The sequence–ensemble relationship in fuzzy protein complexes

Significance

Intrinsically disordered proteins (IDPs) do not possess a stable three-dimensional structure but populate partially folded, transient structures. They are involved in numerous cellular processes, often in cooperation with globular proteins. In such complexes, the protein-bound IDPs can be structurally very heterogeneous ranging from folded, stable conformations to those that are dynamic and disordered. The underlaying principles governing this heterogeneity, known also as fuzziness, are obscure. We developed a simple statistical thermodynamic model based on the helix–coil theory that rationalizes diversity of the target-bound IDP ensembles. The model successfully reproduces bound-state IDP dynamics and explains how ensembles modulate their structure and interaction affinity due to mutations. Collectively, this sheds light on the sequence–ensemble relationship of fuzzy IDP–protein complexes.

Abstract

Intrinsically disordered proteins (IDPs) interact with globular proteins through a variety of mechanisms, resulting in the structurally heterogeneous ensembles known as fuzzy complexes. While there exists a reasonable comprehension on how IDP sequence determines the unbound IDP ensemble, little is known about what shapes the structural characteristics of IDPs bound to their targets. Using a statistical thermodynamic model, we show that the target-bound ensembles are determined by a simple code that combines the IDP sequence and the distribution of IDP–target interaction hotspots. These two parameters define the conformational space of target-bound IDPs and rationalize the observed structural heterogeneity of fuzzy complexes. The presented model successfully reproduces the dynamical signatures of target-bound IDPs from the NMR relaxation experiments as well as the changes of interaction affinity and the IDP helicity induced by mutations. The model explains how the target-bound IDP ensemble adapts to mutations in order to achieve an optimal balance between conformational freedom and interaction energy. Taken together, the presented sequence–ensemble relationship of fuzzy complexes explains the different manifestations of IDP disorder in folding-upon-binding processes.

A large fraction of most proteomes consists of intrinsically disordered proteins (IDPs) that lack a stable three-dimensional structure (1, 2). This is due to the specific amino acid composition of IDPs that is enriched in charged and flexible residues and depleted of hydrophobic ones (3). Even though IDPs do not possess a stable globular structure, they are involved in a wide range of biological functions, particularly in the signal transduction and transcription regulation (2, 4–6), and are key elements for the formation of membrane-less organelles such as stress granules and P-bodies (7–9). The specific amino acid composition enables accurate detection of IDPs from protein sequences using bioinformatic tools (10). In their free state, IDPs adopt different conformations ranging from very disordered, almost random-coil polymers to the molten-globule–like states (11, 12) and may transiently sample conformations that are observed in their target-bound states (13). The conformational properties of free IDPs are influenced by a variety of factors including (lack of) sequence complexity (14), posttranslational modifications (15, 16), proline isomerization (17), and distribution of charged residues (18). Conformational ensembles of free (unbound) IDPs have been characterized in detail by experiment and simulations (19), and a significant insight into the sequence–ensemble relationship has been obtained using a polymer theory (20, 21).

Central to the function of IDPs are, however, their interactions with other macromolecules. Initially, it was observed that many IDPs adopt a stable folded structure upon binding their targets. Most frequently, this entails the formation of an α-helix (22–24). This view was later expanded by observing that IDPs frequently preserve a significant degree of disorder in the bound state and sample multiple conformations (25). The target-bound IDPs possess many structural options ranging from very stable structures (disorder-to-order transitions) to those in which IDPs show significant dynamics and conformational heterogeneity in the bound state (disorder-to-disorder transitions) (26–28). Furthermore, IDPs may adopt different conformations when bound to various targets (context-dependence) (29) or adopt different conformations on the same target (structural polymorphism) (30). Despite the observed IDP dynamics and conformational heterogeneity, such complexes are functionally specific and play crucial roles in a number of physiological processes. Target-bound IDP ensembles optimize allosteric coupling (31, 32) and can be easily modulated by mutations and posttranslational modifications leading to the emergence of new phenotypes (5). Collectively, such complexes have also been described as “fuzzy complexes,” a concept which postulates that the structural heterogeneity and multiplicity is required for their biological function. To account for all the versatile behavior and structural characteristics, a classification of fuzzy complexes has been proposed (33), and recent bioinformatic methods can detect fuzzy-prone regions form the protein sequences (34).

There has been a significant progress in understanding the IDP structural ensembles in the free, unbound state. In contrast, the sequence–ensemble relation of the IDPs bound to their targets is much less understood, and the key parameters determining their structural ensembles remain obscure. Here, we address these fundamental issues by developing a statistical–thermodynamic model describing the target-bound IDP ensembles. We show that these ensembles are defined by a relatively simple code specified by the IDP sequence and the distribution of IDP–target interactions. The model describes quantitatively the conformational phase space of target-bound IDPs and explains to what extent these ensembles are determined by the target or by the IDP sequence. Comparison with the experimental data shows that the model-predicted IDP dynamics agrees with the NMR relaxation experiments, thus validating the presented approach. Furthermore, we show experimentally that changes in the IDP folding energetics can significantly alter their bound-state ensembles, which adapt in a way to optimize unfavorable folding and formation of energetically favorable interactions with the target. Taken together, our analysis deciphers the sequence-ensemble code of fuzzy protein complexes and advances the understanding of physical principles governing versatile functionality of IDPs.

Results

A Statistical–Thermodynamic Model of Target-Bound IDP Ensembles.

We develop a simple statistical–thermodynamic model focusing on IDPs that fold into α-helical conformations in the bound state, which is also the most common IDP secondary structure binding motif (24). In principle, the theoretical framework could be used to describe IDPs forming other types of secondary structures; however, the parameters pertaining helix–coil transitions have been accurately determined for all amino acids (35, 36), enabling a quantitative description of the unbound (37, 38) and, as we show, also the bound-state IDP ensemble. The model is based on the helix–coil theory of Lifson and Roig (LR) (39), which was expanded to include tertiary IDP–target interactions (Materials and Methods). The IDP amino acid sequence defines its folding propensity through the ΔG_CH contributions, which is the free energy for transition of a residue from coil to helix (Fig. 1A). The unbound ensembles can be modeled using the classical LR model (Eqs. 1–3), and since the ΔG_CH for IDPs are generally positive (low folding propensity), these ensembles are largely disordered in their unbound state (Fig. 1B).

Fig. 1.

Statistical–thermodynamic model of target-bound IDP ensembles. (A) The model considers IDPs that fold into the α-helix upon binding their targets. The IDP sequence defines its folding propensity, ΔG_CH, which is the free energy for transition of a residue from coil to helix conformation. The classical helix–coil model describes the unbound IDP ensemble (Eqs. 1–3). Folding of IDP is driven by formation of few strong interactions with the target, termed hotspots (represented by number 1). The hotspot distribution imposes constraints on the possible microstates that are compatible with target binding (Eqs. 4–7). Different hotspot distributions can also be characterized by the coarse-grained parameter N_SEP/N, which is the fraction of IDP bounded by the first and last hotspot (shown as stars). (B) IDP ensembles can be represented by a probability distribution of microstates with varying helicity. While the unbound IDP ensemble is largely disordered (green bars show high probability for the microstates with low helicity), the IDP–target interactions stabilize microstates with higher helicity (orange bars).

Upon binding the target, IDPs may acquire different degrees of structural ordering. Since ΔG_CH > 0, folding of IDP is unfavorable and is therefore driven by the favorable interactions with the target. We treat these interactions in a simplified way by assuming that only a set of strong contacts defines the majority of the interaction energy. Accordingly, a set of hotspot residues will stabilize the helical conformation and effectively nucleate the helix (Fig. 1A). We assume that IDPs may bind the target with all hotspots or only with some of them, resulting in partially bound states, and we use the same average interaction energy, ΔG_INT, for all hotspots (Eq. 5 and Materials and Methods). Based on these assumptions, we derive the partition function of the target-bound ensemble (Eqs. 4–7), which consists of microstates with varying helicities that satisfy the specified hotspot distribution. While the hotspot distribution selects the possible microstates that are compatible with target binding, the IDP folding propensity determines to what extent these microstates will be populated (Fig. 1B). Since the partition function is derived analytically, a range of different parameters can be screened, allowing us to assess various aspects of sequence–ensemble relationship in the fuzzy complexes.

We first present theoretical results for the simple IDP models showing that the hotspot distribution and the IDP folding energetics (ΔG_CH) are the two main parameters defining the conformations of target-bound ensembles. In order to characterize numerous hotspot distributions of hypothetical IDPs in an accessible way, we use the coarse-grained parameter N_SEP, which is simply the separation between the first and last hotspot in the IDP sequence for any number of hotspots. Accordingly, the ratio N_SEP/N defines the fraction of the IDP with N residues bounded by the hotspots. In the second part of the paper, we use the model to calculate different macroscopic observables and compare them with the experimental data (Eqs. 8–11). While the coarse-grained parameters are convenient to analyze the general IDP behavior theoretically, in the second part of the paper we use the exact IDP sequences and hotspot distributions derived from the high-resolution structures of IDP–target complexes as input parameters (Materials and Methods).

Folding Propensity and Hotspot Distribution Determine the Target-Bound Structural Ensemble.

We first analyze a simple theoretical model in which IDPs bind the target with two hotspots positioned at different separations resulting in a range of N_SEP/N parameter values. These IDPs differ in their folding propensity (ΔG_CH) that we initially assume is the same for all residues (effectively a homopolymer IDP). The model predicts that combinations of these two parameters give rise to a vast range of IDP ensembles with varying degrees of disorder. Depending on the number and distribution of helical residues, these ensembles can be classified into four macroscopic types: disordered, helical, clamp, and partially bound ensembles which bind via one hotspot (Fig. 2A). The conformational phase space of the target-bound IDPs shows the most populated type of the ensemble at given parameter combination and illustrates the synergy between hotspot distribution and folding propensity in determining the ensemble conformation (Fig. 2B). For example, when the two hotspots are close in sequence (small N_SEP/N), ensembles tend to be mostly disordered (blue region, Fig. 2B) in a range of ΔG_CH values. As the separation between two hotspots increases, these ensembles transition either into helical ensembles (red region, Fig. 2B) or into ensembles consisting of two short helical segments connected by a flexible linker (clamp ensembles, black region, Fig. 2B). The outcome depends on the folding propensity ΔG_CH: for IDPs with low folding propensity (ΔG_CH > 50 cal mol⁻¹res⁻¹), transition into clamp ensembles occurs, while those with higher folding propensity (ΔG_CH < 50 cal mol⁻¹res⁻¹) transition into helical ensembles. Finally, the ensembles with partially bound IDPs (via a single hotspot) are favored for separated hotspots and low folding propensity (yellow region, Fig. 2B). This part of phase space illustrates a situation when the IDP–target interactions are traded-off to gain higher conformational freedom. It should be noted that each phase is defined according to the most probable ensemble at a given parameter combination; however, the underlaying transitions are not sharp (SI Appendix, Fig. S1). Particularly near the phase boundaries, several ensemble types can coexist and suggest that in these regions, IDPs have a high tendency for polymorphism, which we estimate using a conformational diversity metric (SI Appendix, Fig. S2).

Fig. 2.

Conformational phase space of fuzzy complexes. (A) The target-bound IDPs may acquire different macroscopic ensembles with a varying degree of disorder: disordered, helical, ensembles with clamp conformation, and ensembles with partially formed interactions. Hotspot residues are shown as stars. (B) Phase diagram shows how hotspot distribution characterized simply by N_SEP/N and the average IDP folding propensity (ΔG_CH) determine the target-bound IDP ensemble. Calculation is shown for a hypothetical homopolymer IDP with n = 32, having two hotspots at different separations and the average IDP–target interaction strength ΔG_INT = −3 kcal mol⁻¹ hotspot⁻¹. (C) Bound-state conformations of IDPs MLL, cMyb, and FOXO3a illustrate the structural diversity of fuzzy ensembles (PDB codes 2AGH and 2LQI). Their corresponding coarse-grained parameters N_SEP/N and ΔG_CH place them in the appropriate area of the phase diagram.

Next, we extend this simple model by considering the effects of IDP length, sequence diversity, IDPs with multiple hotspots, and different values of the IDP–target interaction energy (SI Appendix, Figs. S3–S6). The length (N) of the IDP has a small effect on the overall characteristics of the phase diagram and slightly changes the absolute positions of the phase boundaries (SI Appendix, Fig. S3). The hotspot interaction strength (ΔG_INT) effectively determines the phase boundaries between fully and partially bound complexes, and in case of multiple hotspots, ΔG_INT shifts the phase boundaries between helical, clamp, and disordered ensembles (SI Appendix, Fig. S4). We next investigated the effects of IDP sequence diversity by generating random IDP heteropolymer sequences with different folding propensities using the helix–coil parameters specific for each amino acid (35). While the overall trends of phase transitions are identical to those calculated using a homopolymer approximation, there is a significant variation between different hetero-sequences and the corresponding homo-sequences having the same average ΔG_CH (SI Appendix, Fig. S5). This illustrates to what extent the IDP sequence, not only its average folding propensity, determines the properties of the bound-state ensemble. To describe numerous possible hotspot distributions of IDP sequences with multiple hotspots in an accessible way, we use the additional parameter κ_H, similar to the one describing charge pattering in the polyampholyte IDPs (18). For IDPs with an even hotspot distribution κ_H = 0, while for uneven, separated distributions κ_H = 1 (Materials and Methods). The coarse-grained parameters N_SEP/N and κ_H can thus approximately represent different distributions of multiple hotspots, and together with the energetic parameter ΔG_CH, the target-bound ensembles can be represented on a more general three-dimensional phase diagram (SI Appendix, Fig. S6). For IDPs with more separated hotspots (κ_H = 1), the phase diagrams are very similar to ones described for the simple model with two hotspots (Fig. 2B). However, IDPs with more even hotspot distributions (κ_H = 0) tend to have higher populations of helical ensembles and partially bound ensembles at the expense of clamp structure compared to the simple model, but these effects become more pronounced only at higher N_SEP values (SI Appendix, Fig. S6).

Having explored these more complex cases, it appears that the results from a simple IDP model presented in Fig. 2 capture the main characteristics and key parameters that determine the conformational ensemble of target-bound IDPs. Even some realistic IDPs characterized by the simple metrics (e.g., N_SEP/N) can be presented on the phase diagram. For example, MLL, cMyb, and FOXO3a fold upon binding to the KIX domain resulting in structures with varying degrees of disorder (Fig. 2C). Their folding propensities and hotspot distributions characterized by N_SEP/N and ΔG_CH position these ensembles in the appropriate areas of the phase diagram.

Model-Predicted Profile of Bound-State IDP Dynamics Agrees with the NMR Relaxation Data.

To check whether the quantities predicted by the model are consistent with the experiment we first focus on the NMR relaxation data, which provides residue-specific insight into IDP dynamics (40–42). The intrinsically disordered transactivation domains of Hif1-α and CITED2 fold upon binding the TAZ1 domain of the CREB-binding protein (Fig. 3A). Recent relaxation experiments provide a detailed information on the bound-state IDP dynamics (43) (Fig. 3B). The profile of the backbone S² order parameter is rather complex for Hif1-α, while that for CITED2 shows that its central part becomes ordered and is followed by the long, disordered tail. We hypothesized that for the IDPs that fold into α-helix, the value of S² order parameter would correlate with the residue-specific probability for the helical conformation p(w_i). This is because helical states are constrained to a small region of Φ, Ψ phase space, while the S² parameter is sensitive to the spatial restriction of the amide vector (44). The probability p(w_i) can be calculated using our model, provided the IDP amino acid sequence and the hotspots distribution (Eq. 9). To identify hotspots, we tested four different computational methods that use high-resolution structures of the IDP–target complexes as input (Materials and Methods). The predicted hotspots from different methods largely agree with one another and are consistent with the mutational data which is available only for some resides (SI Appendix, Fig. S7). We used the PPCheck tool (45) to identify IDP–target hotspots, which gives best agreement with the data (46) (SI Appendix, Fig. S7). The resulting model-predicted dynamic profiles are in a remarkable agreement with the NMR experiments, suggesting that the presented model can clearly distinguish between a dynamic Hif1-α and more-ordered CITED2 complexes (full line, Fig. 3B). Next, we performed a similar model analysis focusing on the ACTR in complex with CBP (47) and the ARTHEMIS in complex with Ligase IV (48) and obtained the predictions which are consistent with the experimental data (SI Appendix, Fig. S8). This not only validates the presented model but also shows that the dynamics of the IDP in its bound-state is determined by a relatively simple code that combines the hotspot distribution and the IDP sequence.

Fig. 3.

Model-predicted IDP dynamics agrees with the NMR relaxation data. (A) Structures of IDP–target complexes show that IDPs Hif1-α and CITED2 fold upon binding the TAZ domain to different degrees (PDB 1L8C and 1R8U, respectively). (B) NMR relaxation experiments show that these IDPs retain considerable dynamics in the bound state (gray bars show the value of the amide order parameter S²). The model-predicted values of the IDP bound-state dynamics (Eq. 9, shown as full line), based on the distribution of hotspots and the IDP sequence are in reasonable agreement with the experimental values.

Target-Bound Ensembles Modulate Their Conformation in Response to Changes in the Folding Propensity Induced by Mutations.

Even though the hotspot distribution imposes constraints on the microstates that are compatible with target-binding, the microstates in the ensemble can, to some extent, be redistributed while still satisfying the specified hotspot distribution. For example, a move in the vertical direction in Fig. 2B shows that decreasing IDP folding propensity should shift the ensemble toward more disordered states. To test these predictions, we use an intrinsically disordered domain of HigA2 antitoxin that folds upon binding its globular target (Fig. 4A) (49). We changed the HigA2 folding propensity by introducing glycine residues at the solvent-exposed, noninteracting positions, such that the IDP–target interface is not disrupted (SI Appendix, Table S1 lists HigA2 sequences). These polyglycine mutants 2G, 4G, and 6G decrease the HigA2 folding propensity by up to ΔΔG_CH = +4 kcal mol⁻¹ (ΔΔG_CH = ΔG_CH,MUT − ΔG_CH,WT). On the other hand, introduction of hydrocarbon staples S1 and S2 increases the folding propensity up to ΔΔG_CH = −2.7 kcal mol⁻¹. As expected from the classical helix–coil model, these changes in the folding propensity affect the average helicity of the unbound IDP ensemble to different degrees (SI Appendix, Fig. S9). More importantly, CD spectroscopy shows that these mutations also change the average helicity of the target-bound IDP ensemble as expected from our model (SI Appendix, Fig. S9). The polyglycine mutations reduce the IDP folding propensity and thereby decrease the average helicity of the bound-state ensemble (blue dots, Fig. 4B). On the other hand, hydrocarbon staples have little effect on the average bound-state helicity since the wild-type (wt) HigA2 is already highly helical (Fig. 4B). It should be emphasized that the observed decrease in the bound-state helicity is not caused by the residual fraction of unbound IDP due to reduction of binding affinity of polyglycine mutants (SI Appendix, Fig. S9).

Fig. 4.

Target-bound IDPs modulate their ensembles in response to changes in the folding propensity. (A) We studied the model IDP system HigA2 (blue cartoon) that folds into α-helix upon binding its globular target (gray, PDB 5JAA). Solvent-exposed, noninteraction residues (shown as yellow sticks) were substituted for glycine in order to decrease HigA2 folding propensity (ΔΔG_CH >0, mutations 2G, 4G, and 6G) or with hydrocarbon staples to increase the folding propensity (ΔΔG_CH <0, mutations S1 and S2). (B) The average bound-state IDP helicity determined from CD spectroscopy shows that the polyglycine mutations shift the target-bound IDP ensemble toward less helical, more disordered states (blue circles, bars show 1sd). The model-predicted average helicity (Eq. 10) for different mutants reasonably agrees with the data (blue line). Data from a different IDP system shows that noninteracting CcdA mutants induce similar shifts of the bound-state ensemble helicity (red circles) and can also be correctly described by the model (red line).

A Trade-off between Hotspot Interactions and the IDP Conformational Freedom Explains Ensemble Modulation.

To explain conformational fluctuations of the HigA2 ensemble quantitatively, we use the model to calculate its average ensemble helicity (Eq. 10). We identified six mostly hydrophobic hotspot residues in the N-terminal half of the HigA2 using the PPCheck. We assume that HigA2 can bind via all hotspots (fully bound microstates according to the hotspot set h1 listed in SI Appendix, Table S2) or using some of these hotspots (partially bound microstates bound via 1 to 5 hotspots according to the hotspot sets h2–h9). Furthermore, we assume that all hotspots bind with the same interaction energy ΔG_INT = −1.2 kcal mol⁻¹. Based on these assumptions, we calculate the average ensemble helicity for different mutants, which is consistent with the experimental data as it correctly predicts both the decrease in helicity observed for the polyglycine mutants and also minor changes for the hydrocarbon staples (blue line, Fig. 4B). A different model which considers that HigA2 binds only with all hotspots (hotspot set h1) cannot explain a significant decrease in helicity of the 6G mutant (SI Appendix, Fig. S10). Collectively, these results show that the bound-state IDP cannot be considered as one with a static, uniform conformation but that the energetically degenerate, structurally heterogeneous ensemble is a more adequate description. We perform a similar model analysis on the previously reported data from a different IDP–target system CcdA (50). The noninteracting CcdA mutations were designed to reduce its folding propensity and were also observed to reduce the CcdA bound-state helicity (red dots, Fig. 4B). The observed change in the average helicity agrees with the model prediction based on the CcdA hotspot distribution assuming the average hotspot interaction energy ΔG_INT = −1.7 kcal mol⁻¹ (red line, Fig. 4B). Overall, the agreement between model-predicted and measured bound-state IDP helicity for both Hig2A and CcdA systems yields a straight line with R² = 0.95 and the slope close to unity (SI Appendix, Fig. S11). Therefore, the model quantitatively predicts the conformational fluctuations of fuzzy complexes and demonstrates that changes of the IDP folding propensity ΔΔG_CH can dramatically affect the IDP bound-state ensemble.

The model reveals the mechanism governing the modulation of target-bound IDP ensembles as the one involving two processes (Fig. 5A). The first involves only a redistribution of microstates within the same hotspot set (thus retaining all hotspots) according to the given IDP folding propensity. The second process optimizes the fractions of partially bound microstates; thus, some hotspots are traded off to gain higher conformational freedom of the IDP. For example, the wt HigA2 ensemble consists of microstates with high helicity forming practically all hotspots (according to the hotspot sets h1–h3, white bars, Fig. 5B). Introduction of two glycine residues (2G mutant) makes HigA2 folding more unfavorable; therefore, the target-bound ensemble shifts toward microstates with lower helicity but all hotspots remain formed (blue bars, Fig. 5 B, Left). This is accompanied by only a small decrease in the average bound-state helicity. Introduction of additional glycine residues (4G and 6G mutants) decreases the average ensemble helicity even more drastically, not only because low-helicity microstates are preferred but also due the higher fraction of partially bound microstates (hotspot sets h6-h9, green and yellow bars, Fig. 5 B Middle and Right). For example, in the ensemble of 6G mutant, the fraction of microstates bound with all the hotspots is only about 0.2 compared to 0.75 for the wt HigA2 ensemble. Generally, for a small ΔG_CH perturbation, ensembles redistribute microstates without disrupting the IDP–target binding interface, thus retaining all the interaction energy. When the IDP folding becomes even more unfavorable (higher ΔΔG_CH), the gain in conformational freedom outweighs the interaction energy and the fraction of partially bound microstates is increased. The heterogeneity in establishing hotspot interactions and malleability of the bound-state IDP ensemble underline the fuzzy behavior in these systems.

Fig. 5.

Mechanism governing the IDP ensemble modulation. (A) Ensemble modulation in response to the noninteracting mutants is illustrated schematically. As the folding becomes less favorable, mutated IDPs minimize their folding and adopt their bound-state ensembles accordingly, either by redistributing microstates within the same hotspot distribution (2G, 4G) or by exploiting other partially bound states (6G). (B) Probability distribution for microstates with varying helicity are shown for the wt HigA2 (white bars) and different polyglicine mutants (color bars). In response to the 2G and 4G mutations, IDP ensembles adopt by increasing probability of microstates with lower helicity while keeping all the hotspot interactions (h1–h3 hotspot sets). In response to the 6G mutant, the IDP ensemble adopts by increasing fraction of partially bound ensembles (h4–h9 hotspot sets). Indexes h1–h9 denote different hotspot distributions as defined in SI Appendix, Table S2, while the percentage denotes a fraction of microstates that binds the target with a given hotspot set.

Changes in the IDP–Target Affinity Can Be Quantitatively Predicted.

To fully discern the effects of noninteracting mutations, we measured the IDP–target affinity using isothermal titration calorimetry (ITC). The affinity of 4G and 6G could be determined using direct titrations, while for other peptides, the affinities were out of the accessible range for ITC. We therefore used the competition ITC and measured affinities of WT and 2G, while that for hydrocarbon staples were inaccessible even in this setup (SI Appendix, Fig. S12). Affinity for the wt HigA2 is K_D = 3 pM, and the introduction of glycine residues gradually decreases the affinity up to K_D = 0.3 μM for the 6G mutant (SI Appendix, Table S3). Thus, introduction of noninteracting mutations decreases IDP–target interaction strength by four orders of magnitude. Given that folding propensities of amino acids, including glycine, are known, we anticipated that these changes in the interaction affinity could be predicted using our model. The unbound IDP ensemble can be fully described from the IDP sequence using the classical LR model (Eqs. 1–3), while the partition function of the bound state needs an additional input of the hotspot distribution as described in previous examples. Both states then define the change in IDP–target affinity relative to the wt IDP (Eq. 11). We used identical hotspot parameters (distribution and ΔG_INT) as used for the analysis of helicity changes (Fig. 4). The calculated change in the IDP–target affinity (ΔΔG) correlates excellently with the experimental values (ΔΔG_EXP) (Fig. 6, slope of the line = 1.1, R² = 0.99). Previously reported experimental data of IDP–target affinity changes due to the noninteracting mutations from a different IDP–target system CcdA (50) can also be described successfully (red dots, Fig. 6).

Fig. 6.

Changes in the IDP–target affinity due to noninteracting mutations can be quantitatively predicted. The model-predicted free energy changes of the IDP–target interaction (ΔΔG) correlate excellently with the experimental values (ΔΔG_EXP) for both HigA2 and CcdA systems. Points show the data for wt and polyglycine HigA2 mutations but not for hydrocarbon staples in which the affinity was too high and could not be determined. Line shows the liner regression to the data points with the slope = 1.1 and R² = 0.99. Bars show two SDs and originate from the experimental error in determination of IDP–target affinity.

The measured enthalpies also seem to support the proposed mechanism of ensemble modulation as proposed in the previous section. While the binding enthalpies of wt, 2G, and 4G are all very similar, that of 6G is significantly lower (SI Appendix, Table S3). Such a strong reduction of the binding enthalpy for this mutant would be expected to accompany the proposed partial loss of some hotspot interactions, since hotspots are associated with a higher enthalpic contributions compared to those associated with the differences in IDP folding for 2G and 4G. Finally, it is important to emphasize that the prediction of mutational effects depends on correctly accounting for the shifts in the target-bound ensemble. For example, if the target-bound IDP conformation would be considered as static (e.g., as seen in the crystal structure), the predicted ΔΔG values are significantly overestimated (SI Appendix, Fig. S13). Thus, adaptation of the target-bound ensemble due to mutations has a significant influence on the IDP–target affinity since IDPs adjust their ensembles to achieve an optimal balance between conformational freedom and the formation of IDP–target interactions.

Discussion

An analysis of fuzzy protein complexes by simulations, experiments, and bioinformatic methods have identified many different manifestations of disorder and its associated functions. Therefore, it is of considerable interest to elucidate the underlying principles behind this seemingly chaotic behavior. Here, we rationalize the sequence–ensemble relationship of fuzzy complexes by demonstrating that the IDP sequence information translates into the conformational properties by a relatively simple code: IDP sequence defines its structure-forming propensity, while its target defines the distribution of interactions that can be explored by the bound IDP through various conformations (Fig. 2). The presented phase diagrams show how key parameters determine the conformational ensemble of fuzzy complexes and provides a physical principle for the classification of fuzzy complexes proposed previously (33). Furthermore, the simple model correctly captures target-bound IDP dynamics (Fig. 3) and modulation of its ensemble due to perturbations induced by mutations as reflected in the changes in IDP helicity and IDP–target affinity (Figs. 4–6). The described adaptation of IDP bound-state ensemble in terms of structure and energetics profoundly differs from the mutational effects observed for the globular proteins.

Broadly, the model also sheds light on some poorly understood phenomena in fuzzy complexes. For example, the structural polymorphism describes cases in which IDPs adopt different conformations when bound on the same target. This behavior underscores the “chaotic” nature of IDPs. We have shown that for some combinations of system parameters, the probabilities for different types of IDP ensembles are comparable. In fact, the model-predicted transitions between different phases are rather broad, and situations in which several macroscopic IDP conformations may coexist are not so rare (SI Appendix, Fig. S1). We show that the regions near the phase boundaries have a particularly high tendency for conformational polymorphism, which we quantified by a conformational diversity metric (SI Appendix, Fig. S2).

Identification of key parameters shaping the fuzzy ensembles suggests the extent to which the bound-state IDP conformations are templated by the target or encoded by its sequence. This question has been addressed in several studies reaching different conclusions (51–53). For example, IDP regions with helical preference of the p53 protein adopt different conformations on different targets (54), suggesting that the IDP conformation is context dependent and templated by the target. In contrast, some IDPs may adopt practically identical conformations when bound to different targets (52, 55) suggesting that IDP conformations are encoded by its sequence alone. Inspection of our phase diagram (Fig. 2B) suggests that the bound-state structure is encoded both by the IDP sequence (which determines the folding propensity ΔG_CH) and by the target (which defines N_SEP and κ_H) to different degrees. For example, in the area of phase space where the folding propensity is either very high or low (ΔG_CH < 20 or ΔG_CH > 250 cal mol⁻¹ res⁻¹), the bound-state structure is practically independent of the target site (of N_SEP and κ_H); therefore, the ensemble is encoded only by the IDP sequence. On the other hand, in the middle part of the phase space (20 < ΔG_CH < 250 cal mol⁻¹ res⁻¹), the influence of the target becomes more prominent and the IDP conformation is encoded both by the target and the IDP sequence.

Based on the simple thermodynamic arguments, it has been suggested that stabilization (or destabilization) of helical elements in IDPs should directly translate into the changes of IDP–target affinity (56, 57). This hypothesis has been investigated by introducing amino acid mutations away from the IDP–target interface (mutations of noninteracting residues) in order to promote or destabilize helix formation. In the case of ACTR binding to its target NCBD, the changes in the unbound IDP helicity strongly affected IDP–target affinity (58), but in another system in which PUMA binds MCL-1, such a relationship is weak (59). For a series of helicity-modulating mutants in cMyb, only a few resulted in considerable changes in affinity, suggesting that these effects might be position specific (60, 61). Similarly, changes in the average unbound helicity have little effect on the p27–Cdk interaction (62). How general is the link between IDP folding propensity (which can be affected by noninteracting mutations) and the IDP–target affinity? First, what seems to have been overlooked in previous studies is that mutations often also change the bound-state ensembles, not only the unbound IDP ensemble. Second, the mutational effects are position specific due to short-rage cooperativity of helix–coil transitions and are therefore difficult to interpret from the observed average quantities. Our model and experiments show that noninteracting mutations can strongly affect the target-bound conformation since IDPs optimize their ensembles to find a balance between maximal conformational freedom and formation of favorable interactions (Figs. 4–6). Depending on the underlaying hotspot distribution, the resulting effects of IDP mutations can be compensated by modulations of the bound-state ensemble. These results therefore advance the previous view which focused mainly on the unbound state, while the target-bound state was considered static (helical) as observed in the IDP–target structures (50, 58–62). The model we present treats the mutational effects in a position-dependent way both in the unbound and bound state, and upon considering both states, a reliable prediction of IDP–target affinity can be obtained (Fig. 6). Taken together, the model described in this study provides the physical basis for understanding sequence–ensemble relationship in fuzzy complexes and demystifies the seemingly chaotic behavior of fuzzy protein complexes.

Materials and Methods

Further details regarding the ITC and CD measurements and the computational estimation of IDP hotspots are described in SI Appendix, Materials and Methods.

Hotspot Constrained Helix–Coil Model for IDP–Target Interactions.

In contrast to globular proteins in which folding is highly cooperative and may be described by simple two or multi state models, the cooperativity of helix–coil transitions in peptides is lower and governed by interactions that are close in sequence (local). Two Ising-like models describing the helix–coil transitions are those by Zimm and Bragg (63) and by Lifson and Roig (39) (LR). In the LR model, a given residue can exist either in helix (h) or a coil (c) state. The residues with Φ and Ψ dihedral angles associated with the helix geometry are considered to be in h state, and the rest of the Φ, Ψ space is attributed to the c state. Thus, a given N-residue polypeptide may adopt 2^N different microstates. The basic form of LR model assumes only local interactions; that is the formation of hydrogen bonds between i and i+4 residues. Accordingly, residues in the c state will be assigned a statistical weight 1, and those in the h state are assigned weight v or w, depending on the state of neighboring residues. The weight w, also known as propagation constant, is assigned to those residues which neighbor h residues on both sides and it is related to the free energy $\Delta G_{CH}$ for the transition of a resides from c to h state:

$$\Delta G_{CH,i}=-RTln{w}_i.$$ [1]

Propagation constants and corresponding values of $\Delta G_{CH}\hspace{1em}$differ between amino acids; thus, we use index i to denote a particular amino acid residue. Importantly, previous studies have established the values of w_i and $\Delta G_{CH,i}$ for all amino acids (35) which range from w(Ala) = 1.61 (promotes helix) to w(Gly) = 0.05 (helix breaker). Weight v, also known as nucleation constant, is assigned to residues in h conformation that do not form i-i+4 hydrogen bonds, in other words to h residues appearing in the context of coil. A value v = 0.048 is usually adopted for all amino acids (35). These definitions lead to the following rules for assignation of statistical weights to the central residue in the eight triplets: hhh: w, hhc: v, chh: v, chc: v, hch: 1, hcc: 1, cch:1, and ccc: 1. For example, for the microscopic state: cchhhhccchhcccc, the statistical weight is 11vwwv111vv1111. These LR definitions can be more conveniently expressed as a matrix, which adopts its common 3 × 3 form (64):

$$M_i=\left(\begin{array}{ccc}{w}_i& v& 0\\ 0& 0& 1\\ v& v& 1\end{array}\right).$$ [2]

The matrix method enables practical enumeration of all possible microscopic states of a polypeptide with a given amino acid sequence. The partition function $Z_U$ for a given for a polypeptide with N-residues in its free, unbound state is obtained via multiplication of matrices:

$$Z_U=\left(\begin{array}{ccc}0& 0& 1\end{array}\right){\displaystyle \prod }_{i=1}^{i=N}{M}_i\left(\begin{array}{c}0\\ 1\\ 1\end{array}\right)=e{\displaystyle \prod }_{i=1}^{i=N}{M}_i{e}^{+}.$$ [3]

Here, $M_i$ denotes matrix for the ith reside in the sequence with its characteristic w_i weight. Matrices $M_i$ are multiplied in the order of the amino acid sequence till the last (N-th) residue. End vectors ($e$ and $e^{+}$) ensure that the terminal residues will be assigned either 1 or v, but not w weight. This treatment can be also simplified by using the same average value of the w (or the corresponding $\Delta G_{CH}$) for all residues. In this case, the IDP is treated effectively as homopolymer, and the product ${\displaystyle \prod }_{i=1}^{i=N}{M}_i$ is replaced simply by $M^N$.

The versatility of LR theory allows numerous extensions. Several strategies for how to introduce tertiary interactions to describe folding of α-helical proteins have been described previously (65, 66). The key observation that needs to be considered is that IDPs fold upon binding their targets, suggesting that tertiary interactions stabilize helical states and thus nucleate helices. We approach to this as follows. First, all the IDP hotspots are identified and represented in a sequential order. For example, for a hotspot distribution 0001001100, the residues represented with number 1 will form hotspots. Next, different sets of these hotspots are defined. For example, an IDP with three hotspots can bind the target with either one, two, or all three hotspots. More formally, we assume that each of N_H hotspots may bind independently; thus, in addition to one set with all hotspots (fully bound), there are additional sets in which only n hotspots are formed (partially bound). All these constitute a binding set which contains ${\displaystyle \sum }_{1\le n\le N_H}\left(\frac{N_H}{n}\right)=2^{N_H}-1$ hotspot sets. This approach allows a convenient way to implement different degrees of cooperativity. For example, when all partially bound hotspot sets are excluded, then a fully cooperative model is obtained. Next, for every h hotspot set, we derive the contracted partition function (67) $Z_{b,h}$ that corresponds to all microstates with n hotspots formed. Based on the hotspot set h, we first employ a generating function $F\left(h\right)$:

$$F\left(h\right)=e{\displaystyle \prod }_{i=1}^{i=N}\left(\begin{array}{ccc}{w}_i\ast z_i& v& 0\\ 0& 0& 1\\ v& v& 1\end{array}\right)e^{+},$$ [4]

where the matrix is multiplied in the order of amino acid and hotspot sequence. While the amino acid sequence defines the values of $w_i$ weights (in LR model), the corresponding hotspot set h defines the value of parameter $z_i$. When ith residue forms a hotspot, $z_i$ = x, otherwise, $z_i$ = 1. In other words, the hotspot residues will gain and additional weight x which related to the IDP–target interaction energy:

$$\Delta G_{INT}=-RTlnx.$$ [5]

Given the coarse-grained treatment of the IDP–target interactions in this model, we assume the same average interaction energy $\Delta G_{INT}$ for all hotspots. It is important to note that the x weights are assigned only to residues also having the w weight; therefore, this will drive helix folding. The function $F\left(h\right)$ generates all possible microstates and it differs from $Z_U$ (Eq. 3) by microstates that have additional x weights. In order to keep only these states, which interact with the target, we apply a selection procedure to obtain a contracted partition function corresponding to IDP binding with n hotspots according to the hotspot set h:

$$Z_{b,h}=\frac{d^{n}F\left(h\right)}{d{x}^n}{x}^n/n!.$$ [6]

The global partition function which includes all fully bound and partially bound conformations is then obtained by summation of contracted partition functions $Z_{b,h}$:

$$Z_B={\displaystyle \sum }_{h=1}^{h=2^{N_H}-1}{Z}_{b,h}.$$ [7]

From hereon, we can calculate various macroscopic properties. For example, the fraction microstates that are bound according to h-th hotspot set and establish n hotspots:

$$p\left(h\right)=\frac{Z_{B,h}}{Z_B}.$$ [8]

The probability that a given residue i adopts a helical conformation in the bound-state ensemble is:

$$p\left(w_i\right)=\frac{1}{Z_B}\frac{d{Z}_B}{dln{w}_i},$$ [9]

and the average number of residues in the helical conformation in the target-bound ensemble, <n_w>, is obtained by

$$<n_w>\hspace{0.5em}=\frac{dln{Z}_B}{dlnw}.$$ [10]

Normalizing this value with the total number of residues capable of adopting the w state then gives the fractional helicity. The average <n_w> was used to classify ensembles to different macroscopic types as presented in Fig. 2. When <n_w> is greater than N/2 − 1, such an ensemble is helical, otherwise disordered (for both types <n_w> should also be greater than N_SEP+1). In cases where <n_w> is less than N_SEP+1, these ensembles have clamp conformation, while fraction of partially bound ensembles is estimated using Eq. 8.

The changes in the IDP–target interaction affinity have been estimated from the ratio of bound and unbound partition functions of wt and mutated IDP:

$$\Delta \Delta G=\Delta G_{MUT}-\Delta G_{WT}=-RTln\frac{Z_{B,MUT}}{Z_{U,MUT}}+RTln\frac{Z_{B,WT}}{Z_{U,WT}}.$$ [11]

Different hotspot distributions can be described by the coarse-grained parameters N_SEP/N and κ_H. Parameter κ_H describes how evenly multiple hotspots are distributed in the IDP sequences bounded by first and last hotspot (that is in a given N_SEP). First, the distance between each pair of N_H hotspots is calculated $d_i=N_{H,i+1}-N_{H,i}$, where index i denotes residue index of the hotspot. Then the SD of hotspot distances is calculated as: $\text{σ}=\sqrt{\frac{1}{N_H}{\displaystyle \sum }_{i=1}^{N_H}{(d_i-\overline{d})}^2}$. Finally, ${\text{κ}}_H=\frac{\text{σ}}{{\text{σ}}_{MAX}}$ where ${\text{σ}}_{MAX}$ corresponds to maximal $\text{σ}$ observed in a hotspot distribution.

Data Availability

All data are included in the paper and/or SI Appendix. The computer code used for analysis is available in GitHub at https://github.com/sanhadzi/PNAS_fuzzy.