We collected R₂ data for a total of 54 nonhomologous IDPs or IDRs (Table 1; Figure 1). According to indicators from NMR properties, including low or negative NOEs, narrow dispersion in backbone amide proton chemical shifts, and small secondary chemical shifts (SCSs), most of the proteins are disordered with at most transient α-helices. A few are partially folded, including Sev-NT with a well-populated (~80%) long helix (residues 478–491; Jensen et al., 2008), CREB-binding protein fourth intrinsically disordered linker (CBP-ID4) with >50% propensities for two long helices (residues 2–25 and 101–128; Piai et al., 2016), HOX transcription factor DFD (HOX-DFD) with a well-folded domain comprising three helices (Maiti et al., 2019), and Hahellin (apo form) as a molten globule (Patel et al., 2014). In Figure 2, we display representative conformations of five IDPs, ranging from fully disordered MAPK kinase 4 (MKK4; Delaforge et al., 2018) and α-synuclein (Sung and Eliezer, 2007) to Measles virus phosphoprotein N-terminal domain (Mev-P_NTD; Milles et al., 2018) with transient short helices to Sev-NT and CBP-ID4 with stable long helices. The sequences of all the IDPs are listed in Appendix 1.

We used 45 of the 54 IDPs to train and validate SeqDYN and reserved the remaining 9 for testing. The sequence lengths of the training set range from 39 to 406 residues, with an average of 125.3 residues. Altogether R2 data are available for 3966 residues. A large majority (35 out of 45) of the 45 IDPs have mean R2 values (R¯2, calculated among all the residues in a protein) between 2.5 and 5.5 s^–1 (Table 1 and Figure 3A). This R¯2 range is much lower than that of structured proteins with similar sequence lengths. The low R¯2 values and lack of dependence on sequence length (Figure 3—figure supplement 1A) suggest that R2 of the IDPs is mostly dictated by local sequence instead of tertiary interaction.

The most often used temperature for acquiring the R2 data was 298 K, but low temperatures (277–280 K) were used in a few cases (Table 1 and Figure 3—figure supplement 1B). Of the seven IDPs with R¯2 > 6.4 s^–1, four can be attributed to low temperatures (Sólyom et al., 2015; Martin et al., 2016; Janke et al., 2018; Jenner et al., 2018), one is due to a relatively low temperature (283 K) as well as the presence of glycerol (20% v/v; Bafaro et al., 2019), and two can be explained by tertiary structure formation [a folded domain (Maiti et al., 2019) or molten globule (Patel et al., 2014)]. A simple reason for higher R2 values at lower temperatures is the higher water viscosity, resulting in a slowdown in molecular tumbling; a similar effect is achieved by adding glycerol. In some cases, R2 was measured at both low and room temperatures (Abyzov et al., 2016; Martin et al., 2020). To a good approximation, the effect of lowering temperature is a uniform scaling of R2 across the IDP sequence. For Sev-NT, downscaling of the R2 values at 278 K by a factor of 2.0 brings them into close agreement with those at 298 K (Figure 3—figure supplement 1C), with a root-mean-square-deviation (RMSD) of 0.5 s^–1 among all the residues. Likewise, for A1-LCD, downscaling by a factor of 2.4 brings the R2 values at 288 K into good match with those at 298 K (Figure 3—figure supplement 1D), with an RMSD of 0.4 s^–1. Because SeqDYN is concerned with the sequence dependence of R2, a uniform scaling has no effect on model parameter or prediction; therefore mixing the data from different temperatures is justified. The same can be said about the different magnetic fields in acquiring the R2 data (Table 1 and Figure 3—figure supplement 1E). Increasing the magnetic field raises R2 values, and the effect is also approximated well by a uniform scaling (Abyzov et al., 2016; Conicella et al., 2016; Janke et al., 2018).

One measure on the level of sequence dependence of R2 is the standard deviation, σR2, calculated among the residues of an IDP. Among the training set, the R2 values of 30 IDPs have moderate sequence variations, with σR2 ranging from 0.5 to 1.5 s^–1 (Table 1); the histogram of σR2 calculated for the entire training set peaks around 0.75 s^–1 (Figure 3A). There is a moderate correlation between σR2 and R¯2 (Figure 3A, inset), reflecting in part the fact that σR2 can be raised simply by a uniform upscaling, for example as a result of lowering temperature. Still, only two of the five IDPs with high R¯2 attributable to lower temperature or presence of glycerol are among the seven IDPs with high sequence variations (σR2>2 s^–1). Therefore, the sequence variation of R2 as captured by σR2 manifests mostly the intrinsic effect of the IDP sequence, not the influence of external factors such as temperature or magnetic field strength. The mean σR2 value among the training set is 1.24 s^–1.

One way to eliminate the influence of external factors is to scale the R2 values of each IDP by its R¯2; we refer to the results as scaled R2, or sR2. We then pooled the sR2 values for all residues in the training set, and separated them according to amino-acid types. The amino acid type-specific mean sR2 values, or msR2, are displayed in Figure 3B. The seven amino acids with the highest msR2 in descending order are Trp, Arg, Tyr, Phe, Ile, His, and Leu. The presence of all the four aromatic amino acids in this “high-end” group immediately suggests π-π stacking as important for raising msR2; the presence of Arg further implicates cation-π interactions. In the other extreme, the seven amino acids with the lowest msR2 in ascending order are Gly, Cys, Val, Asp, Ser, Thr, and Asn. Gly is well-known as a flexible residue; it is also interesting that all the four amino acids with short polar sidechains are found in this “low-end” group. Pro has an excessively low msR2 [with data from only two IDPs (Murrali et al., 2018; Wong et al., 2020)], but that is due to the absence of an amide proton.

The null model is to assume a uniform R2 for all the residues in an IDP. The root-mean-square-error (RMSE) of the null model is equal to the standard deviation, σR2, of the measured R2 values. The mean RMSE, RMSE¯, of the null model, equal to 1.24 s^–1 for the training set, serves as the upper bound for evaluating the errors of R2 predictors. The next improvement is a one-residue predictor, where first each residue (with index n) assumes its amino acid-specific mean sR2 (msR2) and then a uniform scaling factor Υ is applied:(1)R2n=Υ∙msR2n

This one-residue model does only minutely better than the null model, with a RMSE¯ of 1.22 s^–1.

In SeqDYN, we account for the influence of neighboring residues. Specifically, each residue i contributes a factor fi;n to the R2 value of residue n. Therefore,(2a)R2(n)=Υ∏i=1Nf(i;n)

where N is the total number of residues in the IDP. The contributing factor depends on the sequence distance s=|i−n| and the amino-acid type of residue i:(2b)f(i;n)=1+q(i)−11+bs2

There are 21 global parameters. The first 20 are the q values, one for each of the 20 types of amino acids; the last parameter is b, appearing in the Lorentzian form of the sequence-distance dependence. We define the correlation length, Lcorr, as the sequence distance at which the contributing factor is midway between the values at s=0 and ∞. It is easy to verify that Lcorr=b-1/2. Note that the single-residue model can be seen as a special case of SeqDYN, with Lcorr set to 0 and q set to msR2.

The functional forms of Equation 2a and Equation 2b were adapted from Li et al., 2020; we also used them for predicting residue-specific membrane association propensities of IDPs (Qin et al., 2022). In these previous applications, a linear term was also present in the denominator of Equation 2b. In our initial training of SeqDYN, the coefficient of the linear term always converged to near zero. We thus eliminated the linear term. In addition to the Lorentzian form, we also tested a Gaussian form for the sequence-distance dependence and found somewhat worse performance. The more gradual attenuation of the Lorentzian form with increasing sequence distance evidently provides an overall better model for the R2 data in the entire training set. Others (Cho et al., 2007; Sekiyama et al., 2022; Delaforge et al., 2018) have modeled R2 as the average of some parameters over a window; a window has an extremely abrupt sequence-distance dependence (1 for s<Lcorr and 0 for s>Lcorr).

We parametrized the SeqDYN model represented by Equation 2a and Equation 2b on the training set of 45 IDPs. In addition to the 21 global parameters noted above, there are also 45 local parameters, namely one uniform scaling factor (Υ) per IDP. The parameter values were selected to minimize the sum of the mean-square-errors for the IDPs in the training set, calculated on R2 data for a total of 3924 residues. We excluded the 42 Pro residues in the training set because, as already noted, their R2 values are lower for chemical reasons. We will present validation and test results below, but first let us look at the parameter values.

The q values are displayed in Figure 3B alongside msR2. In descending order, the seven amino acids with the highest q values are Trp, Ile, Tyr, Arg, His, Phe, and Leu. These are exactly the same amino acids in the high-end group for msR2, though their order there is somewhat different. In ascending order, the seven amino acids (excluding Pro) with the lowest q values are Gly, Asn, Ser, Asp, Val, Thr, and Cys. The composition of the low-end group is also identical to that for msR2. The q values thus also suggest that π-π and cation-π interactions in local sequences may raise R2, whereas Gly and short-polar residues may lower R2.

Given the common amino acids at both the high and low ends for msR2 and q, it is not surprising that these two properties exhibit a strong correlation, with a coefficient of determination (R²; excluding Pro) at 0.92 (Figure 4A). Also, because the high-end group contains the largest amino acids (e.g. Trp and Tyr) whereas the low-end group contains the smallest amino acids (e.g. Gly and Ser), we anticipated some correlation of msR2 and q with amino-acid size. We measure the latter property by the molecular mass (m). As shown in Figure 4B, both msR2 and q indeed show a medium correlation with m, with R²=0.67 (excluding Pro) and 0.61, respectively. A bulkiness parameter was proposed as an indicator of sequence-dependent backbone dynamics of IDPs (Cho et al., 2007; Delaforge et al., 2018). Bulkiness was defined as the sidechain volume-to-length ratio, and identified amino acids with aromatic or branched aliphatic sidechains as bulky (Zimmerman et al., 1968). We found only modest correlations between either msR2 or q and bulkiness, with R² just below 0.4 (Figure 4C).

The optimized value of b is 3.164 × 10^–2, corresponding to an Lcorr of 5.6 residues. The resulting optimized RMSE¯ is 0.95 s^–1, a clear improvement over the value 1.24 s^–1 of the null model. To check the sensitivity of prediction accuracy to b, we set b to values corresponding to Lcorr = 0, 1, 2,…, and retrained SeqDYN for b fixed at each value (Figure 4D). Note that the null-model RMSE¯, 1.24 s^–1, sets an upper bound. This upper bound is slowly reached when Lcorr is increased from the optimal value. In the opposite direction, when Lcorr is decreased from the optimal value, RMSE¯ rises quickly, reaching 1.22 s^–1 at Lcorr = 0. The latter RMSE¯ is the same as that of the single-residue model. Lastly we note that there is a strong correlation between the uniform scaling factors and R¯2 values among the 45 IDPs (R²=0.77), as to be expected. For 39 of the 45 IDPs, Υ values fall in the range of 0.8–2.0 s^–1.

As presented next, we evaluate the performance of SeqDYN by leave-one-out cross validation, where each IDP in turn was left out of the training set and the model was trained on the remaining 44 IDPs to predict R2 for the IDP that was left out. The parameters from the leave-one-out (also known as jackknife) training sessions allow us to assess the potential bias of the training set. For this purpose, we compare the values of the 21 global parameters, either from the full training set or from taking the averages of the jackknife training sessions. For each of the q parameters, the values from these two methods differ only in the fourth digit; for example for Leu, they are both 1.1447 from full training and from jackknife training. The values for b are 3.164×10^–2 from full training as stated above and 3.163×10^–2 from jackknife training. The close agreement in parameter values between full training and jackknife training suggests no significant bias in the training set.

Another question of interest is whether the difference between the q parameters of two amino acids is statistically significant. To answer this question, we carried out fivefold cross-validation training, resulting in five independent estimates for each parameter. For example, the mean ±standard deviation of the q parameter is 1.1405 ± 0.0066 for Leu and 1.2174 ± 0.0211 for Ile. A t-test shows that their difference is extremely statistically significant (P<0.0001). In contrast, the difference between Leu and Phe (q=1.1552 ± 0.0304) is not significant. t-test results for other pairs of amino acids are found in Figure 4—figure supplement 1.

We now present leave-one-out cross-validation results. We denote the RMSE of the R2 prediction for the left-out IDP as RMSE(–1). As expected, RMSE(–1) is higher than the RMSE obtained with the IDP kept in the training set, but the increases are generally slight. Specifically, all but eight of the IDPs have increases <0.1 s^–1; the largest increase is 0.35 s^–1, for CBP-ID4. The mean RMSE(–1), or RMSE¯(−1), for the 45 IDPs is increased by 0.05 s^–1 over RMSE¯, to 1.00 s^–1. The latter value is still a distinct improvement over the mean RMSE 1.24 s^–1 of the null model. The histogram of RMSE(–1) for the 45 IDPs is shown in Figure 5A. It peaks at 0.5 s^–1, which is a substantial downshift from the corresponding peak at 0.75 s^–1 for σR2 (Figure 3A). Thirty-four of the 45 IDPs have RMSE(–1) values lower than the corresponding σR2.

To further illustrate the performance of SeqDYN, we present the comparison of predicted and measured R2 values for five IDPs: MKK4, α-synuclein, Mev-P_NTD, Sev-NT, and CBP-ID4 (Figure 5B–F). A simple common feature is the falloff of R2 at the N- and C-termini, resulting from missing upstream or downstream residues that otherwise would be coupled to the terminal residues, as first recognized by Schwalbe et al., 1997. Representative conformations of the five IDPs are displayed in Figure 2, with residues colored according to the predicted R2 values. For four of these IDPs, the RMSE(–1) values range from 0.44 to 0.76 s^–1 and are scattered around the peak of the histogram, while the RMSE(–1) for the fifth IDP, namely CBP-ID4, the RMSE(–1) value is 2.01 s^–1 and falls on the tail of the histogram (Figure 5A). Figure 5B displays the measured and predicted R2 for MKK4. SeqDYN correctly predicts higher R2 values in the second half of the sequence than in the first half. It even correctly predicts the peak around residue Arg75. The sequence in this region is H₇₂IERLRTH₇₉; six of these eight residues belong to the high-end group. In contrast, the lowest R2 values occur in the sequence S₇GGGGSGGGSGSG₁₉, comprising entirely of two amino acids in the low-end group.

R2 values for α-synuclein are shown in Figure 5C. Here, SeqDYN correctly predicts higher R2 near the C-terminus and a dip around Gly68. However, it misses the R2 peaks around Tyr39 and Asp121. MD simulations Dey et al., 2022 have found that these R2 peaks can be explained by a combination of secondary structure formation (β-sheet around Tyr39 and polyproline II helix around Asp121) and local (between Tyr39 and Ser42) or long-range (between Asp121 and Lys96) interactions. SeqDYN cannot account for long-range interactions (e.g. between β-strands and between Asp121 and Lys96). Figure 5D shows that SeqDYN gives excellent R2 predictions for Mev-P_NTD. It correctly predicts the high peaks around Arg17, Glu31, Leu193, and lower peaks around Arg235 and Trp285, but does underpredict the narrow peak around Tyr113.

The overall R2 profile of Sev-NT is predicted well by SeqDYN, but the peak in the long helical region (residues 478–491) is severely underestimated (green curve in Figure 5E). A similar situation occurs for CBP-ID4, where the peak in the second long helical region (around Glu113) is underpredicted (green curve in Figure 5F). While the measured R2 exhibits a higher peak in the second helical region than in the first helical region (around Arg16), the opposite is predicted by SeqDYN. When the R2 data were included in the training set (i.e., full training), the second peak is higher than the first one, but that is not a real prediction because the R2 data themselves were used for training the model. It merely means that the SeqDYN functions can be parameterized to produce any prescribed R2 profile along the sequence. Indeed, when the R2 data of CBP-ID4 alone were used to parameterize SeqDYN, the measured R2 profile is closely reproduced (Figure 5—figure supplement 1). The reversal in R2 peak heights between the two helical regions is the reason for the aforementioned unusual increase in RMSE when CBP-ID4 was left out of the training set.

It is apparent that SeqDYN underestimates the R2 of stable long helices. Transient short helices does not seem to be a problem, since these are present, for example in Mev-P_NTD, where transient helix formation in the first 37 residues and between residues 189–198 (Milles et al., 2018) coincides with R2 peaks that are correctly predicted by SeqDYN. SeqDYN can treat coupling between residues within the correlation length of 5.6 residues, but a much longer helix would tumble more slowly than implied by an Lcorr of 5.6, and thus it makes sense that SeqDYN would underestimate R2 in that case.

Our solution then is to apply a boost factor to the long helical region. To do so, we have to know whether an IDP does form long helices and if so what the constituent residues are. Secondary structure predictors tend to overpredict α-helices and β-strands for IDPs, as they are trained on structured proteins. One way to counter that tendency is to make the criteria for α-helices and β-strands stricter. We found that, by filtering PsiPred (http://bioinf.cs.ucl.ac.uk/psipred; McGuffin et al., 2000) helix propensity scores (pHlx) with a very high cutoff of 0.99, the surviving helix predictions usually correspond well with residues identified by NMR as having high helix propensities. For example, for Mev-P_NTD, PsiPred plus filtering predicts residues 14–17, 28–33, and 191–193 as helical; all of them are in regions that form transient helices according to chemical shifts (Milles et al., 2018). Likewise long helices are also correctly predicted for Sev-NT (residues 477–489) and CBP-ID4 (residues 6–17 and 105–116; Jensen et al., 2008; Piai et al., 2016).

We apply a boost factor, BHlx, to helices with a threshold length of 12:(3)BHlx=1+αpHlxΘ(pHlx≥0.99;LHlx≥12)

The Θ function is 1 if the helix propensity score is above the filtering cutoff and the helix length (LHlx) is above the threshold, and 0 otherwise With a boost amplitude α at 0.5, the boosted SeqDYN prediction for Sev-NT reaches excellent agreement with the measured R2 (Figure 5E, red curve). The RMSE(–1) is reduced from 0.76 s^–1 to 0.38 s^–1 upon boosting. Applying the same helix boost to CBP-ID4 also results in a modest reduction in RMSE(–1), from 2.01 to 1.90 s^–1 (Figure 5F, red curve). The only other IDP for which PsiPred plus filtering predicts a long helix is the N-terminal region of lysyl-tRNA synthetase (KRS-NT). The authors who studied this protein did not report on secondary structure (Cho et al., 2014), but feeding their reported chemical shifts to the TALOS +server (https://spin.niddk.nih.gov/bax/nmrserver/talos/; Shen et al., 2009) found only short stretches of residues that fall into the helical region of the Ramachandran map. The SeqDYN prediction for KRS-NT is already good [RMSE(–1)=0.83 s^–1]; applying a helix boost would deteriorate the RMSE(−1) to 1.16 s^–1.

We have reserved nine IDPs for testing SeqDYN (parameterized on the training set of 45 IDPs). The level of disorder in these test proteins also spans the full range, from absence of secondary structures [ChiZ N-terminal region (Hicks et al., 2020), Pdx1 C-terminal region (Cook et al., 2019), and TIA-1 prion-like domain (Sekiyama et al., 2022)] to presence of transient short helices [synaptobrevin-2 (Lakomek et al., 2019), α-endosulfine (Thapa et al., 2022), YAP (Feichtinger et al., 2022), angiomotin-like 1 (AMOTL1) (Vogel et al., 2022)] to formation of stable long helices [FtsQ Smrt et al., 2023 and CAHS-8 Malki et al., 2022]. For eight of the nine test IDPs, the RMSEs of SeqDYN predictions are lower than the experimental σR2 values, by an average of 0.66 s^–1. For the ninth IDP (Pdx1), the SeqDYN RMSE is slightly higher, by 0.06 s^–1, than the experimental σR2. Together, the nine test IDPs have a mean RMSE of 1.13 s^–1, close to the RMSE¯(−1) of 1.00 s^–1 for the training set in the leave-one-out cross-validation.

The comparison of predicted and measured R2 profiles along the sequence is presented in Figure 6A–I. For ChiZ, SeqDYN correctly predicts the major peak around Arg25 and the minor peak around Arg46 (Figure 6A). The R2 profile of Pdx1 is largely featureless, except for a dip around Gly216, which is correctly predicted by SeqDYN (Figure 6B). Correct prediction is also obtained for the higher R2 in the first half of TIA-1 prion-like domain than in the second half (Figure 6C). SeqDYN gives an excellent prediction for synaptobrevin-2, including a linear increase up to Arg56 and the major peak around Trp89 (Figure 6D).

The prediction is also very good for α-endosulfine, including elevated R2 around Glu34, which coincides with the presence of a transient helix, and depressed R2 in the last 40 residues (Figure 6E). The only miss is an underprediction for the peak around Lys74. SeqDYN also predicts well the overall shape of the R2 profile for YAP, including peaks around Asn70, Leu91, Arg124, and Arg161, but severely underestimates the peak height around Asn70 (Figure 6F). NOE signals indicate contacts between Met86, Leu91, Fhe95, and Fhe96 (Feichtinger et al., 2022); evidently this type of local contacts is captured well by SeqDYN. The R2 elevation around Asn70 is mostly due to helix formation: residues 61–74 have helix propensities up to 40% (Feichtinger et al., 2022). PsiPred predicts helix for residues 62–73, but only residues 65–68 survive the filtering that we impose, resulting in a helix that is too short to apply a helix boost. The prediction for AMOTL1 is mostly satisfactory, including peaks around Phe200 and Arg264 and a significant dip around Gly292 (Figure 6G). However, whereas the two peaks have approximately equal heights in the measured R2 profile, the predicted peak height around Phe200 is too low. SCSs indicate helix propensity around both R2 peaks (Vogel et al., 2022). PsiPred also predicts helix in both regions, but only five and two residues, respectively, survive after filtering, and are too short for applying a helix boost.

For FtsQ, SeqDYN correctly predicts elevated R2 for the long helix [residues 46–74 Smrt et al., 2023] but underestimates the magnitude (RMSE = 2.32 s^–1; green curve in Figure 6H). PsiPred plus filtering predicts a long helix formed by residues 47–73. Applying the helix boost substantially improves the agreement with the measured R2, with RMSE reducing to 1.71 s^–1 (red curve in Figure 6H). SeqDYN also gives a qualitatively correct R2 profile for CAHS-8, with higher R2 for the middle section (residues 95–190; RMSE = 2.36 s^–1; green curve in Figure 6I). However, it misses the extra elevation in R2 for the first half of the middle section (residues 95–145). According to SCS, the first and second halves have helix propensities of 60% and 30%, respectively (Malki et al., 2022). PsiPred plus filtering predicts helices for residues 96–121, 124–141, 169–173, and 179–189. Only the first two helices, both in the first half of the middle section, are considered long according to our threshold. Once again, applying the helix boost leads to marked improvement in the predicted in R2, with RMSE reducing to 1.92 s^–1 (red curve in Figure 6I).

SeqDYN is trained on IDPs, what if we feed it with the sequence of a structured protein? The prediction using the sequence of hen egg white lysozyme, a well-studied single-domain protein, is displayed in Figure 7A. It shows remarkable agreement with the R2 profile measured by Klein-Klein-Seetharaman et al., 2002 in the unfolded state (denatured by 8 M urea at pH 2 and reduced to break disulfide bridges), including a major peak around Trp62, a second peak around Trp111, and a third peak around Trp123. Klein-Seetharaman et al. mutated Trp62 to Gly and the major peak all but disappeared. This result is also precisely predicted by SeqDYN with the mutant sequence (Figure 7B).

SeqDYN also predicts well the R2 profiles of other proteins in the unfolded state. For unfolded apomyoglobin (8 M urea; pH 2.3), Schwarzinger et al., 2002 claimed that depressed R2 corresponded to stretches of small amino acids (Gly and Ala), whereas elevated corresponded to local hydrophobic interactions. SeqDYN reproduces all the observed peaks and valleys in the R2 profile (Figure 7C). The deepest valley indeed occurs over a Gly/Ala-rich stretch, G₁₂₅ADAQGA₁₃₁, but the highest peak occurs over a stretch, I₁₀₂KYLEFI₁₀₈, that contains both hydrophobic and charged residues, all of which are on the high end of the q parameters (Figure 3B). The R2 profile of unfolded ubiquitin (8 M urea; pH 2) is relatively flat, which Wirmer et al., 2006 attributed to lack of residual secondary structure, based on the assumption that β-sheets (major elements of folded ubiquitin) are less resistant to denaturation than α-helices. SeqDYN predicts a relatively flat R2 profile (Figure 7D), but the reason is that the ubiquitin sequence lacks a contiguous stretch of high-q amino acids.

Starting from six nonhomologous IDPs in our previous MD study (Dey et al., 2022), we obtained R2 data for eight IDPs from the Bimolecular Magnetic Resonance Data Bank (BMRB; https://bmrb.io); data for two other IDPs were from our collaborators (Hicks et al., 2020; Smrt et al., 2023). Most of the 54 IDPs studied here were from searching the literature. Disorder was judged by dispersion in backbone amide proton chemical shifts, NOE, and SCS. R2 data that were not available from the authors or BMRB were obtained by digitizing R2 plots presented in figures of published papers, using WebPlotDigitizer (https://automeris.io/WebPlotDigitizer; Rohatgi, 2022) and further inspected visually.

Homology of IDPs was checked by sequence alignment using Clustal W (http://www.clustal.org/clustal2; Larkin et al., 2007), and presented as a clock-like tree using the ‘ape’ package (http://ape-package.ird.fr; Paradis et al., 2004). IDPs that had discernible homology with the selected training set were removed. Removed IDPs included HOX-SCR and β-synuclein from our previous MD study (Dey et al., 2022), due to homology with HOX-DFD and α-synuclein, respectively.

The training of SeqDYN was coded in python, similar to our previous work for predicting residue-specific membrane association propensities (ReSMAP; https://zhougroup-uic.github.io/ReSMAPidp/; Qin et al., 2022). The cost function was the sum of mean-squared-errors for the IDPs in the training set. We used the least_squares function in scipy.optimize, with Trust Region Reflective as the minimization algorithm and all parameters restricted to the positive range. For the web server (https://zhougroup-uic.github.io/SeqDYNidp/; Qin and Zhou, 2024), we rewrote the prediction code javascript.