What happens to hydrophobic “bonds†as the temperature increases?
PLoS Comput Biol. 2015 May; 11(5): e1004277.
The Hydrophobic Temperature Dependence of Amino Acids Directly Calculated from Poly peptide Structures
Erik van Dijk
Computer Scientific discipline Department, Centre for Integrative Bioinformatics (IBIVU), VU University, Amsterdam, Netherlands,
Arlo Hoogeveen
Information science Section, Centre for Integrative Bioinformatics (IBIVU), VU Academy, Amsterdam, Netherlands,
Sanne Abeln
Computer Scientific discipline Department, Centre for Integrative Bioinformatics (IBIVU), VU University, Amsterdam, Netherlands,
Helmut Grubmüller, Editor
Received 2014 Oct 3; Accepted 2015 Apr 12.
- Supplementary Materials
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S1 Information: Data file containing counts. Counts of the different parameters, for each PDB-structure, in a tab-separated format.
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S1 Text: Description raw data, contained in S1 Data. (PDF)
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S2 Text: Lodge of magnitude estimation for temperature dependence of protein stability. (PDF)
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S1 Fig: Surface based free energy estimates for classes of amino acids reference corrected. Points testify the free energy estimates for each temperature bin, lines are fitted with a parabola, consistent with the potentials plant in [ten].
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S2 Fig: Surface based gratis free energy estimates for all amino acids reference corrected. Points bear witness the gratuitous energy estimates for each temperature bin, lines are fitted with a parabola, consistent with the potentials found in [10].
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S3 Fig: Surface based free energy estimates for all amino acids non corrected. Points show the free energy estimates for each temperature bin, lines are fitted with a parabola, consistent with the potentials plant in [10].
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S4 Fig: Contact based gratis energy estimates for classes of amino acids reference corrected. Points show the costless energy estimates for each temperature bin, lines are fitted with a parabola, consistent with the potentials found in [10].
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S5 Fig: Contact based free energy estimates for all amino acids reference corrected. Points show the costless energy estimates for each temperature bin, lines are fitted with a parabola, consistent with the potentials found in [10].
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S6 Fig: Contact based complimentary energy estimates for all amino acids not corrected. Points bear witness the free free energy estimates for each temperature bin, lines are fitted with a parabola, consistent with the potentials found in [10].
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S7 Fig: Area based free free energy estimates for classes of amino acids reference corrected. Points show the free energy estimates for each temperature bin, lines are fitted with a parabola, consistent with the potentials institute in [ten].
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S8 Fig: Area based gratis energy estimates for classes of amino acids not corrected. Points show the free energy estimates for each temperature bin, lines are fitted with a parabola, consistent with the potentials found in [10].
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S9 Fig: Area based gratuitous energy estimates for all amino acids reference corrected. Points show the gratuitous energy estimates for each temperature bin, lines are fitted with a parabola, consistent with the potentials found in [10].
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S10 Fig: Area based gratuitous energy estimates for all amino acids non corrected. Points evidence the gratis energy estimates for each temperature bin, lines are fitted with a parabola, consequent with the potentials institute in [10].
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S11 Fig: Scaled surface area based free energy estimates for classes of amino acids reference corrected. Points testify the complimentary energy estimates for each temperature bin, lines are fitted with a parabola, consistent with the potentials plant in [x].
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S12 Fig: Scaled expanse based free free energy estimates for classes of amino acids not corrected. Points show the free energy estimates for each temperature bin, lines are fitted with a parabola, consistent with the potentials constitute in [ten].
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S13 Fig: Scaled expanse based free energy estimates for all amino acids reference corrected. Points show the costless energy estimates for each temperature bin, lines are fitted with a parabola, consistent with the potentials constitute in [10].
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S14 Fig: Scaled expanse based free free energy estimates for all amino acids not corrected. Points show the free energy estimates for each temperature bin, lines are fitted with a parabola, consistent with the potentials constitute in [10].
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S15 Fig: Quadratic fits of the temperature dependence of LCW-theory for various sizes. The colored, dashed lines testify theoretical predictions based on adding from LCW-theory [10, 42]. The grey, solid lines, show a quadratic fit to these theoretical predictions.
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- Data Availability Argument
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Data are available at http://www.few.vu.nl/~abeln/hydrophobicT/
Abstract
The hydrophobic result is the principal driving force in poly peptide folding. Ane can estimate the relative strength of this hydrophobic effect for each amino acid past mining a large prepare of experimentally adamant protein structures. However, the hydrophobic force is known to be strongly temperature dependent. This temperature dependence is idea to explain the denaturation of proteins at low temperatures. Here we investigate if it is possible to extract this temperature dependence directly from a large set of protein structures determined at dissimilar temperatures. Using NMR structures filtered for sequence identity, we were able to excerpt hydrophobicity propensities for all amino acids at five different temperature ranges (spanning 265-340 K). These propensities prove that the hydrophobicity becomes weaker at lower temperatures, in line with current theory. Alternatively, 1 can conclude that the temperature dependence of the hydrophobic consequence has a measurable influence on protein structures. Moreover, this piece of work provides a method for probing the private temperature dependence of the different amino acid types, which is hard to obtain by direct experiment.
Author Summary
In general, proteins go functional once they fold into a specific globular structure. On folding, hydrophobic amino acids go buried inside the protein such that they are shielded from the h2o; this hydrophobic effect makes a protein fold stable. However, the forcefulness of the hydrophobicity is known to exist strongly temperature dependent, leading for example to lower stability at lower temperatures (common cold denaturation). Nevertheless, information technology is difficult to quantify the temperature dependence for hydrophobic amino acids. Here we are able to estimate the strength of the hydrophobic effect, by analysing the positions of a large number of amino acids from poly peptide structures experimentally determined at different temperatures. For each amino acid blazon, we apply the ratio between the number of residues at the inside and at the surface of the folded structures every bit a measure for its hydrophobicity. This approach shows that the hydrophobic upshot becomes weaker at lower temperatures, equally expected from theoretical predictions. Agreement the temperature dependence for amino acids, can assistance to make proteins (or enzymes) stable at a specific temperature range. For instance, the design of enzymes that are stable and functional at depression temperatures may benefit from this work.
Introduction
When a protein folds, hydrophobic amino acids get buried inside the protein to form a hydrophobic core. Within this core the hydrophobic side chains are shielded from the water. The tendency of hydrophobic groups to cluster together when they are put into water—or the hydrophobic effect—is the virtually important driving force in protein folding. Notation that there are several factors that contribute to the overall stability of a folded protein: for example the formation of hydrogen bonds between backbone atoms (secondary structure) and side bondage; the formation of salt bridges between charged amino acids and the burial of hydrophobic side bondage upon folding. It is thought that this hydrophobic strength gives the single largest contribution to the stability of near protein folds [1]. Moreover, the positioning of hydrophobic clusters in the sequence may impact the folding pathway and dynamics e.g. [2, 3]. Note that these stabilizing forces are partially compensated past the decrease in chain entropy upon folding.
Hydrophobicity is a consequence of the collective behaviour of the h2o molecules and 'oily' groups. In essence the water-hydrophobe interface is unfavourable compared to water-water or hydrophobic-hydrophobic interactions. The free energy difference upon burial of hydrophobic groups is partially entropic and partially enthalpic, causing a singled-out temperature dependence [4, 5]. Even though the verbal molecular cause for these enthalpic and entropic contributions is the focus of agile research [six, 7] and can change depending on the type of protein [vii], the resultant temperature dependence can be measured experimentally for several different non-polar substances [eight, 9]. From such measurements, models and theory we know that the hydrophobic force peaks between 30–80°C and becomes weaker at both lower and higher temperatures, see Fig 1A.
Length scale dependence of hydrophobic effect from calculations by Huang and Chandler [x] (A).
The cost of making a cavity in the water with a radius of the given size against temperature is plotted. The position of the maximum depends on the size (radius) of the solute. Pocket-size solutes with a radius of 4 Å have a peak at around 70°C, whereas larger particles with a radius of ten Å accept a top around 40°C. An example poly peptide structure: PDB-ID: 2K5I (B). We estimate free energies of transfer from the hydrophobic core to the surface of the poly peptide past comparing the number of hydrophobic amino acids on the surface (small yellow spheres), to the number of cached hydrophobics (big yellow spheres), to the number of polar amino acids on the surface (small bluish spheres) and to the number of buried polar amino acids (big blueish spheres).
Since hydrophobicity is such a large contributor to protein stability, the temperature dependence of the hydrophobic effect has of import consequences. Firstly, some proteins do non only unfold at loftier temperatures, equally can be explained through the entropy of the chain, but also at low temperatures (common cold denaturation) [11]. This effect is idea to exist a event of hydrophobicity becoming weaker at low temperatures [12]. Secondly, alternating states of intrinsically disordered proteins may become more than favourable at different temperatures due to this upshot [13]. Thirdly, poly peptide-protein and protein-substrate interactions—if dominated by hydrophobic interactions—may likewise be sensitive to temperature changes.
It is essential to quantify the temperature dependence if one wants to model and predict the stability of folded proteins and protein interactions over a large range of temperatures. For industrial purposes, proteins or enzymes that can be agile over a wide temperature range are of crucial importance. To achieve this, proteins from species that live at extreme temperatures, thermophiles and psychrophiles, have been used and adapted extensively for biocatalysis [14, 15]. Agreement and quantifying the hydrophobic temperature dependence for specific amino acids is essential if ane wants to predict thermostability of proteins.
Earlier, Folch et al. [16] showed that temperature dependent pairwise potentials for amino acids tin can help to predict the melting temperature of homologous pairs of proteins. More than recently, this study was extended to as well predict stability at low temperatures [17]. In this work nosotros focus on the temperature dependence of the effective interactions between hydrophobic amino acids and water.
Even though this temperature dependence has of import consequences, information technology is often non considered due to practical concerns. The temperature dependence is typically not included in interaction potentials for poly peptide construction prediction or fibroid grained simulations; such potentials do not model the water molecules explicitly or in enough detail to capture this consequence. It is hard to measure the temperature dependence for specific amino acids by experiments, nether physically relevant conditions. In other words, it is hard to measure the difference in free energy between the folded and unfolded chain for separate amino acids. In this piece of work we bear witness that it is possible to obtain this temperature dependence for specific amino acids by mining a large set of protein structures resolved by Nuclear Magnetic Resonance (NMR).
Physically or chemically relevant quantities can be obtained past averaging over a large set of structures. For example, specific bond lengths, the virtually favourable dihedral angles or estimate hydrophobicities for different amino acrid types can exist obtained past taking an ensemble boilerplate over a set of protein structures. More specifically, hydrophobicity scales for the dissimilar amino acrid types may exist obtained using physicochemical properties [18], or past computing how often we notice each residue type exposed to the solvent at the surface of a protein [18–21]. Different approaches give slightly different results—and a somewhat unlike ranking between the residues—but do agree overall. Hydrophobicity scales are useful for a broad range of problems involving construction prediction: from predicting the severity of a mutation to disorder prediction and full structure prediction e.g. [22–27].
Estimates for pairwise free energies between amino acrid types have been obtained by mining protein structures. A pairwise interaction potential may be calculated by counting the number of contacts made betwixt different types of amino acids [16, 28, 29]. More recently, this method has been farther developed to allow the extraction of interactions between the solvent and the different types of residues, likewise as the pairwise interactions [thirty]. Knowledge-based amino acid pair-potentials are used in construction prediction [31], coarse-grained protein simulations [32–35] and protein-poly peptide docking methods [36]. Recently, a cognition based amino acid pair potential with a temperature dependence has also been used to predict the thermostability of proteins [17].
In this piece of work, we estimate the hydrophobic result as the gratis energy cost for transferring a hydrophobic amino acrid from the core of the poly peptide to the water exposed surface, see Fig 1B. We employ three singled-out approaches to estimate these transfer free energies. Firstly, we apply a previously validated approach to derive a statistical pair potential between amino acids to extract free energy estimates for the hydrophobic interaction. This contact based method has been shown to yield hydrophobicity estimates that give physically realistic results upon simulation. Secondly, nosotros use a more direct approach that calculates propensities for surface accessibility for each of the amino acids; this method is similar to other approaches that derive knowledge based hydrophobicity scales [18–twenty]. Thirdly, nosotros use an area based approach that considers the amount of exposed surface surface area per amino acid. The three approaches give similar results, and show significant temperature dependence for hydrophobic amino acids in line with expectations from theory and measurements on small hydrophobic particles.
Results/Discussion
In order to extract the hydrophobic temperature dependence from experimentally adamant protein structures, it is important to cull the set of structures carefully. Firstly, nosotros explored the contents of the Protein DataBank (PDB), [37], containing over 96k structures. Fig 2 shows the temperature distribution of bachelor protein structures determined by X-ray crystallography and nuclear magnetic resonance (NMR). For this study we merely use structures adamant by NMR, every bit these experiments tin can be performed on soluble proteins at the temperature range of involvement for the hydrophobic effect. This makes it possible to probe temperature dependent furnishings in proteins; for case a temperature induced transition [38] and common cold denaturation [11] have been observed using this technique.
Distribution of temperatures at which experimental protein structures were resolved.
All acquisition temperatures of structures as of April 2014 available in the PDB are shown. The 80,662 X-ray diffraction structures are centred around 100 K, while the ten,969 NMR structures show a pinnacle at room temperature (300 K). Annotation that the small peak of NMR information just above absolute goose egg may exist temperatures entered in celsius instead of kelvin; this information is non used in this study. Temperature bins, equally given in Tabular array 1, are indicated in different shades of grey.
In order to obtain estimates for the solvation free energies of different types of amino acids at different temperatures, we divided the data into 5 temperature bins, run into Fig ii and Tabular array 1. The bins were chosen symmetrically around the tiptop at room temperature (300 Thousand), to rest the number of structures in each bin.
Table 1
Selected protein structures.
| temperature range | chains in PDB | bondage select-25 | chains after filters |
|---|---|---|---|
| 265–290 | 1421 | 259 | 207 |
| 291–296 | 1440 | 378 | 344 |
| 297–299 | 4689 | 1095 | 1033 |
| 300–305 | 1864 | 618 | 560 |
| 306–340 | 1361 | 470 | 412 |
We prepare out to explore if nosotros can detect the temperature dependence of the hydrophobic outcome by analysing this filtered set of protein structures. Protein structures determined past NMR at different temperatures were used to obtain gratuitous energy estimates for the transfer of amino acids from the core of the protein to the surface. Nether the assumption of random mixing, the transfer free energy estimates can be estimated through statistical methods [16, 28–30]. We investigate three methods, 1) a contact based calculation which has been shown to give a reasonable attraction [30], 2) a direct adding of propensities to surface exposure 3) an area based calculation that incorporates the accessible expanse in a continuous measure of hydrophobicity, run across Methods for details.
Firstly, we investigate whether the raw free energy estimates are dependent on the temperature. To further increase the statistical accurateness, amino acids are divided into v classes: hydrophobic, charged, polar, aromatic and other, encounter Table two. Fig 3 shows a surprisingly clear temperature dependence for the different hydrophobic amino acids: at lower temperatures the hydrophobic effect becomes weaker. This is in line with expectations from experiments and theory [iv, 5]. The results for the area based potential are very like to the results of the contact based potential (come across S7–S14 Figs).
Table 2
Amino acid class definition.
| Form | Amino Acids |
|---|---|
| Hydrophobics | ALA, ILE, LEU, MET, VAL |
| Aromatics | HIS, PHE, TRP, TYR |
| Charged | ARG, ASP, GLU, LYS |
| Polar | ASN, GLN, SER, THR |
| Other | CYS, GLY, PRO |
Raw gratuitous energies of transfer for classes of amino acids.
Contact based (A) and surface based (B) free energies are shown for different classes of amino acids. Points show the free energy estimates for each temperature bin, lines are fitted with a parabola, consequent with the potentials found in [10]. Arrows point the bins used to exam the significance of the temperature dependence.
To test if this temperature dependence is indeed significant, we resampled the poly peptide structures using random temperature labels. From this procedure p-values were calculated to decide the significance of the gratuitous energy departure. Tabular array 3 shows the difference in transfer energy (ΔΔThou) and p-values between the everyman temperature bin (265–290K) and room temperature (297–299K). Clearly, the temperature tendency for the hydrophobic residues is significantly stronger than one would expect from random fluctuations. The standard error to the mean is estimated from the deviations in the potential obtained as indicated in the results by splitting the data ready into five parts and recalculating the potentials for each role.
Table 3
Significance of hydrophobic temperature dependence pooled.
| amino acid class | p-value contacts | p-value surface | ΔΔThou contacts | ΔΔThou surface |
|---|---|---|---|---|
| hydrophobic | < 0.01 | < 0.01 | 0.ten | 0.32 |
| polar | < 0.01 | 0.23 | -0.05 | 0.thirteen |
| charged | < 0.01 | 0.lxxx | -0.06 | -0.04 |
| aromatic | 0.04 | < 0.01 | 0.06 | 0.32 |
| other | 0.32 | < 0.01 | 0.02 | 0.41 |
Fig 3 besides shows that the surface based potentials give larger absolute differences in costless energies than the contact based potentials. This tin can well-nigh likely be explained by the strict cutoff (seven% accessible surface area) in the surface based potential compared to the more gradual calculation of the contact based potential; charged and polar amino acids are rarely entirely buried and give therefore a very strong point for the surface based measure out. The relative hydrophobicity, however, is consistent betwixt the 3 methods, showing our results are qualitatively independent of the method of derivation for the potential.
The results in Fig iii show a slight temperature dependence for charged (and polar) amino acids. For the surface based potential, however, this outcome is not significant (Tabular array 3).
Our transfer free energy estimates are calculated under the assumption of a random mixing model; this provides us with relative transfer free energies for each type of amino acids. This ways it is not trivial to compare the free energy differences between different temperature bins. The temperature dependence of the hydrophobic residues could cause the shift of the polar and charged amino acids. In order to enable comparison at different temperatures, we set a reference state for the free energy estimates. The reference country is an important part of the potential, and can make up one's mind the accuracy of a potential in structure validation [39].
As we are here especially interested to compare the transfer free energies between different temperatures it is desirable that our reference does not accept whatever temperature dependent interaction with the solvent. Betancourt and Thirumalai [29] and Buchete et al. [27] utilise Threonine, a small-scale water-similar polar amino acid, as a reference in the calculation for their amino acid pair-potential. In our example, every bit the number of structures bachelor is limited, choosing a single amino acid as reference will propagate noise through the results. Instead, we pool all the charged and hydrophilic amino acids for each temperature bin, and use those as a reference potential (see Table two). Even though information technology is known that polar and charged residues can accept a temperature dependent interaction with the solvent and that this interaction can have consequences for protein structure and stability (run into for example Refs. [xl, 41]), comparison raw estimates (Fig 3) with reference corrected estimates (S1 and S4 Figs) shows that this correction does not modify the relative trends, see Methods for further details.
Fig 4 shows estimates for the corrected transfer free energies for all hydrophobic and aromatic amino acids individually, with the polar and charged amino acids as a reference. Results for all amino acids, with and without reference correction are shown in S2, S3, S5 and S6 Figs. The hydrophobicity becomes weaker at lower temperatures, showing the results from the 'raw' estimates hold upwardly. Once again, the significance of the temperature dependence of each hydrophobic amino acrid blazon is examined. For almost all hydrophobic amino acids the gratuitous energy estimates have a significant temperature dependence (Table 4). Note that the correction to a reference of polar and charged amino acids was as well performed in the resampling procedure to obtain statistical significance
Reference corrected gratuitous energies of transfer for hydrophobic amino acids.
Contact based (A) and surface based (B) free energies are shown for hydrophobic and aromatic amino acids. The gratis energies are corrected by setting a reference of the polar and charged amino acids. Points show the costless energy estimates for each temperature bin and lines are fitted with a parabola. Arrows bespeak the bins used to test the significance of the temperature dependence.
Table 4
Significance of hydrophobic temperature dependence.
| amino acid (class) | p-value contacts | p-value surface | ΔΔG contacts | ΔΔChiliad surface |
|---|---|---|---|---|
| ALA | 0.03 | < 0.01 | 0.12 | 0.38 |
| CYS | < 0.01 | < 0.01 | 0.32 | 0.67 |
| GLY | 0.42 | 0.12 | 0.03 | 0.eleven |
| ILE | < 0.01 | < 0.01 | 0.20 | 0.19 |
| LEU | < 0.01 | < 0.01 | 0.19 | 0.32 |
| MET | 0.68 | 0.09 | 0.03 | 0.19 |
| PHE | < 0.01 | < 0.01 | 0.23 | 0.36 |
| PRO | 0.62 | < 0.01 | 0.07 | 0.46 |
| TRP | 0.23 | 0.21 | 0.xv | 0.16 |
| TYR | 0.34 | 0.08 | 0.10 | 0.nineteen |
| VAL | 0.02 | < 0.01 | 0.xv | 0.16 |
Fig iv as well shows that the estimated transfer free energies show a very like trend with respect to temperature to those that have been measured for hydrophobic particles [iv] or obtained by calculation according to LCW-theory [10, 42]. For clarity, we fitted parabolas through the estimated transfer complimentary energies, which is a reasonable approximation for trends calculated from theory and observed in experiment (see S15 Fig). Information technology can be observed that the complimentary energies for the hydrophobic amino acids prove a maximum of around 310–350 kelvin for both the surface and contact based free energy estimates; this is slightly lower than what is expected from theory (see for comparison Fig 1A)
Due to the lack of data at higher temperatures (T > 320Thousand), information technology is hard to estimate a precise maximum for the transfer free energies. Even so, an interesting trend may be observed from Fig 4. Larger amino acids, for example Tryptophan, have a maximum at lower temperatures compared to smaller amino acids such equally Alanine. Again, this trend is consistent with theory and experiments [ten], where the transfer costless free energy of larger particles shows a maximum at lower temperatures.
Overall, we tin conclude that the temperature dependence of the hydrophobic outcome has a measurable influence on protein structures adamant by NMR. The upshot nosotros discover appears to be on the right social club of magnitude in comparing with theory for the hydrophobic effect and known cold denaturating behaviour of proteins (see S2 Text). The results show that structures determined at lower temperature have more exposed hydrophobic expanse. This suggests that at these temperatures the structures already go more open up, every bit has been observed for some specific proteins (e.yard. [43]). It would be very interesting to investigate if these low temperature structures are more than flexible and dynamic than the same structures obtained at room temperature.
Conclusion
In this piece of work we set out to investigate whether the hydrophobic temperature dependence could be obtained past mining a big set of poly peptide structures resolved by NMR. We used a contact based, an area based and a surface based approach to obtain gratis free energy estimates for the transfer of an amino acid out of the hydrophobic protein core onto the water exposed surface. We find a surprisingly clear tendency for the complimentary energy estimates with respect to the temperature: the hydrophobic issue becomes weaker at lower temperatures, every bit is expected based on theory, simulations and experiments. Alternatively, one tin conclude that the temperature dependence of the hydrophobic effect has indeed a measurable influence on poly peptide structures. Despite the sparseness of the data, and the inconsistencies in reporting of experimental temperatures, we find that the observed trend holds and is significant regardless of the precise method used to gauge the transfer free energies, the specific groupings of amino acids or the chosen reference.
Methods
Data collection
The temperature (in kelvin) at which the experiment is performed tin be found in the mandatory 'conquering data' department of PDB files. Several filters were practical. Some structures were filtered out because no temperature was entered or because they were given several temperatures from multiple information collection sessions. In order to get representative statistics for amino acid composition, it is important to remove any bias in the PDB for big sequence families. To take out this redundancy we used PDB filter-select 25% [44–46]. Table 1 shows the number of remaining structures in each bin afterward these filterings. A few further PDB files had to exist removed due to their incompatibility with DSSP. After these steps, each PDB-file was divide into multiple models, and the accessible surface area was adamant using DSSP for each model. For each residue in the protein concatenation, the boilerplate attainable area over all models was used. The final counts for each PDB-construction are shown in S1 Information. The format is explained in S1 Text.
Calculation of contact based potential
To obtain estimates for the free energies of transferring specific amino acrid types from the outside of the protein to the hydrophobic core, we used 2 approaches. The starting time approach is based on contacts betwixt amino acids, and betwixt amino acids and the solvent every bit in the work of Abeln and Frenkel [30]. This potential has been shown to requite an appropriate distinction betwixt the protein cadre and surface by simulation. The 2nd approach uses the presence or absence of amino acids on the surface of the protein, providing a more straight fashion to obtain the hydrophobicty of each amino acids.
In the contact based approach, we calculate knowledge-based pair-potentials over the gear up of structures described higher up. The free energy estimates ϵ i,j between amino acid types i and j can be calculated as:
where c i,j are the number of contacts between amino acids blazon i and j, and where ω i,j is the expected number of contacts. Note that here we are specifically interested in the example where one of the interaction partners is the solvent, i.e. ϵ i,solvent.
Nosotros can calculate the expected number of contacts, ω i,j , past considering the distribution of the amino acid types i and j in the set up of protein structures:
here n i q i is the full amount of contacts for type i, where north i is the number of amino acid of type i and q i is the coordination number, which we prepare to iv for all amino acids to remain consequent with Abeln and Frenkel [30]. Note that the sum in denominator loops over all the amino acids and water (k). In exercise the full number of contacts for an amino acid blazon n i q i tin be calculated directly from the data.
The number of h2o contacts is estimated through the size of the surface accessible expanse for a residuum as calculated past DSSP [47]. Note that for the water contact points, we do not consider real water molecules, only a surface expanse similar to the size of an amino acid. Nosotros estimate the number of contacts every bit the product between q = iv and the fraction of exposed surface area α r for residue r. Hence, based on the assumption that a residual tin interact with four other residues, h2o contact points can be created. The fraction of exposed area, α r , is given by:
S r is the solvent accessible area, calculated with the DSSP program, and a(r) is the amino acid blazon of residual r; is the maximum attainable area in an unfolded concatenation for that amino acid type.
Calculation of surface based potential
An alternative mensurate for hydrophobicity can exist obtained by calculating the propensity for an amino acid to be on the surface. Classic amino acid propensities, which are for instance used to describe the analogousness for a sure secondary construction type, can be calculated through a unproblematic ratio of fractions eastward.yard. Chapter 12 of Ref. [48]. Here we employ the structural classes buried and non-cached. To decide whether a residue (r) is cached, we use a cutoff: α r < 7% [49]. We tin can calculate the propensity (P) for amino acids to be buried equally:
where P a,b stands for the propensity for an amino acrid type, a, to be buried every bit indicated by the subscript b. Translating this into counts yields:
where Northward a,b is the total number of amino acids of type a that are cached, and N a,nb is the total number of amino acids of type a that are non-cached. Similarly,
where N b is the total number of buried amino acids, and North nb is the full number of amino acids that are not buried.
When propensities are used to estimate transfer gratis energies, through ΔF a,b = −kT log(P a,b ) it has the disadvantage that:
This can exist seen by substituting the formula for P a,nb in the formula for the complimentary energy, ΔF.
Here we define our propensities in an alternative way to overcome this problem similar to Shatyan et al. [21]. If we define our culling propensities, P*, analogous to a partitioning coefficient, we obtain:
(viii)
which does have the desired property summarized in Eq 7.
Adding of surface area based potential
While the contact based potential is established, some of the assumptions are particularly useful in the context of a coarse grained lattice simulation. On the other mitt, the surface based potential uses the assumption that a residue is buried when less and then seven% of its surface is exposed. To test the robustness of our results with regards to these assumptions, we investigated ii additional potentials, based on the exposed area. The showtime one corresponds to the contact based potential, with very large (infinite) coordination numbers. This area based potential is calculated by comparison the amount of exposed surface area, S r , for an amino acrid type a to that of the average amino acid.
(9)
S r is the solvent accessible area, calculated with the DSSP program, and a(r) is the amino acid type of residuum r; is the maximum accessible surface area in an unfolded chain for that amino acid type.
A similar potential, but scaled with the maximum solvent accessible area, is also calculated. Nosotros will refer to this potential equally the scaled expanse based potential, C a,s = C a max(Due south a ). The interactions of each rest are multiplied by its maximum accessible surface expanse. The results for this potential are very similar. Big residues have a higher interaction score when compared to smaller residues. The results for this potential are shown in S11, S12, S13 and S14 Figs.
Significance of temperature dependence
The estimated error to the mean for each data point was obtained past splitting the data into five parts each containing an equal number of structures. The potential was recalculated for each of the five parts, and a standard deviation was calculated from each of them. This allows united states of america to estimate a 95% confidence interval past taking two standard errors on each side of the hateful. These are the error bars shown in the plots.
The significance of the temperature dependence of the potentials was adamant through a resampling procedure for two different temperature bins: the lowest temperature range and room temperature. We resampled our data past shuffling the temperature labels of the protein structures and recalculating the contact based and surface based potentials for a prepare of 1000 random samples. P-values for the difference in hydrophobicity between the two temperature bins were determined as the fraction of resampled free energy differences that were larger in size than the original adding.
Plumbing fixtures procedure
To obtain an estimate for the temperature dependence of the potential, we need to assign a unmarried temperature for the structures inside a temperature bin. The average temperature of the structures is taken to be the temperature of the bin. A weighted least squares fitting procedure was used to fit a parabola to the potential as a function of temperature, which is a reasonable approximation to the relation found in both theory and experiment. In a weighted to the lowest degree squares fit, the sum is minimized. Here, the i indicates the index of the temperature bin, w i is the weight, and r i is the difference between observations and the model. The number of residues of type south in bin i was used as weight.
Supporting Information
S1 Information
Data file containing counts.
Counts of the different parameters, for each PDB-structure, in a tab-separated format.
(TXT)
S1 Text
Clarification raw data, contained in S1 Data.
(PDF)
S2 Text
Guild of magnitude estimation for temperature dependence of protein stability.
(PDF)
S1 Fig
Surface based free free energy estimates for classes of amino acids reference corrected.
Points evidence the gratuitous free energy estimates for each temperature bin, lines are fitted with a parabola, consequent with the potentials found in [10].
(EPS)
S2 Fig
Surface based free energy estimates for all amino acids reference corrected.
Points testify the free energy estimates for each temperature bin, lines are fitted with a parabola, consequent with the potentials plant in [x].
(EPS)
S3 Fig
Surface based complimentary energy estimates for all amino acids non corrected.
Points testify the free energy estimates for each temperature bin, lines are fitted with a parabola, consequent with the potentials institute in [ten].
(EPS)
S4 Fig
Contact based gratuitous free energy estimates for classes of amino acids reference corrected.
Points testify the free free energy estimates for each temperature bin, lines are fitted with a parabola, consistent with the potentials found in [10].
(EPS)
S5 Fig
Contact based costless energy estimates for all amino acids reference corrected.
Points evidence the free free energy estimates for each temperature bin, lines are fitted with a parabola, consistent with the potentials found in [10].
(EPS)
S6 Fig
Contact based free free energy estimates for all amino acids not corrected.
Points show the complimentary energy estimates for each temperature bin, lines are fitted with a parabola, consistent with the potentials found in [10].
(EPS)
S7 Fig
Expanse based free energy estimates for classes of amino acids reference corrected.
Points testify the free free energy estimates for each temperature bin, lines are fitted with a parabola, consistent with the potentials found in [x].
(EPS)
S8 Fig
Area based free energy estimates for classes of amino acids not corrected.
Points bear witness the complimentary energy estimates for each temperature bin, lines are fitted with a parabola, consequent with the potentials establish in [x].
(EPS)
S9 Fig
Expanse based gratis free energy estimates for all amino acids reference corrected.
Points bear witness the free energy estimates for each temperature bin, lines are fitted with a parabola, consistent with the potentials found in [10].
(EPS)
S10 Fig
Area based free free energy estimates for all amino acids not corrected.
Points prove the free energy estimates for each temperature bin, lines are fitted with a parabola, consistent with the potentials found in [10].
(EPS)
S11 Fig
Scaled area based free energy estimates for classes of amino acids reference corrected.
Points show the free free energy estimates for each temperature bin, lines are fitted with a parabola, consistent with the potentials found in [x].
(EPS)
S12 Fig
Scaled area based gratis energy estimates for classes of amino acids non corrected.
Points bear witness the costless free energy estimates for each temperature bin, lines are fitted with a parabola, consistent with the potentials found in [10].
(EPS)
S13 Fig
Scaled area based free free energy estimates for all amino acids reference corrected.
Points evidence the energy estimates for each temperature bin, lines are fitted with a parabola, consistent with the potentials institute in [10].
(EPS)
S14 Fig
Scaled surface area based gratis free energy estimates for all amino acids not corrected.
Points show the free energy estimates for each temperature bin, lines are fitted with a parabola, consistent with the potentials found in [x].
(EPS)
S15 Fig
Quadratic fits of the temperature dependence of LCW-theory for diverse sizes.
The colored, dashed lines show theoretical predictions based on calculation from LCW-theory [10, 42]. The grayness, solid lines, evidence a quadratic fit to these theoretical predictions.
(EPS)
Acknowledgments
We would like to thank Michele Vendruscolo, Aleksandr B. Sahakyan and Massimo Sandal for helpful comments and suggestions.
Funding Statement
SA has been supported by a Veni grant on the project 'Understanding toxic protein oligomers through ensemble characteristics' from Netherlands Organisation for Scientic Research (NWO)'. The funders had no function in report blueprint, data drove and assay, decision to publish, or preparation of the manuscript.
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