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Biophys J. Dec 2005; 89(6): 3714–3720.
Published online Sep 8, 2005. doi:  10.1529/biophysj.105.062125
PMCID: PMC1366941

Predicting the Tolerance of Proteins to Random Amino Acid Substitution

Abstract

We have recently proposed a thermodynamic model that predicts the tolerance of proteins to random amino acid substitutions. Here we test this model against extensive simulations with compact lattice proteins, and find that the overall performance of the model is very good. We also derive an approximate analytic expression for the fraction of mutant proteins that fold stably to the native structure, Pf(m), as a function of the number of amino acid substitutions m, and present several methods to estimate the asymptotic behavior of Pf(m) for large m. We test the accuracy of all approximations against our simulation results, and find good overall agreement between the approximations and the simulation measurements.

INTRODUCTION

A protein's tolerance to random amino acid substitutions is of fundamental importance both in protein engineering and molecular evolution. In molecular evolution, a protein's neutrality, that is, the fraction of single amino acid substitutions that do not disrupt the protein's function, has a substantial influence on how this protein evolves and accumulates mutations (16). In protein engineering, the knowledge of a protein's tolerance to mutations helps one to optimize the mutagenesis conditions in directed protein evolution (7); several groups have characterized experimentally a protein's loss of function under random mutations (811).

Protein mutagenesis studies suggest that a large fraction of deleterious amino acid substitutions disrupt a protein's structure rather than specifically affecting functional residues (1214). Therefore, the fraction of substitutions that disrupt a protein's structure is a reasonable lower bound to the fraction of substitutions that will disrupt a protein's function.

We (8) have recently proposed a thermodynamic model that allows one to calculate the probability Pf(m) with which a protein retains its structure after m amino acid substitutions. This model uses as input the distribution of free energy changes ΔΔG for individual amino acid substitutions. It is based on the idea that the free energy change caused by one amino acid substitution is independent of the change caused by another such substitution, and that the protein continues to fold correctly as long as its free energy of folding remains below some threshold level. If the protein's free energy of folding is initially a distance C from the threshold, then the fraction of sequences with m substitutions that still fold correctly is given by the fraction of sums equation M1 that are less than C, where the Xi are independent, identically distributed random variables taken from the ΔΔG distribution. For a small set of both simulated lattice proteins and real proteins, we (8) have shown that this model has excellent predictive power. Here, we are interested in three questions:

  1. How well does this model hold up for a more extensive data set of lattice proteins?
  2. Can one make general statements about how Pf(m) behaves for large m, and how is this behavior influenced by the ΔΔG distribution?
  3. How can the neutrality be calculated from the distribution of ΔΔG values?

METHODS

Lattice protein simulations

We implemented a maximally compact, 5 × 5 two-dimensional square lattice model, as previously described (15,5). In short, we folded simulated polypeptide chains of length L = 25 residues into a maximally compact structure, representing one of the 1081 possible (16) self-avoiding compact walks of length 25 not related by rotational or reflection symmetry. (We neglected the vanishingly small fraction of palindromic sequences.) We used an alphabet of 20 amino acids, and calculated the contact energies between nonbonded neighboring residues according to Table 3 of Miyazawa and Jernigan (17). We calculated a lattice protein's free energy of folding ΔGf as described by Taverna and Goldstein (15), and considered the protein to be stably folded if ΔGf was below a cutoff ΔGcut. We carried out all analyses for three different cutoffs, ΔGcut = −4.0 kcal/mol, −5.0 kcal/mol, and −6.0 kcal/mol.

We first analyzed a dataset of 300 randomly chosen sequences, 100 at each cutoff. We generated these sequences in the following way: First, we generated random sequences and tried to fold them. We kept all those sequences whose free energy of folding was below ΔGcut = −4.0 kcal/mol, and whose native conformation was different from the native conformations of all stably folding sequences we had encountered so far. We repeated this procedure until we had 100 sequences that could stably fold into 100 unique conformations at ΔGcut = −4.0 kcal/mol. For the remaining two cutoffs, we used hill climbing and subsequent neutral evolution to obtain, at each cutoff, 100 additional sequences that could stably fold into the same 100 conformations as the original sequences. Under hill climbing, we repeatedly mutated a sequence, and accepted all mutations that increased the protein's stability without changing the native conformation. Under neutral evolution, we repeatedly mutated a sequence, and accepted all mutations that did not destabilize the protein beyond the chosen cutoff and did not change the native conformation. We always repeated neutral evolution until we had accepted 1000 mutations.

For all 300 sequences, we estimated Pf(m), the fraction of mutant proteins that fold stably to the original native conformation after m amino-acid substitutions, by randomly sampling mutants according to the following procedure: We carried out all single-point mutations, and sampled 104, 5 × 104,105,…107 multiple-point mutations for m = 2,3,4,…,8. We then calculated Pf(m) by dividing the number of correctly folded sequences that we found at the given mutational distance m by the total number of mutants we tried at that distance. We defined a protein as correctly folded if its minimum free energy was below the chosen cutoff ΔGcut and if its native conformation was identical to that of the starting sequence. In the vast majority of these 300 replicates, we found between several hundred and several thousand correctly folded proteins at each mutational distance m. Consequently, our estimate for Pf(m) in lattice proteins is highly accurate.

We measured the ΔΔG distribution of each of the 300 sequences by carrying out all possible single-point mutations, and then calculating the differences between the minimum free energy of the original sequence and the mutated sequences.

We calculated the prediction for Pf(m) from the ΔΔG distribution as described (8). In short, we first binned the ΔΔG distribution into bins of width 0.01 kcal/mol, and then calculated the m-fold convolution of this binned distribution using the fast Fourier transform of the software package R, version 1.9.1 (18). Finally, we numerically integrated the convolved distribution from −∞ to C to obtain Pf(m).

We carried out a second set of simulations to determine the influence of the starting sequence on the neutrality left angle bracketνright angle bracket. We selected the sequences of 10 representative conformations (among the 100 unique conformations of the first data set), and generated, through neutral evolution as before, for each conformation at each cutoff nine additional sequences folding stably into this conformation. We measured then both Pf(m) and the ΔΔG distribution for these additional 270 sequences as described above.

Calculation of left angle bracketνright angle bracket

Pf(m) decays approximately as left angle bracketνright angle bracketm for large m. We estimated left angle bracketνright angle bracket from the measured Pf(m) by carrying out a linear regression of ln Pf(m) versus m, where we restricted the range of m from 4 to 8 to capture the asymptotic behavior of Pf(m). The neutrality left angle bracketνright angle bracket followed then as left angle bracketνright angle bracket = ea, where a is the slope of the regression line.

We also calculated left angle bracketνright angle bracket in the context of a number of approximation schemes, described in Appendices A–D, and summarized in Results, below. For the Cramér approximation (Appendix B), we numerically minimized the moment-generating function [var phi](t) of the ΔΔG distribution. Let {ΔΔGi} be the set of free energy changes caused by all single point mutations. Then, equation M2 and its derivative equation M3 We numerically found the value t* at which [var phi]′(t*) = 0, and then set left angle bracketνright angle bracket = [var phi](t*).

For the Markov chain approximation (Appendix C), we constructed the matrix Wij using bins of width 0.015 kcal/mol, and spanning a range of 25.0 kcal/mol, from ΔGcut to ΔGcut −25.0 kcal/mol. We calculated the largest eigenvalue of this matrix by repeatedly multiplying Wij to a vector (with all components initially set to one), and then renormalizing the vector to unit length, until the vector had converged to the dominant eigenvector of Wij. We then obtained the quantity left angle bracketνright angle bracket from the change in length in the dominant eigenvector of Wij after a single multiplication with Wij.

RESULTS

First, we assess how well our method to predict Pf(m) works in a large data set. We (8) have previously studied only a handful of noncompact lattice proteins and three real proteins. Overall, we find that the method works very well for the compact lattice proteins we study here. Fig. 1 shows several typical examples. In many cases, we find that the prediction of Pf(m) is highly accurate up to m = 8, which is the largest number of mutations we consider (Fig. 1, AD). In those cases that show some discrepancy between the predicted and the measured Pf(m), we typically find that the prediction works well up to m = 3 or 4, but starts to deviate from the measured results for larger m. There is no clear tendency toward either over- or underestimation of the measured results by the prediction (Fig. 1, EH). Note that our data set covers a wide range of different conformations, as all 100 sequences at a given cutoff fold into a unique conformation.

FIGURE 1
Fraction of correctly folded mutants Pf as a function of the number of mutations m, for eight lattice proteins that stably fold at ΔGcut = −5.0 kcal/mol. Points indicate measurement results, and solid lines indicate the prediction ...

We can quantify the performance of our prediction using the root-mean-squared (RMS) deviation of the log-transformed Pf(m). Let equation M4 be the predicted fraction of mutants that fold correctly, and Pf(m) the corresponding measured value. Then, we define the logarithmic RMS deviation ρ as

equation M5
(1)

For our data, cases in which the prediction agrees well with the measured Pf(m) have RMS values well below 1.0, whereas cases in which the prediction shows some clear deviation have RMS values of ~1.0 or higher (Fig. 1). When we consider all 300 replicates, we find that the majority of the cases have an RMS value below 1.0, and only rarely does the RMS value exceed 2.0 (Fig. 2). There seems to be a slight tendency for the RMS value to increase as the cutoff value becomes more stringent (i.e., from ΔGcut = − 4 kcal/mol to − 6 kcal/mol).

FIGURE 2
Histogram of RMS values for 100 randomly chosen sequences each, at three cutoff levels.

Next, we are interested in asymptotic expressions of Pf(m) for small and large m. For small m, we can approximate Pf(m) using the Edgeworth expansion (Appendix A). The Edgeworth expansion provides correction terms to the central limit theorem for finite sums of random variables. These correction terms take into account successively higher moments of the ΔΔG distribution. Fig. 3 shows how the Edgeworth expansion provides an increasingly more accurate approximation of Pf(m) as higher-order terms are included. However, whereas in some cases the Edgeworth expansion works very well with only three additional moments beyond mean and variance (Fig. 3 A), in other cases the Edgeworth expansion deviates significantly from Pf(m) in all orders we have considered (Fig. 3 B). Furthermore, because the Edgeworth expansion leads to a normal distribution function multiplied by a polynomial (Eq. 3), it must inevitably break down as m becomes large.

FIGURE 3
Prediction of Pf(m) according to the Edgeworth expansion. Points indicate the measured Pf(m), and lines indicate the Edgeworth expansion to various orders (dotted lines, normal term only; dot-dashed lines, normal term plus first-order corrections; dashed ...

For large m, empirical observations show that Pf(m) decays approximately as left angle bracketνright angle bracketm ((810) and Fig. 1). The value left angle bracketνright angle bracket can vary substantially among sequences, but generally tends to increase with the cutoff (Fig. 4). We can interpret left angle bracketνright angle bracket intuitively as the average neutrality of all sequences that stably fold into the given structure. We give a formal argument for this interpretation in Appendix C. An exponential decay of the form Pf(m) ≈ left angle bracketνright angle bracketm follows from the Gaussian term in the Edgeworth expansion (Appendix A). However, the value of left angle bracketνright angle bracket predicted by this term is not very accurate (data not shown). The Gaussian approximation fails because, for large m, Pf(m) is extremely sensitive to small deviations from normality in the tail of the m-fold convolved ΔΔG distribution.

FIGURE 4
Asymptotic neutralities left angle bracketνright angle bracket (measured) at the three different cutoffs, sorted by magnitude and plotted against their rank.

Numerically, we can estimate left angle bracketνright angle bracket by first calculating the prediction for Pf(m) using the m-fold convolution of the ΔΔG distribution, and then obtaining left angle bracketνright angle bracket from a log-linear regression in the same way in which we estimate it from the measured Pf(m) (see Calculation of left angle bracketνright angle bracket, above). In the following, we refer to this method as the convolution method. The convolution method does not generate any new insight into what determines the value of left angle bracketνright angle bracket, but it serves as a useful test case. First, by comparing for a large set of proteins the measured left angle bracketνright angle bracket to the left angle bracketνright angle bracket predicted by the convolution method, we obtain an overall estimate of how well our model performs. Second, the convolution method is the correct benchmark for all other methods of estimating left angle bracketνright angle bracket: Because any deviation between the prediction from the convolution method and the measured left angle bracketνright angle bracket is an inherent shortcoming of our model, we can only expect that any approximate method to estimate left angle bracketνright angle bracket will work at most as well as the convolution method, and will generally perform worse. Fig. 5 A shows that the left angle bracketνright angle bracket predicted by the convolution method correlates strongly with the measured (overall R2 for all 300 data points R2 = 0.789, p < 10−15), in agreement with our earlier observation that, overall, our model works very well.

FIGURE 5
Prediction for left angle bracketνright angle bracket according to three different methods. The solid lines indicate perfect agreement between predicted and measured left angle bracketνright angle bracket. Symbols indicate ΔGcut = −4.0 kcal/mol (solid ...

A straightforward method to predict left angle bracketνright angle bracket from the ΔΔG distribution follows from large-deviation probability theory. Cramér's theorem implies that Pf(m) must decay exponentially, and implies that left angle bracketνright angle bracket is approximately given by the unique minimum of the moment-generating function of the ΔΔG distribution (Appendix B). In Fig. 5 B, we compare the left angle bracketνright angle bracket predicted by the Cramér approximation to the measured left angle bracketνright angle bracket. We see that the Cramér approximation performs almost as well as the convolution method. The correlation between the left angle bracketνright angle bracket values predicted according to the convolution method and the Cramér approximation is very strong (overall R2 for all 300 data points R2 = 0.971, p < 10−15).

The intuitive explanation for why Pf(m) decays approximately as left angle bracketνright angle bracketm is that each correctly folded sequence has, on average, a fraction left angle bracketνright angle bracket of correctly folded single-point neighbors, so that with each mutational step the total Pf(m) is reduced by a factor of left angle bracketνright angle bracket. We can make this reasoning more precise with the Markov chain approximation. The Markov chain approximation is based on the assumption that single-point mutants to sequences at distance m that do not fold correctly do not contribute to Pf(m + 1). With this assumption, left angle bracketνright angle bracket turns out to be the largest eigenvalue of a matrix Wij that contains the transition probabilities from any stable protein to any other stable protein under single-point mutations (Appendix C). We do not present results from the Markov chain approximation in Fig. 5, because they are very similar to those found with the Cramér approximation (overall R2 for all 300 data points R2 = 0.9992, p < 10−15). However, the left angle bracketνright angle bracket values predicted by the Markov chain approximation tend to be slightly smaller than those predicted by the Cramér approximation, the reason being that the Markov chain approximation neglects mutations that stabilize previously unstable sequences (Appendix C).

The last method we consider is the mean-field approximation. The mean-field approximation is based on the idea that we can replace the distribution of proteins with different neutralities by a single protein with an effective neutrality that equals left angle bracketνright angle bracket, and is extremely simple to calculate (Appendix D). Fig. 5 C shows that the mean-field approximation performs only slightly worse than the Cramér approximation. The correlation between the left angle bracketνright angle bracket values predicted from the convolution method and the mean-field approximation is also strong (overall R2 for all 300 data points R2 = 0.939, p < 10−15).

Finally, we have generated an additional data set of 10 × 10 sequences that fold into the same structure, to assess to what extent left angle bracketνright angle bracket depends on the initial sequence or the structure. We find that although there is some spread in the estimated left angle bracketνright angle bracket for different sequences folded into the same structure, the left angle bracketνright angle bracket values for the different starting sequences clearly cluster around a mean value equation M6 that is determined by the structure. Fig. 6 shows data for a representative five of the 10 structures we considered for this additional data set. We carried out a pairwise t-test for all 45 possible pairings of the 10 structures, at each cutoff, and found that (after applying the false-discovery-rate correction for multiple testing (19)) only 12, 9, and 5 of the 45 pairs at cutoffs ΔGcut = −4.0 kcal/mol, −5.0 kcal/mol, and −6.0 kcal/mol do not have a statistically significant (at the 5% level) difference in equation M7

FIGURE 6
Asymptotic neutralities left angle bracketνright angle bracket (measured) for five different structures. Symbols indicate ΔGcut = – 4.0 kcal/mol (solid circles), ΔGcut = −5.0 kcal/mol (open squares), and Δ ...

DISCUSSION

We have extensively tested a model introduced earlier to describe and explain the tolerance of proteins to amino-acid substitutions (8). These tests were performed on an array of 100 structures and three cutoff levels. The model performs well across this data set, which gives strong support for the model's central claims, its generality, and its theoretical underpinnings. The predicted emergence of an exponential decline in the Pf(m) that is parameterized by the mean neutrality left angle bracketνright angle bracket is both observed and estimated by several independent methods, and the preliminary finding that left angle bracketνright angle bracket is principally a structural property receives computational support through tests across 10 structures. Using a Markov chain method, we also explain why the rate of the asymptotic decay of Pf(m), as measured by left angle bracketνright angle bracket, is in fact related to the average neutrality of all sequences that can stably fold into the native conformation.

For computational efficiency, we have used maximally compact two-dimensional lattice proteins (with the full amino-acid alphabet). Compact lattice proteins have the drawback that the additional constraint of maximal compactness allows many more sequences to stably fold than otherwise would; also, noncompact lattice proteins rarely fold into maximally compact formations (20, 21). However, in previous work (8), we had tested the model against a small set of two-dimensional noncompact lattice proteins, as well as two real proteins, and found the model to perform well in these cases. It therefore seems unlikely that the results that we report here are artifacts of the additional constraint of maximal compactness. Likewise, three-dimensional lattice proteins have substantially more conformations at the same sequence length than two-dimensional lattice proteins, and our model could, in principle, break down in three dimensions. We have no specific reason to believe that our model would perform substantially worse for three-dimensional lattice proteins than for two-dimensional lattice proteins, but this hypothesis remains to be tested.

A key advantage of our model is its extreme simplicity. Our finding that left angle bracketνright angle bracket can be trivially computed with reasonable accuracy using either a mean-field approximation or a generating function approach that extends the model's utility. Our finding that the Gaussian term in the Edgeworth expansion cannot accurately describe the data suggests that a Gaussian approximation for the initial ΔΔG distribution is simply not adequate for the estimation of left angle bracketνright angle bracket. Thus our model, although simple, is sensitive to the detailed form of the ΔΔG distribution, rather than just its mean and variance.

Whether these results extend to an equally broad class of naturally occurring proteins remains an open question. A useful feature of our model is that it depends, in a direct and relatively simple manner, on the distribution of the ΔΔG values, which are routinely measured in natural proteins and can be computationally estimated from crystal structures. In general, we do not know the difference C between the native stability of proteins and their minimum free energy cutoff. However, the existence of a cutoff is indicated by diverse observations such as the abundance of temperature-sensitive mutations and the steep (exponential) dependence on stability of the folded and unfolded protein concentrations at equilibrium. We do not know whether the cutoff is consistent across proteins or varies, like left angle bracketνright angle bracket, from structure to structure.

An important practical implication of our model is that the fraction of mutant proteins retaining fold can be increased in a predictable fashion by modest increases in wild-type protein stability. Mutagenesis experiments aimed at discovering functionally improved proteins may thus have stability-dependent optimal mutation rates (7) which, at least in principle, may be estimated using our model. Our results here offer strong support to the suggestion (8) that stability is a critical, but generally overlooked, parameter in directed evolution.

Acknowledgments

This work was supported by National Institutes of Health NRSA No. 5 T32 MH19138 to D.A.D., and by a Howard Hughes Medical Institute predoctoral fellowship to J.D.B. C.O.W. was supported in part by National Institutes of Health grant AI 065960.

APPENDIX A: EDGEWORTH EXPANSION

We wish to estimate the probability equation M8 where Xi are independent, identically distributed random variables distributed according to the ΔΔG distribution, and C is the distance to the free-energy cutoff beyond which the protein does not stably fold. It is convenient to introduce the standardized random variable equation M9 where equation M10 and μ and σ are the mean and standard deviation of the ΔΔG distribution, respectively. Let κn be the nth cumulant (see Appendix E) of the ΔΔG distribution. We define equation M11 and write the standard normal distribution function

equation M12
(2)

Then, the Edgeworth expansion of the distribution function F(z) of the random variable Z is given by (22)

equation M13
(3)

An efficient algorithm to generate higher-order terms of the expansion has been presented by Blinnikov and Moessner (23). From F(z), we obtain Pf(m) via

equation M14
(4)

The zeroth-order term of the Edgeworth expansion (the Gaussian term, which takes into account only the mean and variance of the ΔΔG distribution) is

equation M15
(5)

where erfc(t) is the complementary error function. This expression predicts for large m that Pf(m) ≈ left angle bracketνright angle bracketm with left angle bracketνright angle bracket = exp [−μ2/(2σ2)].

APPENDIX B: CRAMÉR APPROXIMATION

We can calculate the asymptotic behavior of Pf(m) for large m from large-deviation theory. According to the central limit theorem, for large m the sum equation M16 (as introduced in Appendix A) is approximately normally distributed with mean and variance 2, where μ and σ2 are the mean and variance of the ΔΔG distribution. The probability equation M17 is therefore a tail probability that becomes vanishingly small as m approaches infinity. Cramér's theorem (24) for large deviation probability states that, for a < μ,

equation M18
(6)

where [var phi](t) is the moment-generating function of the distribution of Xi, and t* is the value of t at which [var phi](t) – at attains its minimum.

Cramér's theorem can be used as a basis for approximating the asymptotic behavior of Prob(Sm/ma), namely, for large m,

equation M19
(7)

The theorem therefore gives a theoretical justification for the exponential decay of tail probabilities. For the case of interest, we have a = C/m, which is small when m is large. For large m, we may therefore consider an approximation to Pf(m) of the form

equation M20
(8)

and therefore estimate the average neutrality as left angle bracketνright angle bracket[var phi](t*).

Further refinements to Cramér's theorem, especially in the context of placing bounds on tail probabilities for finite m, have been the subject of recent advances in large deviation probability theory (see, for example, Hahn and Klass (25) and references therein) and may be used to obtain more accurate estimates. For our purposes, Cramér's theorem gives a simple and reasonably accurate estimate of Pf(m).

APPENDIX C: MARKOV CHAIN APPROXIMATION

An alternative method to estimate the asymptotic slope left angle bracketνright angle bracket of Pf(m) is based on calculating the steady-state solution of a suitable Markov process. First, we subdivide the range of free energies of folding into discrete bins of width b. We number the bins consecutively and in such a way that all bins with index i ≥ 0 represent stable proteins, and all other bins represent unstable proteins. Now, let pi(m) be the fraction of proteins at mutation distance m in bin i. Clearly, we have equation M21 Next, we introduce the matrix Mij, which gives the probability that a single mutation to a protein in bin j moves that protein into bin i. (Note that under the assumptions of our theory, Mij does not depend on m, and furthermore depends only on the difference ij, but not on the specific values of i or j. The first assumption is necessary for the development of the Markov approximation; the second assumption of stationarity of the transition matrix could be, in principle, relaxed.) Then, we can write Pf(m + 1) as

equation M22
(9)

Our goal is to express the right-hand side in terms of Pf(m), so that we obtain a recursion relation for Pf(m). Unfortunately, the second sum on the right-hand side spans all values of j, positive as well as negative, whereas Pf(m) contains only information about pj(m) with positive index j. Therefore, we now make the approximation that

equation M23
(10)

This approximation is based on the assumption that mutations that stabilize an unstable protein are rare in comparison to mutations that do not destabilize a stable protein, and is the main difference between the Markov chain approximation and the m-fold convolution of the ΔΔG distribution. From here on, we will refer to the submatrix of Mij for which the indices i and j are non-negative as Wij. The distinction between Mij and Wij will become important when we discuss eigenvectors and eigenvalues of Wij.

We can interpret equation M24 as the neutrality of a protein in bin j (for j ≥ 0), and the average neutrality of all proteins at distance m is given by

equation M25
(11)

Since we can replace equation M26 with Pf(m), we find that Pf(m + 1) and Pf(m) are related to each other via

equation M27
(12)

and, assuming that left angle bracketν(m)right angle bracket approaches a limiting value left angle bracketνright angle bracket for large m, we have

equation M28
(13)

This equation implies that for large m, Pf(m) is proportional to left angle bracketνright angle bracketm.

From equation M29 we see that for large m, the pi are proportional to the dominant eigenvector of Wij, by virtue of the Frobenius-Perron theorem (26). (The Frobenius-Perron theorem holds if Wij is primitive—the case whenever there is a path of mutations that leads from any bin i to any other bin j, and Wii > 0 for at least one i.) Furthermore, Eq. 11 implies

equation M30
(14)

where pi* is the dominant eigenvector of Wij. Consequently, left angle bracketνright angle bracket corresponds to the dominant eigenvalue of Wij.

APPENDIX D: MEAN-FIELD APPROXIMATION

A third method to calculate left angle bracketνright angle bracket is the mean-field approximation. The idea of this approximation is that we can replace the distribution of proteins of different stabilities with a single protein of typical stability. The neutrality of this protein should correspond to the average neutrality of all stable proteins. We choose the stability of this protein such that its free energy of folding is identical to the average free energy of folding of all possible single-point mutants that fold correctly. In other words, the average change in free energy of a single mutation that does not destroy the protein's ability to fold is zero. The neutrality of this protein is then the fraction of mutations that cause a change in free energy below a certain cutoff, where the cutoff is chosen such that the average change in free energy for all mutations below the cutoff is as close as possible to zero. We can formalize this condition as follows. Assume that the set {ΔΔGi} contains the free-energy changes caused by all possible single-point mutations (of which there are n), and that the set is ordered such that ΔΔGi < ΔΔGi+1 for all i. Then, we have

equation M31
(15)

APPENDIX E: UNBIASED ESTIMATORS OF CUMULANTS

Let equation M32 be a set of n measurements, and define

equation M33
(16)

According to Dressel (27), the following are unbiased estimators for the first five cumulants κ1κ5 (note that κ1 is the sample average, and κ2 is the sample variance):

equation M34
(17)
equation M35
(18)
equation M36
(19)
equation M37
(20)
equation M38
(21)

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