Analysis of tandem repeat protein folding using nearest-neighbor models

1.1. A thermodynamic classification of linear repeat proteins

Here we will expand this definition, focusing not only on whether repeats can fold independently, but also whether adjacent repeats stabilize (or in principle, destabilize) one another. We will use ΔG_i to represent the free energy of folding of an individual repeat (for autonomously stable repeats, ΔG_i < 0)², and ΔG_i-1,i to represent the free energy of coupling with its immediate N-terminal neighbor (for stabilizing interfaces, ΔG_i-1,i < 0). From this bipartite definition, three useful classes emerge (Figure 1). On one end of the spectrum are tandem repeat proteins where the repeats fold autonomously (ΔG_i < 0) and are uncoupled from their neighbors (ΔG_i-1,i > 0). Proteins in this class, which we refer to as “fully-autonomous repeat proteins” (FARPs), should adopt “beads on a string” structures, corresponding to Kajava’s class V. On the other end of the spectrum are proteins where the repeats cannot fold autonomously (ΔG_i > 0), requiring favorable coupling with their neighbors (ΔG_i-1,i < 0) to drive their folding. Proteins in this class, which we refer to as “nonautonomous repeat proteins” (NARPs), should adopt rigid elongated structures (rods, arcs, or superhelices), corresponding to Kajava’s class III.

The bipartite definition in Figure 1 generates two additional classes of linear repeat proteins. In one, repeats do not fold autonomously, and they are uncoupled from their neighbors (ΔG_i > 0, ΔG_i-1,i ≥ 0). This combination of free energies describes an intrinsically disordered polypeptide, but does not provide a means to study folding and cooperativity. However, the fourth class, where repeats fold autonomously and are favorably coupled to their neighbors (ΔG_i < 0, ΔG_i-1,i < 0), provides a rich opportunity to explore cooperativity in folding, as will be discussed below. These proteins, which we refer to as “semiautonomous repeat proteins” (SARPs), should also adopt rigid elongated structures. In a sense, SARPs are part way in between Kajava’s class III (exhibiting coupling between repeats) and class V (where Kajava classified them since individual spectrin repeats can fold in isolation).

1.2. Examples of proteins composed of tandem folded repeats.

Here we will describe the general properties of tandem repeat proteins that are amenable to nearest-neighbor analysis. These proteins are composed of folded repeats and have no obvious non-nearest neighbor interactions (which excludes globular and closed structures like TIM barrels). Some repeat proteins that match these criteria compiled in Table 1, along with some relevant features extracted from the Pfam database (12). Lengths of repeats selected in Table 1 range from around 20 to 100 residues. Most of these repeat protein families are represented by a large number of sequences (often in the tens of thousands), permitting precise bioinformatics analysis and sequence-based protein engineering. Within each type of repeat protein, sequences of repeats are quite variable, with pairwise identities typically in the low 20 percent range. This variability provides a rich source of variation to connect sequence and structural features to nearest-neighbor energy terms, yet conservation is adequate to create sequences with identical repeats if required for analysis (see section 3.2).

Some structures of repeat proteins are given in Figure 2. Ankyrin repeats are rather small helical repeats that form extensive interfaces with their neighbors (16, 30), placing them in Kajava’s class III. Spectrin repeats are much larger helical repeats that form comparatively small interfaces with their neighbors (38, 18), placing them in Kajava’s class V; as has been noted extensively, adjacent spectrin repeats share a single continuous α-helix, which may couple adjacent repeats. Immunoglobulin repeats of some monomeric proteins such as titin are globular β-sheet domains that form elongated structures with limited nearest-neighbor contacts, suggesting largely autonomous and independent folding (10). Like spectrin repeats, the IgG binding repeats (E-, D-, A-, B-, and C-domains) of protein A fold into three-helix bundles (35); SAXS studies indicate that tandem B-domain (BdpA) repeats are structurally uncorrelated, and are best described by an excluded volume pearl necklace model (9).

2.1. Nearest-neighbor models and their energy terms

The energy terms that are used to make up nearest-neighbor models for repeat protein folding are the intrinsic folding (ΔG_i) and interfacial coupling free energies (ΔG_i-1,i) introduced above (Figure 3). When repeat i folds and its nearest-neighbors (i-1 and i+1) are not folded (reactions i. and ii. Figure 3), the equilibrium constant and free energy for folding are κ and ΔG_i. Equilibrium constants and free energies are related through the standard expression

Here we will typically omit the standard state symbol, but all free energies here are at standard state concentrations (one molar reactant and product).

When repeat i folds and one of its nearest-neighbors is folded (for example, repeat i-1), an interface can be formed (reaction iii., Figure 3). The equilibrium constant for this coupled folding and interface formation is κ_iτ_i-1,i, where κ_i is as defined above. Expressed in this way, τ_i-1,i is an equilibrium constant for forming an interface between folded repeats i-1 and i.

Alternatively, it is possible that repeat i can fold next to a folded repeat but not form an interface (reaction iv. above); this is likely when the interface is weakly stabilizing or destabilizing, as is the case for FARPs. In such cases, the equilibrium constant for folding is κ_i, the same as for folding with unfolded neighbors. In addition to providing a means to analyze FARP unfolding, reaction iv. provides a clear definition of the equilibrium constant for interface formation, τ_i-1,i (vertical transitions, Figure 3). Because interfaces have contributions from two repeats, representing the type of interface requires two repeat types be specified. For example, for repeat types R and X, four types of interfaces can be formed: homopolymeric interfaces between R repeats and between X repeats (with equilibrium constants τ_RR and τ_XX), and heteropolymeric interfaces between R and X repeats (with equilibrium constants τ_RX and τ_XR, depending on the order of the repeats). When relevant, the type of interfacial free energy will be specified using labels such as ΔG_R−1,X, which indicates an interface between an X repeat at position i and an R repeat at position i-1 (Figure 3B).

These equilibrium constants and free energies can be used to construct a partition function for a given repeat array. Here, the partition function is a sum of statistical weights for the fully folded state, each of the different partly folded states, and the unfolded state (which we use as a reference and assign a statistical weight of one). For each state, the statistical weight is simply the product of all the equilibrium constants that are needed to get from the reference state to that state (Figure 3, right-most column). The number of intrinsic κ constants in the product is equal to the number j of folded repeats (i.e., κ^j). However, the number of interfacial τ constants depends on the model and on the arrangement of the folded repeats.

2.2. Partition functions for different nearest-neighbor models

Here we will present several partition functions that model repeat protein folding and can be used to fit equilibrium folding data (Table 3). The models for these partition functions differ in the types of partly folded states they admit (see Appendix 1), and represent different levels of cooperativity. As such, some partition functions are more appropriate for FARPs, and others are more appropriate for NARPs (Table 3).

For the nearest-neighbor models presented here, partition functions are best represented as the product of a series of two-by-two correlation matrices W, with one matrix for each repeat. For a protein with ℓ repeats, the partition function ρ can be written

where n = [0 1] and c = [1 1]^T are row and column vectors that convert the matrix product to a scalar and select the appropriate terms of the partition function. If the repeat array is homopolymeric (that is, if is composed of identical repeats), the partition function becomes

Details of this approach are presented elsewhere (32, 1).

The structure of the correlation matrix is shown in Table 2. The rows of matrix W_i represent whether or not repeat i-1 is folded, and the columns represent whether or not repeat i is folded. Thus, each matrix captures four i-1, i configurations, and the four elements are expressed as equilibrium constants for repeat i relative to the unfolded reference:

Because the left column represents repeat i in the folded state, both entries include the equilibrium constant κ_i. In the top row, the i-1 repeat is folded; although this does not modify the stability of the unfolded state of repeat i (right entry), it likely modifies the stability of the folded state of repeat i (left entry). This modidfication is represented in equation 4 by a general factor ϕ_i-1,i. The form of ϕ varies for different models, as described below.

3.1. Expressions to fit repeat-protein folding transitions using nearest-neighbor models

The partition functions above describe the relative populations of all of the partly folded states along with the fully folded and fully unfolded states for a particular repeat protein array, given a set of nearest-neighbor (intrinsic and interfacial) free energies. However, these free energies are unknowns, and must be determined by analyzing experimental folding data. This requires an expression that gives the value of the observable used to monitor unfolding (Y_obs below, often a spectroscopic observable such as far-UV circular dichroism or tryptophan fluorescence) as a function of the repeat protein conformations in solution. Typically, the populations of folded and unfolded conformations are modulated by a solution variable such as denaturant concentration or temperature, resulting in an equilibrium folding transition (colloquially, a “melt”). Thus, the equation used to fit a melt has the form

where the sum is over each of the c conformations in the set {s} of allowed states. Y_c is the spectroscopic signal from conformation c, and p_c is its population; p_c depends on the intrinsic and interfacial free energies, which in turn depends on the solution variable x. When x represents denaturant concentration, the free energy terms are linearly dependent on denaturant concentration (see Greene & Pace, 1974; Marold et al., 2020).

To use equation 9 to analyze unfolding transitions, the populations p_c must be given explicitly in terms of ΔG_i and ΔG_i-1,i. From statistical thermodynamics, the population of a particular configuration is given by the statistical weight divided by the partition function, such that

Because the partition function ρ is the same for all terms, it can be taken outside the sum. ΔG_c is the free energy difference between conformation c and the unfolded reference state,⁴ and can be written as the sum of ΔG_i and ΔG_i-1,i values, weighted by the number of repeats folded (n_f) and interfaces (n_int) formed:

When Y_c is proportional to the number of repeats that are folded, which is usually the case due to the high degree of structural similarity among repeats, a form of equation 9 can derived that depends on the fraction of repeats that are folded (f_f):

where Y_n and Y_d are the spectroscopic signals from the fully-folded and fully-unfolded, arrays, and

In equation 13, the index j represents the different types of repeats (e.g., N, R, X, C). In the analyses below, data are fitted with equations 12 and 13, using whichever partition function (1D-Ising, fractured Ising, or binomial) is most appropriate. Because, as described in the next section, multiple folding transitions of different constructs are required, a global fit is performed in which different versions of equations 12 and 13, containing shared thermodynamic parameters, are fitted to transitions of different constructs.

3.2. Constructs required for determination of nearest-neighbor thermodynamic parameters

In its simplest form, nearest-neighbor analysis involves only two free energies: ΔG_i and ΔG_i-1,i. This occurs when all repeats are identical, as is sometimes the case with NARPS composed of consensus repeats. To extract values of these two parameters from experimental data, a minimum of two constructs that differ in repeat number are needed. However, homopolymeric consensus NARP arrays are often insoluble, and must be capped with N- and C-terminal repeats containing polar substitutions. This sequence heterogeneity increases the number of thermodynamic parameters that must be determined, and as a result, the number and types of constructs that need to be included in analysis (see (27).

Because individual repeats from NARPs are unstable, there are limits to the amount of heterogeneity that can be accommodated using nearest-neighbor analysis. However, the individual repeats of SARPS and FARPS are stable, allowing fully heterogeneous repeat arrays to be analyzed. In one approach, folding transitions of each individual repeat in an array is analyzed, along with transitions of overlapping pairs of adjacent repeats. For example, for a SARP composed of three repeats ABC, analysis of folding transitions of single-repeat constructs A, B, and C and two-repeat constructs AB and BC is sufficient to determine the five Ising parameters (ΔG_A, ΔG_B, ΔG_C, ΔG_A-1,B, ΔG_B-1,C).

3.3. An example of a non-autonomous repeat protein: consensus ankyrin arrays

One of the first nearest-neighbor studies of a tandem repeat protein was that of an ankyrin repeat protein. Deletion studies using an ankyrin domain from the Drosophila Notch receptor demonstrated that at least three or four repeats were required for folding (29), indicating that ankyrin repeat proteins are NARPs. Thus, a 1D Ising model is appropriate for modeling ankyrin repeat protein unfolding. Though the Notch deletion study was not able to generate enough constructs to determine the Ising parameters for each repeat and interface as a result of the sequence variation among repeats, it did demonstrate that repeats were intrinsically unstable (ΔG_i ≈ +7 kcal/mol) and that interfaces were strongly stabilizing (ΔG_i−1,i ≈ −9 kcal/mol; (29).

Elegant studies using consensus ankyrin repeats confirmed and extended this thermodynamic partitioning (37, 2). An example of a global fit of folding transitions of consensus ankyrin repeat proteins with a 1D-Ising model is shown in Figure 4A. The data set includes eighteen melts for nine constructs that differ in repeat number and capping structure (see Aksel et al., 2011; Marold et al., 2020). The model contains four free energies (the intrinsic folding energies of the N-and C-terminal caps and the internal R repeats, ΔG_N, ΔG_R, and ΔG_C, and an interfacial coupling energy, ΔG_i-1,i) along with a shared denaturant dependence (m) for the three intrinsic free energy terms. Overall, the 1D-Ising model fits the folding transitions of these nine constructs very well, and determines the fitted Ising parameters with tight confidence intervals (2, 27).

3.4. An example of a semiautonomous repeat protein: naturally occurring spectrin arrays

Spectrin repeats are significantly larger (105 residues) than ankyrin repeats, and are known to fold autonomously. Therefore, depending on whether adjacent spectrin repeats interact thermodynamically, spectrin repeat proteins should either be classified as SARPs or FARPs. Jane Clarke’s laboratory has analyzed the folding of single spectrin repeats along with pairs of adjacent repeats and found the pairs to be more stable than the single-repeat constructs, demonstrating that spectrin arrays behave as SARPs (5, 6).

The folding transitions of three adjacent spectrin repeats, R15, R16, and R17, along with the two-repeat pairs, R15R16 and R16R17, are reproduced in Figure 4B. The three constructs involving R16, R17, and the tandem pair R16R17 are well-fitted by a 1D-Ising model, with a reduced sum of square residuals (RSSR)⁵ of 2.5×10⁻⁴. A fitted interfacial ΔG_16,17 value of −3.32 kcal mol⁻¹ is consistent with the classification of this repeat pair as a SARP, as is the goodness of fit. However, the transitions of R15, R16, and R15R16 are not as well-fitted by a 1D-ising model, with an RSSR of 4.8×10⁻⁴ and a nonrandom distribution of residuals (Figure S1A). Although the folding transition of the R15R16 tandem is centered at higher denaturant concentrations, indicating a favorable interfacial interaction, the transition is broad, which is inconsistent with a coupled two-repeat unfolding transition, and thus, inconsistent with a 1D-Ising model.

Although a variety of more complicated models can be fitted to the R15, R16, and R15R16 melts, a particularly good fit is obtained with a model that includes an interaction in which folded repeat R15 is stabilized by unfolded R16. This interaction can be introduced to the partition function for R15R16 using an equilibrium constant ω_15f,16u as follows:

Using this model, the fit of R15, R16, and R15R16 melts gives a significantly improved RSSR of 2.2×10⁻⁴, and the resulting residuals appear more random (Figure S1B). A global fit of the five spectrin folding transitions in Figure 4B using the 1D Ising partition functions to fit R15, R16, R17, and R16R17, along with equation 14 to fit R15R16, gives a low RSSR (2.5×10⁻⁴; Table 4). The fitted free energy of stabilization of folded R15 by unfolded R16⁶ is −2.0 kcal mol⁻¹. This value is consistent with the observation by Batey and Clarke that the rate constant for unfolding of R15 is decreased by a factor of 28 in the context of unfolded R16 (5). Combined with a modest decrease in the folding rate constant, the analogous free energy deduced from the rate constants is −1.9 kcal mol⁻¹, nearly the same as the value determined from the modified Ising fit. It should be noted that although it seems like the inclusion of the additional parameter might lead to an under-parameterization problem (six free energies are extracted from five curves), this problem is made less severe by the fact that an intermediate is populated in the unfolding transition of R15R16, directly constraining ω_15f,16u.

3.5. An example of a fully autonomous repeat protein: BdpA arrays

In full-length Staphylococcus aureus protein A, BdpA is one of five repeated domains with high sequence identity, sharing ~90% sequence identity with its nearest neighbors. Oas and coworkers have studied the folding of a single BdpA repeat and a tandem construct with two adjacent BdpA repeats (3). The equilibrium folding transitions of BdpA and BdpA₂ are reproduced in Figure 4C. The two folding transitions are nearly identical, suggesting that BdpA₂ behaves as a FARP. A 1D-Ising model fits well to the BdpA/BdpA₂ folding transitions, and the fitted interfacial free energy is very close to zero (Table 4).

While a ΔG_BdpA,BdpA value of zero is consistent with an interfacial interaction that is neither stabilizing nor destabilizing, it is inconsistent with the number of states in the 1D Ising model . An interfacial free energy of zero (equilibrium constant of one) would mean that when both repeats are folded, half of the population has an interface formed, and half has does not. However, the 1D Ising model does not allow for fractured interfaces. To account for this missing state, we fitted the BdpA curves using the fractured 1D-ising model. This model fits about as well as the standard 1D Ising model, and give a nearly identical ΔG_BdpA (Table 4). However, ΔG_BdpA,BdpA is poorly defined; though it has a lower bound of around +1 kcal/mol, it is essentially unbounded from above. This reflects the fact that unstable interfaces are not formed and thus have no influence on the folding transitions, regardless of whether interfacial stability is +1 or +10 kcal/mol. This is a manifestation of the thermodynamic maxim that things that are energetically unfavorable do not happen.

Although the fractured Ising partition function is a more appropriate description of the states of BdpA than the simpler 1D-Ising model, its poorly determined interfacial free energy is rather ungainly. The binomial model, which has the same form as the 1D-Ising model but treats adjacent folded repeats as unpaired, fits with about the same RSSR as the fractured 1D-Ising model, and gives identical ΔG_BdpA and m-values to three significant figures (Table 4). The goodness-of-fit of the binomial model further supports the assignment of BdpA arrays to FARPs.