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How accurate are polymer models in the analysis of Forster resonance energy transfer experiments on proteins?

Single molecule Forster resonance energy transfer (FRET) experiments are used to infer the properties of the denatured state ensemble (DSE) of proteins. From the measured average FRET efficiency,, the distance distribution P(R) is inferred by assuming that the DSE can be described as a polymer. The single parameter in the appropriate polymer model (Gaussian chain, Wormlike chain, or Self-avoiding walk) for P(R) is determined by equating the calculated and measured. In order to assess the accuracy of this"standard procedure,"we consider the generalized Rouse model (GRM), whose properties [and P(R)] can be analytically computed, and the Molecular Transfer Model for protein L for which accurate simulations can be carried out as a function of guanadinium hydrochloride (GdmCl) concentration. Using the precisely computedfor the GRM and protein L, we infer P(R) using the standard procedure. We find that the mean end-to-end distance can be accurately inferred (less than 10% relative error) usingand polymer models for P(R). However, the value extracted for the radius of gyration (Rg) and the persistence length (lp) are less accurate. The relative error in the inferred R-g and lp, with respect to the exact values, can be as large as 25% at the highest GdmCl concentration. We propose a self-consistency test, requiring measurements ofby attaching dyes to different residues in the protein, to assess the validity of describing DSE using the Gaussian model. Application of the self-consistency test to the GRM shows that even for this simple model the Gaussian P(R) is inadequate. Analysis of experimental data of FRET efficiencies for the cold shock protein shows that at there are significant deviations in the DSE P(R) from the Gaussian model.

Introduction: Much of our understanding of how proteins fold comes from experiments in which folding is initiated from an ensemble of initially unfolded molecules whose structures are hard to characterize [1]. In many experiments, the initial structures of the denatured state ensemble (DSE) are prepared by adding an excess amount of denaturants or by raising the temperature above the melting temperature (T m ) of the protein [2]. Theoretical studies have shown that folding mechanisms depend on the initial conditions, i.e. the nature of the DSE [3]. Thus, a quantitative description of protein folding mechanisms requires a molecular characterization of the DSE -a task that is made difficult by the structural diversity of the ensemble of unfolded states [4,5].
In an attempt to probe the role of initial conditions on folding, single molecule FRET experiments are being used to infer the properties of unfolded proteins. The major advantage of these experiments is that they can measure the FRET efficiencies of the DSE under solution conditions where the native state is stable. The average denaturant-dependent FRET efficiency E has been used to infer the global properties of the polypeptide chain in the DSE as the external conditions are altered. The properties of the DSE are inferred from E by assuming a polymer model for the DSE, from which the root mean squared distance between two dyes attached at residues i and j along the protein sequence (R ij = |r i − r j | ), the distribution of the end-to-end distance P (R) (where R = |r N − r 0 |), the root mean squared end-to-end distance (R ee = R 2 1/2 ), the root mean squared radius of gyration (R g = R 2 g 1 2 ), and the persistence length (l p ) of the denatured protein [6][7][8][9][10][11][12][13][14][15] can be calculated.
In FRET experiments, donor (D) and acceptor (A) dyes are attached at two locations along the protein sequence [4,16], and hence can only provide information about correlations between them. The efficiency of energy transfer E between the D and A is equal to (1 + r 6 /R 6 0 ) −1 , where r is the distance between the dyes, and R 0 is the dye-dependent Förster distance [4,16]. Because of conformational fluctuations, there is a distribution of r, P (r), which depends on external conditions such as the temperature and denaturant concentration.
As a result, the average FRET efficiency E is given by under most experimental conditions, due to the central limit theorem [17]. If the dyes are attached to the ends of the chain, then P (r) = P (R). Even if E is known accurately, the extraction of P (R) from the integral equation (Eq. 1) is fraught with numerical instabilities.
In experimental applications to biopolymers, a functional form for P (r) is assumed in order to satisfy the equality in Eq. 1. The form of P (r) is based off of a particular polymer model which depends only on a single parameter (see Table I): The Gaussian chain (dependent on the Kuhn length a), the Wormlike Chain (WLC; dependent on the persistence length l p ), and the Self Avoiding Walk (SAW; dependent on the average end-to-end distance R ee ). For the chosen polymer model meant to represent the biopolymer of interest, the free parameter (a, l p , or R ee ) is determined numerically to satisfy Eq. 1. Using this method (referred to as the "standard procedure" in this article), several researchers have estimated R g and l p as a function of the external conditions for protein L [11,14], Cold Shock Protein (CspTm) [13], and Rnase H [16]. The justification for using homopolymer models to analyze FRET data comes from the anecdotal comparison of the R g measured using X-ray scattering experiments and the extracted R g from analysis of Eq. 1 [4].
Here, we study an analytically solvable generalized Rouse model (GRM) [18] and the Molecular Transfer Model (MTM) for protein L [19] to assess the accuracy of using polymer models to solve Eq. 1. In the GRM, two monomers that are not covalently linked interact through a harmonic potential that is truncated at a distance c. The presence of the additional length scale, c, which reflects the interaction between non-bonded beads, results in the formation of an ordered state as the temperature (T ) is varied. A more detailed discussion of these models can be found in the Methods section. For the GRM, P (R) can be analytically calculated, and hence the reliability of the standard procedure to solve Eq. 1 can be unambiguously established. We find that the accuracy of the polymer models in extracting the exact values in the GRM depends on the location of the monomers that are constrained by the harmonic interaction. Using coarse-grained simulations of protein L, we show that the error between the exact quantity and that inferred using the standard procedure depends on the property of interest. For example, the inferred end-to-end distribution P (R) is in qualitative, but not quantitative agreement with the exact P (R) distribution obtained from accurate simulations. In general, the DSE of protein L is better characterized by the SAW polymer model than the Gaussian chain model.
We propose that the accuracy of the popular Gaussian model can be assessed by measuring E with dyes attached at multiple sites in a protein [13,20,21]. If the DSE can be described by a Gaussian chain, then the parameters extracted by attaching the dyes at position i and j can be used to predict E for dyes at other points. The proposed self-consistency test shows that the Gaussian model only qualitatively accounts for the experimental data of CspTm, simulation results for protein L, and the exact analysis of the

Results and Discussion
We present the results in three sections. In the first and second sections we examine the accuracy of the standard procedure (described in the introduction) in accurately inferring the properties of the denatured state of the GRM and protein L models. The third section presents results of the Gaussian Self-consistency Test applied to these models. We also analyze experimental data for CspTm to assess the extent to which the DSE deviates from a Gaussian chain.
I. GRM: The Generalized Rouse model (GRM) is a simple modification of the Gaussian chain with N bonds and Kuhn length a 0 , which includes a single, non-covalent bond between two monomers at positions s 1 and s 2 (Fig. 1). The monomers at s 1 and s 2 interact with a truncated harmonic potential with spring constant k, with strength κ = kc 2 /2, where c is the distance at which the interaction vanishes (Eq. 4). The GRM minimally represents a two state system, with a clear demarcation between ordered (with |r(s 2 ) − r(s 1 )| ≤ c) and disordered (with |r(s 2 ) − r(s 1 )| > c) states. Unlike other polymer models (see Table I), which are characterized by a single length scale, the GRM is described by a 0 and the energy scale κ. For βκ → 0 (the high temperature limit, where β = 1/k B T ), the simple Gaussian chain is recovered (see Methods for details). By varying βκ, a disorder → order transition can be induced (see Fig. 1). The presence of the interaction between monomers s 1 and Because E can be calculated exactly for the GRM (see Eq. 5), it can be used to quantitatively study the accuracy of solving Eq. 1 using the standard procedure [6,10,11,13,14].
Given the best fit for the Gaussian chain (Kuhn length a), WLC (persistence length l p ), and SAW (average end-to-end distance R ee ), as described in Table I, many quantities of interest can be inferred (P (R) or R g , for example), and compared with the exact results for the GRM. The extent to which the exact and inferred properties deviate, due to the additional single energy scale in the GRM, is an indication of the accuracy of the standard procedure used to analyze Eq. 1.
P (R) is accurately inferred using the Gaussian polymer model: If the interacting monomers are located near the endpoints of the chain, the end-to-end distribution function is bimodal, with a clear distinction between the ordered and disordered regions [18]. However, if the monomers s 1 and s 2 are in the interior of the chain, the two-state behavior is obscured because the distribution function becomes unimodal. In Fig. 1 The accuracy of the inferred R g depends on the location of the interaction: The two-state nature of the GRM is obscured by the relatively long unstructured regions of the chain, similar to the effect seen in laser optical tweezer experiments with flexible handles [18]. As a result, P (R) is well represented by a Gaussian chain, with a smaller inferred Kuhn length, a ≤ a 0 (Fig. 2). For large βκ, where the ordered state is predominantly occupied and r(s 2 ) ≈ r(s 1 ), the end-to-end distribution function is well approximated by a Gaussian chain with N * = N − ∆s bonds. Consequently, the single length scale for the Gaussian chain, decreases to a ∼ a 0 1 − ∆s/N ≈ 0.71a 0 for large values of βκ (Fig. 2).
Because the two-state nature of the chain is obscured for certain values of |s 2 − s 1 |, the Gaussian chain gives an excellent approximation to the end-to-end distribution function.
However, the radius of gyration R g is not as accurately obtained using the Gaussian chain model, as shown in Fig. 3. The exact R g for the GRM reflects both the length scale a 0 and the energy scale βκ, which can not be fully described by the single inferred length scale a in the Gaussian chain. For the GRM, R g depends not only on the separation between the monomers ∆s, but also explicitly on s 1 (i.e. where the interaction is along the chain; see the inferred R g is in excellent agreement with the exact result (Fig. 3). The relative error in R g (the difference between the inferred and exact values, divided by the exact value) is no less than -2%. However, for interactions near the endpoint of the chain, with s 1 = 0 and the same ∆s = 31, the relative error between the inferred and exact values of R g is ∼ −14%.
The large errors arise because the radius of gyration depends on the behavior of all of the monomers, so that the energy scale βκ plays a much larger role in the determination of R g than R ee .
II. MTM for protein L: Protein L is a 64 residue protein ( Fig. 4A) whose folding has been studied by a variety of methods [11,14,[22][23][24]. More recently, single molecule FRET experiments have been used to probe changes in the DSE as the concentration of GdmCl is increased from 0 to 7 M [11,14]. From the measured GdmCl-dependent E , the properties of the DSE, such as R ee , P (R), and R g , were extracted by solving Eq. 1, and assuming a Gaussian chain P (R) [11,14]. To further determine the accuracy of polymer models in the analysis of E , we use simulations of protein L in the same range of the concentration of denaturant, [C], as used in experiments [6,9].
The average end-to-end distance is accurately inferred from FRET data: In a previous study [19], we showed that the predictions based on MTM simulations for protein L are in excellent agreement with experiments. From the calculated E with the dyes at the endpoints (solid black line in Fig. 4B), which is in quantitative agreement with experimental measurements [19], we determine the model parameter R ee or l p by assuming that the exact P (R) can be approximated by the three polymer models in Table I. Comparison of the exact value of R ee to the inferred value R F , obtained using the simulation results for E , shows good agreement for all three polymer models (Fig. 5A). There are deviations between R ee and R F at [C] > C m , the midpoint of the folding transition. The maximum relative error (see inset of Fig. 5A) we observe is about 10% at the highest concentration of GdmCl. The SAW model provides the most accurate estimate of R ee at GdmCl concentrations above C m , with a relative error ≤ 0.05, and the Gaussian model gives the least accurate values, with a relative error ≤ 0.10 ( Fig. 5A). Due to the relevance of excluded volume interaction in the DSE of real proteins, the better agreement using the SAW is to be expected.
Polymer models do not give quantitative agreement with the exact P (R): The inferred distribution functions, P F (R)'s, obtained by the standard procedure (as described in the introduction) at [C]=2 M and 6 M GdmCl differ from the exact results ( Fig. 5B). Surprisingly, the agreement between P (R) and P F (R) is worse at higher Inferred R g and l p differ significantly from the exact values: The solution of Eq. 1 using a Gaussian chain or WLC model yields a and l p , from which R g can be analytically calculated (Table I). Figs. 6A and 6B, which compare the FRET inferred R g and l p with the corresponding values obtained using MTM simulations, show that the relative errors are substantial. At high [C] values the R F g deviates from R g by nearly 25% if the Gaussian chain model is used (Fig. 6A). The value of R g ≈ 26Å at [C]= 8 M while R F g using the Gaussian chain model is ≈ 31Å. In order to obtain reliable estimates of R g , an accurate calculation of the distance distribution between all the heavy atoms in a protein is needed. Therefore, it is reasonable to expect that errors in the inferred P (R) are propagated, leading to a poor estimate of internal distances, thus resulting in a larger error in R g . A similar inference can be drawn about the persistence length obtained using polymer models (Fig. 6B). Plotting l F p as a function of [C] (Fig. 6B), against l p = R ee /2L, shows that l p is overestimated at concentrations above 1 M GdmCl, with the error increasing as [C] increases. The error is less when the Gaussian chain model is used.
III. Gaussian Self-consistency test shows the DSE is non-Gaussian: The extent to which the Gaussian chain accurately describes the ensemble of conformations that are sampled at different values of the external conditions (temperature or denaturants) can be assessed by performing a self-consistency test. A property of a Gaussian chain is that if the average root mean square distance, R ij , between two monomers i and j is known then R kl , the distance between any other pair monomers k and l, can be computed using Thus, if the conformations of a protein (or a polymer) can be modeled as a Gaussian chain, then R ij inferred from the FRET efficiency E ij should accurately predict R kl and the FRET efficiency E kl , if the dyes were to be placed at monomers k and l. We refer to this criterion as the Gaussian self-consistency (GSC) test, and the extent to which the predicted R kl from Eq. 2 deviates from the exact R kl reflects deviations from the Gaussian model description of the DSE.
GRM: For the GRM, with a non-bonded interaction between monomers s 1 and s 2 , we calculate E ij using Eq. 8 with j fixed at 0 and for i = 20, 40, and 60. Using the exact results for E ij , the values of R ij are inferred assuming that P (r) is a Gaussian chain.
From the inferred R ij the values of E kl and R kl can be calculated using Eqs. 1 and 2, respectively. We note that, since R kl /R ij = |k − l|/|i − j| (Eq. 2) for any pair (k, l) using the Gaussian chain model, the prediction of the Gaussian chain will be independent of the particular choices of k and l, as long as their difference is held constant. We first apply the GSC test to a GRM in which f O ≈ 0.75 due to a favorable interaction between monomers s 1 = 16 and s 2 = 47. There are discrepancies between the values of the Gaussian inferred (R G kl ) and exact R kl distances, as well as the inferred ( E G kl ) and exact E ij efficiencies when a Gaussian model is used (Fig. 7). The relative errors in the predicted values of the FRET efficiency and the inter-dye distances can be as large as 30-40%, depending on the choice of i and j (see insets in Fig. 7). We note that the relative error in the end-to-end distance is small for dyes near the endpoints (the green line in Fig. 7b), in agreement with the results shown in Fig. 1. The errors decrease as f O decreases, with a maximum error of 20% when f O = 0.5, and 10% when f O = 0.25 (data not shown). By construction, the GRM is a Gaussian chain when f O = 0 and therefore the relative errors will vanish at sufficiently small βκ (data not shown). These results show that even for the GRM, with only one nonbonded interaction in an otherwise Gaussian chain, its DSE cannot be accurately described using a Gaussian chain model. Thus, even if the overall end-to-end distribution P (r) for the GRM is well approximated as a Gaussian (as seen in Fig. 1), the internal R kl monomer pair distances can deviate from predictions of the Gaussian chain model.  (1,14) pair. This is due to the fact that the native state has a beta-strand between these two residues; as the protein denatures, they come closer together, increasing the FRET efficiency. We use these values for E ij in the GSC test. The results are shown in Figs. 8A and 8B. Relative errors in E kl as large as 36% at 2.0 M GdmCl and 50% at 7.5 M GdmCl are found, with the lowest errors generally seen for residues close to one another along the backbone, in agreement with the results from the GRM (Fig. 7a inset).
In addition, the number of data points that underestimate E kl increases as [C] is changed from 7.5 M to 2.0 M for |k − l| < 20. Despite these differences, the gross features in Figs.
8A and 8B are concentration independent. Because the error does not vanish for all (k, l) pairs (Figs. 8A and 8B), we conclude that the DSE of protein L cannot be modeled as a Gaussian chain.
The GSC test for CspTm: In an interesting single molecule experiment, Schuler and coworkers have measured FRET efficiencies by attaching donor and acceptor dyes to pairs of residues at five different locations of a CspTm [13]. They analyzed the data by assuming that the DSE properties can be mimicked using a Gaussian chain model. We used the GSC test to predict E kl for dyes separated by |k − l| along the sequence using the experimentally measured values E ij .
The relative error in E kl (Eq. 2) should be zero if CspTm can be accurately modeled as a Gaussian chain. However, there are significant deviations (up to 17%) between the predicted and experimental values (Fig. 9). The relative error is fairly insensitive to the denaturant concentration (compare Figs. 9A and 9B). It is interesting to note that the trends in Fig. 9 are qualitatively similar to the relative errors in the GRM at f O > 0.
Based on these observations we conclude tentatively that whenever the DSE is ordered to some extent (i.e., when there is persistent residual structure) then we expect deviations from a homopolymer description of the DSE of proteins. At the very least, the GSC test should be routinely used to assess errors in the modeling of the DSE as a Gaussian chain.

Conclusions
In order to assess the accuracy of polymer models to infer the properties of the DSE of proteins from measurement of FRET efficiencies, we studied two models for which accurate calculations of all the equilibrium properties can be carried out. Introduction of a non-bonded interaction between two monomers in a Gaussian chain (the GRM) leads to an disorder-order transition as the temperature is lowered. The presence of 'residual structure' in the GRM allows us to clarify its role in the use of the Gaussian chain model to fit the accurately calculated FRET efficiency. Similarly, we have used the MTM model for protein L to calculate precisely the denaturant-dependent E from which we extracted the global properties of the DSE by solving Eq. 1 using the P (R)'s for the polymer models in Table I. Quantitative comparison of the exact values of a number of properties of the DSE (obtained analytically for the GRM and accurately using simulations for protein L) and the values inferred from E has allowed us to assess the accuracy with which polymer models can be used to analyze the experimental data. The major findings and implications of our study are listed below.
(1) The polymer models, in conjunction with the measured E , can accurately infer values of R ee , the average end-to-end distance. However, P (R), l p , and R g are not quantitatively reproduced. For the GRM, R g is underestimated, whereas it is overestimated for protein L. The simulations show that the absolute value of the relative error in the inferred R g can be nearly 25% at elevated GdmCl concentration.
(2) We propose a simple self consistency test to determine the ability of the Gaussian chain model to correctly infer the properties of the DSE of a polymer. Because the Gaussian chain depends only on a single length scale, the FRET efficiency can be predicted for varying dye positions once E is accurately known for one set of dye positions. The GSC test shows that neither the GRM, simulations of protein L, nor experimental data on CspTm can be accurately modeled using the Gaussian chain. The relative errors between the exact and predicted FRET efficiencies can be as high as 50%. For the GRM, we find that the variation in the FRET efficiency as a function of the dye position changes abruptly if one dye is placed near an interacting monomer. Taken together these findings suggest that it is possible to infer the structured regions in the DSE by systematically varying the location of the dyes. This is due to the fact that the FRET efficiency is perfectly monotonic using the Gaussian Chain model. An experiment that shows non-monotonic behavior in E ij as the dye positions i and j are varied is a clear signal of non-Gaussian behavior, and sharp changes in the FRET efficiency as a function of |i − j| may indicate strongly interacting sites (see Fig. 7a).
(3) The properties of the DSE inferred from Eq. 1 become increasingly more accurate as Theory and computational methods GRM model: In order to understand the effect of a single non-covalent interaction between two monomers along a chain, we consider a Gaussian chain with Kuhn length a 0 and N bonds, with a harmonic attraction between monomers s 1 ≤ s 2 , which is cutoff at a distance c. The Hamiltonian for the GRM is where k is the spring constant that constrains r(s 2 ) − r(s 1 ) to a harmonic well. The Hamiltonian in Eq. 3 allows the exact determination of many quantities of interest. Defining x = r(s 2 ) − r(s 1 ) and ∆s = s 2 − s 1 , we can determine most averages of interest for the GRM using · · · = d 3 r 1 d 3 xd 3 r N (· · ·)G(x, r N ; ∆s, N) d 3 r 1 d 3 xd 3 r N G(x, r N ; ∆s, N) C α -SCM protein model and GdmCl denaturation: We use the coarse-grained C αside chain model (C α -SCM) to model protein L (for details see the supporting information in [19]). In the C α -SCM each residue in the polypeptide chain is represented using two interaction sites, one that is centered on the α-carbon atom and another that is located at the center-of-mass of the side chain [26]. Langevin dynamics simulations [27] are carried out in the underdamped limit at zero molar guanidinium chloride. Simulation details are given in [19].
We model the denaturation of protein L by GdmCl using the molecular transfer model (MTM) [19]. MTM combines simulations at zero molar GdmCl with experimentally measured transfer free energies, using a reweighting method [28][29][30] to predict the equilibrium properties of proteins at any GdmCl concentration of interest.

Analysis:
GRM: The average squared end-to-end distance can be computed directly from Eq. 5, using R 2 ee = Na 2 0 + ( x 2 − ∆s a 2 0 ). The exact expression for x 2 is easily determined, but somewhat lengthy, and we omit the explicit result here. Also of interest is the end-to-end distribution function, P (R) = δ[r N − R] , which can be obtained from Eq. 5. In order to determine the probability of an interior bond being in the 'ordered' state (i.e. the fraction of residual structures, see the inset for Fig. 1a), we compute the interior distribution, P I (X) = δ[x − X] , so that f O = |x|≤c d 3 x P I (x). The radius of gyration requires a more complicated integral than the one found in Eq. 5, but we find Note that, unlike the average end-to-end distance, the radius of gyration depends not only on ∆s, but also on s 1 .
The FRET efficiency for a system with dyes attached to r(j = 0) = 0 and r(i), E = [1 + (|r(i)|/R 0 ) 6 ] −1 , is determined from Eq. 5 as where E G (i) is the FRET efficiency for a Gaussian chain with i bonds, and This result allows us to compute the Gaussian Self-consistency test, after a numerical integral over r.
Protein L: Averages and distributions were computed using the MTM [19] which combines experimentally measured transfer free energies [31], converged simulations and the WHAM equations [28][29][30]. The WHAM equations use the simulation time-series of potential energy and the property of interest at various temperatures and gives a best estimate of the averages and distributions of that property. The native state ensemble (NSE) and DSE subpopulations were defined as having a structural RMSD (root mean squared deviation), after least squares minimization, of less than or greater than 5Å relative to the crystal structure for the NSE and DSE respectively. The exact values of l p are computed using the average R from simulations and the relationships listed in Table I. for quantities that are extracted by solving Eq. 1 using the polymer models listed in Table I.
a The average end-to-end distance R ee = R 2 P (R)dR 1/2 b L and l p are the contour length and persistence length respectively. C 1 = (π 3/2 e −α α −3/2 (1 + 3α −1 + 15 4 α −2 )) −1 where α = 3L/(4l p ). C 2 = 1/(2l p ). c Using the simulated R 2 , l p was solved for numerically using this equation.          The color of each point corresponds to the color of each line in Fig. 4b, except for the 1-64 pair, which is not shown here. Figure 9: The Gaussian Self-consistency test (GSC) using experimental data from CspTm. One dye was placed at one endpoint, and the location of the other was varied.