and the catalytic effect on a chemical reaction, respectively. The open-ended arrow indicates a degradation reaction. The rate constants of chemical reactions are indicated on top of the respective chemical reaction.

binding of ligand to the receptor. In disordered binding, the binding of the first and second ligand molecules does not follow a specific order. Therefore, when a single ligand binds to a dimeric receptor, the resulting “conformers” are not distinguishable from one other. In general,57−59 for disordered binding of a multimeric receptor with N binding sites, the number of ways ith binding can happen is given by the binomial factor Ni . For a dimeric receptor this factor becomes 2 (N = 2 and i = 1) for a first binding event. Similarly there are two possible ways by which a ligand can unbind from the fully engaged receptor R2L2. Therefore, the rate of formation singly engaged receptor increases by a factor of 2 and similarly the rate of dissociation of a ligand from a fully engaged dimeric receptor increases by a factor of 2. In the kinetic equations of the model (Table 1), we introduced the parameter ν (= 2) to represent the disordered nature of ligand binding. Whereas in ordered binding, the engagement of ligands happens in a specific order with ν = 1. In the main text, we report the results with disordered binding of the ligands to the receptors. The dissociation constant of receptor−ligand binding is a typical measure of binding affinity of the ligand to the receptor. The dissociation constant of the first and the second steps of ligand engagement are given by K1 and K2, respectively, for which the dissociation constants are defined as the ratio of rate constants corresponding to unbinding and binding reactions

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RESULTS AND DISCUSSIONS The basic framework of our model consists of a dimeric receptor in which each monomer is capable of engaging one ligand molecule. The doubly engaged receptor is the active form of the receptor−ligand complex and regulates the downstream signaling pathway. We have considered two different cases of downstream signaling pathways. In one case the active receptor upregulates the gene responsible for synthesizing the ligand molecule. This module creates a positive feedback loop between the ligand and the receptor− ligand complex. In the other module we considered that the 1295

DOI: 10.1021/acssynbio.8b00517 ACS Synth. Biol. 2019, 8, 1294−1302

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~bard/xpp/xpp.html). In Figure 2 we present one-parameter bifurcation diagrams of the model with a basal synthesis rate of

Table 1. Mathematical Equations for the Model with Positive Feedback Loop in the Ligand Production dL = sL + kLR 2L 2 + k1bR 2L1 + νk 2bR 2L 2 − νk1f R 2·L − k 2f R 2L1·L − γLL dt dR 2 = k1bR 2L1 − νk1f R 2·L dt

dR 2L1 = νk1f R 2·L + νk 2bR 2L 2 − k1bR 2L1 − k 2f R 2L1·L dt dR 2L 2 = k 2f R 2L1·L − νk 2bR 2L 2 dt R T = R 2 + R 2L1 + R 2L 2

Figure 2. (a) One parameter bifurcation analysis of the model with basal synthesis rate (sL) as the bifurcation parameter. The solid and broken lines represent stable and unstable steady states of the system. The upper and lower stable branches are termed as the “ON” and “OFF” states, respectively. The two saddle-node bifurcation points are indicated as SNON and SNOFF in the bifurcation diagram for C = 1. The shaded region indicates the region of bistability for C = 1. Different colored lines represent bifurcation diagrams with indicated values of cooperativity achieved by varying K1 keeping K2 fixed. (b) One parameter bifurcation analysis with different cooperativities achieved by varying K2 keeping K1 fixed. The parameter values are k1b = 0.2, k2b = 0.25, k1f = 0.1; k2f = 0.5, kL = 2.66, and γL = 1.0. To change cooperativity by varying K1, k1b was modified accordingly keeping k1f fixed at 0.1. Similarly to change cooperativity by varying K2, k2b was modified keeping k2f fixed at 0.5.

(K1 = k1b/νk1f and K2 = νk2b/k2f). Following Ha et al.,33 we define the ratio K1/K2, as a measure of cooperativity (C = K1/ K2) in the receptor−ligand binding. The numerical value of C determines the cooperativity in the binding kinetics. We identify three regimes of cooperativity based on the value of C: negative cooperativity, C < 1 (K1 < K2); positive cooperativity, C > 1 (K1 > K2); and noncooperativity C = 1 (K1 = K2). Cooperativity can be made positive or negative by adjusting the values of the binding (k1f and k2f) and unbinding rate constants (k1b and k2b) as C = k1bk2f/(4k1f k2b). The factor 4 originates from the binomial factor ν = 2 in the binding and unbinding steps in the case of disordered binding. We point out here that in the case of disordered binding, our definition of C does not contradict the definition provided earlier by Thordarson18 in which the measure of cooperativity was denoted as α. In terms of rate constants (k1f, k1b, k2f, k2b) our measure of cooperativity “C” is same as “α” defined there. Positive Feedback in Ligand Production. We incorporated the ligand−receptor complex module in a positive feedback loop, where the active form of the ligated receptor (doubly engaged receptor, R2L2) triggers production of the ligand that activates the receptor (Figure 1). Active cell surface receptors regulate the expression of target genes by activating or deactivating cytoplasmic signaling molecules that shuttle into the nucleus to trigger gene regulation. In the current model we have not considered these cytoplasmic molecules that transmit information from the cell surface to the nucleus of the cell. We reduced the complexity of the pathway by assuming that the active receptor directly regulates synthesis of the ligand. However, the generality will not be lost if one includes many intermediate signaling molecules that transmit information in a unidirectional way, from cell surface to nucleus. In addition to the regulated production of the ligand by the active receptor, we also introduced an unregulated basal production of ligand (sL) to initiate the signaling feedback loop. Therefore, our module is a simple representation of an autocrine signaling module.60,61 The rate constant for the regulated ligand production (kL) serves as the strength of the positive feedback loop in the model. The ligand degrades exponentially in time with an average lifetime of 1/γL. In this model the total amount of the receptor, RT = R2 + R2L1 + R2L2, was assumed to be constant. All the chemical reactions in the model follow the mass action rate law and the dynamical equations of the model are listed in the Table 1. As the total receptor is fixed in this model one can reduce the number of equations in the model to three using the receptor conservation relation. We first carried out one-parameter bifurcation analysis of the model using XPPAUT software (http://www.math.pitt.edu/

ligand (sL) as the bifurcation parameter for different values of C. We found that the model exhibits bistabilitytwo stable steady states that are separated by an unstable steady state. The bistability is created by the two saddle-node bifurcation points (SNOFF and SNON in Figure 2a). We call the “upper” and “lower” branches in the bifurcation diagrams as ON and OFF states, respectively. At the saddle-node bifurcation point the node (stable steady state) and the saddle (the unstable steady state) mutually annihilate each other and the steady state disappears beyond the bifurcation point. As our aim here was to investigate the effect of negative and positive cooperativity on the bistability, we varied the cooperativity, C, by modifying the values of K1 (Figure 2a) and K2 (Figure 2b). To modify the value of C in Figure 2a, we kept K2 (= 1) fixed and varied K1. Particularly we changed k1b, keeping k1f fixed to modify the value of K1. We found that negative cooperativity (C = 0.7) resulted in a bigger region of bistability (SNON−SNOFF) as compared to noncooperativity (C = 1) or positive cooperativity (C = 1.3). In fact both bifurcation points moved to a higher value of signal (sL) as compared to negative cooperativity. For bigger positive cooperativity (C > 1.3) the model did not exhibit bistability anymore, whereas with increased negative cooperativity (C < 0.7) the model showed irreversible bistable behavior where the SNOFF point moved to the negative region of the input signal (S1 Supplementary Figure). It is important to note that the critical signaling required for the system to respondthe “ON” threshold (SNON)is much lower in negative cooperativity than in positive cooperativity. This occurs because as we decreased K1 to obtain negative cooperativity the stability (or abundance) of singly occupied receptor (R2L1) increased requiring less amount of ligand (sL) to push the equilibrium to the doubly engaged state (R2L2) that triggers the feedback cycle. In the case of positive cooperativity, increasing K1 resulted in lower abundance of R2L1 and thus the SNON moved to the right as it required more ligand to trigger the feedback loop. Similarly 1296

DOI: 10.1021/acssynbio.8b00517 ACS Synth. Biol. 2019, 8, 1294−1302

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ACS Synthetic Biology when the signal was removed from the ON state, due to more abundant R2L1 a large amount of ligand needed to be removed to obtain the left bifurcation point (SNOFF) in negative cooperativity. On the other hand, due to less abundance of R2L1 in positive cooperativity, a small amount of ligand removal drove the system from the ON to the OFF state pushing the left bifurcation point to the far right. Further, the extent of responsethe steady state value of active receptor (R2L2)was higher in negative cooperativity than positive cooperativity. Therefore, negative cooperativity in the signaling module can induce stronger response at the expense of low signal strength as compared to positive cooperativity when K1 is lowered to achieve negative cooperativity. Next we investigated the effect of the dissociation constant, K2, on the bistability in positive and negative cooperativity. Here we kept K1 fixed and varied K2 by changing the value of k2b. Here we found a reverse trend where positive cooperativity leads to robust bistability as compared to negative cooperativity (Figure 2b). Therefore, the qualitative behavior of the model was opposite to the qualitative behavior observed with variation of K1. Here, as we increased K2 to get negative cooperativity, the abundance of the doubly engaged receptor (R2 L 2 ) decreases requiring more ligand to push the equilibrium to the right. Therefore, in negative cooperativity the right bifurcation point (SNON) moved to the right. For the same reason the left bifurcation point (SNOFF) also moved to the right while the ligand was removed from the “ON” state. In the case of positive cooperativity on the other hand, with a low value of K2, R2L2 was abundant and thus both bifurcation points shifted to the left. Therefore, with a variation of ligand synthesis rate (sL) negative cooperativity generates robust bistability as compared to positive cooperativity when the intermediate state R2L1 is stabilized by reducing K1. However, positive cooperativity results in a bigger bistability when the terminal state R2L2 is stabilized by reducing K2. Further, to determine the bifurcation behavior of the model for different values of K1 and K2, we performed two-parameter bifurcation analysis again using XPPAUT software. In Figure 3a we present the two-parameter bifurcation analysis of the model with K1 as the second parameter with a fixed value of K2 (= 1). The two-parameter phase diagram of the model exhibits a cusp-shaped region where the region bounded by the two lines represents bistability and the region outside represents monostability of the model. The lines indicate the loci of saddle-node bifurcation points. We added a horizontal dashed line (C = 1) to distinguish the region of positive and negative cooperativities in the plot. The two-parameter phase diagram suggests that the model exhibited “stronger” bistability in negative cooperativity when K1 was reduced. We repeated the bifurcation analysis for different values of K2 (Figure 3b) and in this case too negative cooperativity leads to robust bistability. Further we found that the region of bistability increases with reduction in the value of K2. This highlights the fact that a strong binding of the ligand is a necessary condition for ultrasensitivity in the negative cooperativity of receptor− ligand binding.33 Similar calculations with K2 as a secondary bifurcation parameter in a two-parameter bifurcation analysis indicated an opposite behavior where positive cooperativity resulted in “stronger” bistability as compared to negative cooperativity (Figure 3c,d). Again the stabilization of the intermediate receptor−ligand complex R2L1 by reducing K1 leads to robust bistability in negative cooperativity, whereas increasing K2 leads to destabilization of the terminal complex

Figure 3. Two-parameter bifurcation analyses of the model. The solid line is the locus of saddle-node bifurcation points. The region bounded by the solid lines represents bistability, and the region outside represents monostability. The noncooperativity, C = 1, is indicated by the horizontal dashed line in order to distinguish the regime of negative and positive cooperativity. Top row: twoparameter bifurcation diagrams with a fixed value of K2 (left) and K1 (right). Bottom row: two-parameter bifurcation diagrams with indicated values of K2 (left) and K1 (right).

R 2 L 2 thus resulting in weaker bistability in negative cooperativity. Therefore, the stabilities of the two complexes determine the robustness of bistability in negative or positive cooperativity. The strength of positive feedback is an important parameter in determining the region of bistability in any positive feedback regulated system.62 We performed a two-parameter bifurcation analysis of the model with respect to the strength of the positive feedback loop (kL) (Figure 4a). As anticipated the

Figure 4. Two parameter bifurcation analysis of the model showing the effect of feedback strength (kL) and the number of receptor (RT) on bistability. With increasing feedback strength the bistable region increases both in negative and positive cooperativity. Negative cooperativity generates bistable behavior with low feedback strength (a) and low total amount of receptor (b) as compared to positive cooperativity. To change cooperativity (C), K1 was varied keeping K2 fixed at 1.0.

robustness in bistability increases with an increase in feedback strength. However, the feedback strength needed to achieve bistability is smaller in negative cooperativity than in positive cooperativity. A two-parameter bifurcation with respect to total receptor (RT) (Figure 4b) indicates that bistability can be observed with a relatively smaller total receptor amount in negative cooperativity than that in positive cooperativity. Therefore, negative cooperativity allows the system to generate bistability in a weak positive feedback strength and low receptor abundance. 1297

DOI: 10.1021/acssynbio.8b00517 ACS Synth. Biol. 2019, 8, 1294−1302

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Similarly when negative cooperativity was achieved by increasing K2 (K1 < K2) in Figure 5b, R2L2 abundance decreased; however, the ligand present pushed the equilibrium to the R2L2 side and it allowed the system to maintain the OFF state for larger removal of the receptor. We also found similar observations in the case of ordered (ν = 1) binding of the ligand to the receptor (S2 Supplementary Figure). Positive Feedback in Both Ligand and Receptor. In the previous model, we considered a positive feedback loop in the ligand production. However, in many receptor−ligand systems receptor abundance is upregulated through a positive feedback loop mediated by the active receptor.61,63−66 We extended the described model by introducing a positive feedback loop in the production of the receptor. To include the positive feedback in receptor, we introduced explicit production and degradation of cell surface receptor. The production of receptor consists of two terms: a basal production rate of receptor (sR) and production by the active receptor (kR) representing the positive feedback loop on the receptor. The free and ligated receptors degrade exponentially with time and have an average lifetime of 1/γR. In the degradation of the receptor−ligand complex, only the receptor degrades but the ligand gets replenished. This mechanism represents the internalization of cell surface receptor. To simplify the model, we did not consider the dimerization step of the receptor as it may introduce additional nonlinearity into the dynamics and thereby interfere with the cooperative nature of receptor−ligand binding. We present the network of this model in Figure 6, and we listed the corresponding dynamical

So far we have considered the amount of ligand (sL) as the bifurcation parameter in the model. Next we investigated the bistable behavior of the model by choosing the amount of total receptor (RT) as bifurcation parameter (Figure 5). Here also

Figure 5. Bifurcation analysis for the model with the total number of receptors (RT) as the bifurcation parameter. One-parameter bifurcation diagrams with varying K1 (a) and K2 (b). Two-parameter bifurcation diagrams with fixed K2 (c) and K1 (d). K1 and K2 were kept fixed at 1.0 in the appropriate cases.

we explored the nature of bistability in positive and negative cooperativity generated either by varying K1 (Figure 5a,c) or K2 (Figure 5b,d). In both the cases negative cooperativity allowed the model to generate bistability with a significant range of total receptor concentration; however, in this case the effect of positive and negative cooperativities on the model are opposite to that of the case where the ligand was the bifurcation parameter. Here positive cooperativity resulted in robust bistability when K1 was varied and negative cooperativity resulted in robust bistability when K2 was varied. It is surprising to find that the qualitative nature of the bifurcations are not similar with respect to the variation of the components of the receptor−ligand complex. We noticed that the behavior of right saddle-node bifurcation points is similar to that of the variation of RT (Figure 5a,b) and sL (Figure 2a,b). This occurs because the reduction of K1 (for negative cooperativity) favors the intermediate complex R2L1; therefore, it requires less amount of ligand (in the case of ligand as the bifurcation parameter) or receptor (in the case of total receptor as the bifurcation parameter) to stabilize R2L2. Thus, the ON threshold (SNON) in negative cooperativity was smaller than that in positive cooperativity (Figure 5a,c). Alternately, the reduction of K2 (for positive cooperativity) favors the full complex R2L2; thus, in positive cooperativity the ON threshold was less than that in negative cooperativity (Figure 5b,d). However, the behavior of the OFF thresholds (SNOFF) is not similar to the variation of ligand and total receptor. In Figure 5a the increased region of bistability in positive cooperativity resulted as the SNOFF moved to the left. Once R2L2 is formed, it is the positive feedback loop that helps maintain the ON state, and therefore the amount of ligand present is also high. Although in positive cooperativity (K1 > K2) R2L1 is less abundant, the large amount of ligand present forces the equilibrium to the right. This allows the ON state to “survive” the removal of receptors resulting in a smaller value of SNOFF.

Figure 6. Network diagram of the model with two positive feedback loops. In addition to the positive feedback loop in ligand upregulation, the active receptor upregulates production of the dimeric receptor. sR, kR, and γR represent rate constants associated with the basal production, upregulated production and degradation rate of the receptor, respectively.

equations in Table 2. Here our aim was to investigate the nature of bistability when an additional positive feedback loop Table 2. Mathematical Equations for the Model with Positive Feedback Loop in Receptor Production dL = sL + kLR 2L 2 + k1bR 2L1 + νk 2bR 2L 2 − νk1f R 2·L − k 2f R 2L1·L − γLL dt + γRR 2L1 + 2γRR 2L 2

dR 2 = sR + kRR 2L 2 + k1bR 2L1 − νk1f R 2·L − γRR 2 dt dR 2L1 = νk1f R 2·L + νk 2bR 2L 2 − k1bR 2L1 − k 2f R 2L1·L − γRR 2L1 dt dR 2L 2 = k 2f R 2L1·L − νk 2bR 2L 2 − γRR 2L 2 dt 1298

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via the receptor is fused with the positive feedback loop involving the ligand creating a dual independent positive feedback system (L ⇄ R2L2 ⇄ R). In protein interaction networks, fused positive feedback loop motifs are known to be present in the cell cycle and cell differentiation networks. As we did in the previous model, here also we carried out two-parameter bifurcation analysis to determine the effect of the second feedback loop on the bistability created by the negative or positive cooperativity. We must mention here that due to the lack of nonlinearity the model without the positive feedback loop in ligand production (kL = 0) did not produce bistability with respect to the synthesis rate of the receptor or the ligand (S3 Supplementary Figure). To determine the possibility of multistability in the model with positive feedback only in the receptor but not in the ligand, we used Chemical Reaction Network Theory (CRNT)67,68 toolbox version 2.35 (https://crnt.osu.edu/CRNTWin). The CRNT toolbox predicts that there is no possibility of multistability for any positive values of rate constants in the model if only receptor upregulation is present. However, when both positive feedback loops are present CRNT predicted multistability. In Figure 7

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CONCLUSION

Ultrasensitivity is a key ingredient to generate multistability in a feedback regulated network.22,62 Theoretical modeling and subsequent biochemical experiments have recently shown that negative cooperativity in dimeric receptor−ligand binding can generate a significant signaling threshold with an ultrasensitive switch-like response under the condition of strong ligand binding.33 In this work, we have incorporated a dimeric receptor−ligand binding module in a positive feedback loop to investigate the effect of negative cooperativity on bistable signal response. As negative cooperativity (C < 1) can be achieved by adjusting either of the dissociation constants, K1 or K2, we explored the effect of negative cooperativity on the bistability by adjusting either K1 or K2. Our modeling and calculations point out that, similar to positive cooperativity, under certain conditions negative cooperativity can also generate robust bistability (Figure 3). Further, the bistability in the regime of negative cooperative receptor−ligand binding can be observed both with respect to the amount of ligand or with respect to the amount of receptor (Figure 4). With the amount of ligand as bifurcation parameter, negative cooperativity generated robust bistability as compared to positive cooperativity when K1 was reduced and the opposite happened when K2 was increased (Figure 3). On the other hand, with the total amount of receptor as bifurcation parameter, negative cooperativity resulted in robust bistability when K2 was increased and the behavior flipped when K1 was reduced (Figure 5). The variation of different rate constants to achieve cooperativity resulted in the dichotomous nature of the bistability. Our two-parameter bifurcation analyses indicate that the relative stabilities of the singly engaged (R2L1) and doubly engaged (R2L2) receptors control the region of bistability in the feedback regulated system. With the amount of ligand as bifurcation parameter, stabilization of the intermediate complex R2L1 and terminal complex R2L2 favors bistability in the negative and positive cooperativity, respectively. Conversely, with the amount of receptor as bifurcation parameter, stabilization of R 2 L 1 and R 2 L 2 complexes favor bistability in positive and negative cooperativity, respectively. Our calculations further suggest that, negative cooperativity can generate bistability with low ligand and low receptor (Figure 4) concentrations as compared to positive cooperativity in receptor−ligand binding dynamics. All together our modeling of a simple feedback regulated system with cooperative receptor−ligand binding dynamics explored the conditions of generating robust bistability with negative cooperativity in receptor−ligand dynamics. Our modeling and bifurcation analyses highlights that the robustness of the bistability, defined in terms of the region of bistability, is quite opposite and depends on the bifurcation parameter. For example with the basal synthesis rate of the ligand as the bifurcation parameter, if the dissociation constant of first binding step was chosen to modify cooperativity then negative cooperativity generated robust bistability as compared to positive cooperativity. On the other hand, if the dissociation constant of second binding step was varied to modify cooperativity then positive cooperativity generated robust bistability. This trend however became completely opposite when total receptor concentration or basal synthesis rate of receptor was chosen as the bifurcation parameter. Thus our work showcases a contrasting behavior of the model depending

Figure 7. Two-parameter bifurcation analyses showing the effect of the strength of the receptor upregulation (kR) on the bistable region. The horizontal dashed line indicates C = 1. K1 and K2 were kept fixed at 1.0 in the appropriate cases. The parameter values are k1b = 0.2; k2b = 0.25; k1f = 0.1; k2f = 0.5; γL = 0.18; γR = 0.06; kL = kR = 0.03; sR = 1.0 when sL is the bifurcation parameter and sL = 0.5 when sR is the bifurcation parameter.

we present all the two-parameter bifurcation analyses where we explored the effect of receptor upregulation rate (kR) on the cooperative binding of the model. For a given value of receptor upregulation rate, the qualitative nature of the two parameter bifurcation patterns are similar to the previous model having only one positive feedback loop. However, the two parameter bifurcations of K1 vs sL or K2 vs sL are sensitive to the strength of the positive feedback loop in receptor upregulation. With the increase in the receptor upregulation rate the region of bistability increases significantly (Figure 7a,b). Particularly the SNOFF bifurcation point is quite sensitive to kR. An increased value of kR leads to more receptors forcing the equilibrium to the right which allows the model to maintain its ON state for increased removal of ligand; thus, the SNOFF moves to the lower value of sL. On the other hand the bifurcations with respect to the unregulated synthesis rate of the receptor (sR) are very weakly dependent on kR (Figure 7c,d). 1299

DOI: 10.1021/acssynbio.8b00517 ACS Synth. Biol. 2019, 8, 1294−1302

Research Article

ACS Synthetic Biology upon the choice of parameter used to modify cooperativity and the choice of bifurcation parameters. In our models, we analyzed the tunability of bistability generated by negative cooperativity in the receptor−ligand binding. To modify the cooperativity we modified the dissociation constants of the first and second binding steps in the receptor−ligand engagement. The dissociation constants are typically intrinsic to the receptor−ligand pair and in in vitro experimental conditions modifications of binding constants are hard to achieve. Nevertheless by introducing mutations in the proteins, a significant alteration of dissociation constants has been achieved in the recent past.69 On the other hand, in order to determine the regime of bistability with the tuning of other parameters, we varied basal and regulated synthesis rates of the ligand and receptor (sL, kL, sR, and kR) and the degradation rates of the ligand and receptor (γL and γR). These oneparameter bifurcation analyses showcase the robustness of bistability generated by cooperative receptor−ligand binding. With the advancement of molecular biology, modification of these rates is not impossible. On the basis of feasibility and convenience it is possible for the experimentalists to design protocols to vary the parameters of these rates. Most of these variations in rates have been achieved in synthetic biology experimental protocols. For example, protein syntheses have been varied either by controlling the transcription rate using an inducible promoter54 or by modifying the translation rate by introducing point mutations in the ribosome binding site in the target gene.70,71 Protein synthesis rates also have been regulated by tuning the signaling pathways of target proteins.72,73 Further tunability in protein degradation rates also has been achieved in S. cerevisiae by targeting its degradation pathway using synthetic biology approaches.74 In our models we assumed that the receptor attains its full activity only when it is doubly engaged by the ligand. However, many receptors, such as cytokine receptor and growth hormone receptors, have a single occupancy whereby a ligand binding leads to activation of the receptor.75 To generate negative cooperativity, here we relied on the asymmetry in the binding affinities of the ligand to the divalent receptor. Therefore in our model it is crucial that two ligand molecules bind successively. We assumed that the singly engaged receptor is inactive; however, one can assign weak activity to this receptor as compared to the fully engaged receptor. Under that situation we expect that one would get similar results as we reported throughout, although the region of bistability may decrease due to the loss of sharpness in the active receptor vs total ligand plot.33 While traditionally the Hill function for positive cooperativity, the Goldbeter−Koshland switch in enzyme kinetics,76 multisite phosphorylation,57,77−79 and more recently, molecular titration80,81 all have provided suitable mechanisms to generate the much needed nonlinearity for system-level complex phenomena, we show here that negative cooperativity in receptor−ligand binding also has the potential to generate robust bistability. Particularly in the case of synthetic biology settings negative cooperativity can be explored further to establish its potential to develop new devices.

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One-parameter bifurcation analysis showing an irreversible bistable switch with smaller value of C; twoparameter bifurcation analyses of the model with ordered binding of the ligand; one-parameter bifurcation analyses showing feedback loop only on receptor does not generate bistability; ode codes (PDF)

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Debashis Barik: 0000-0003-1681-1273 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The work was supported by funding from the Science and Engineering Research Board, Department of Science and Technology (India), Grant No. EMR/2015/001899, to DB. AD acknowledges fellowship from the INSPIRE program of Department of Science and Technology, India.

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REFERENCES

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acssynbio.8b00517. 1300

DOI: 10.1021/acssynbio.8b00517 ACS Synth. Biol. 2019, 8, 1294−1302

Research Article

ACS Synthetic Biology

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