Correction: Finkelstein et al. How Can Ice Emerge at 0 °C? Biomolecules 2022, 12, 981.

We regret to state that our article "How Can Ice Emerge at 0 °C?" [...].

Physics of Ice Nucleation and Antinucleation: Action of Ice-Binding Proteins.

Ice-binding proteins are crucial for the adaptation of various organisms to low temperatures. Some of these, called antifreeze proteins, are usually thought to inhibit growth and/or recrystallization of ice crystals. However, prior to these events, ice must somehow appear in the organism, either coming from outside or forming inside it through the nucleation process. Unlike most other works, our paper is focused on ice nucleation and not on the behavior of the already-nucleated ice, its growth, etc. The nucleation kinetics is studied both theoretically and experimentally. In the theoretical section, special attention is paid to surfaces that bind ice stronger than water and thus can be "ice nucleators", potent or relatively weak; but without them, ice cannot be nucleated in any way in calm water at temperatures above -30 °C. For experimental studies, we used: (i) the ice-binding protein mIBP83, which is a previously constructed mutant of a spruce budworm Choristoneura fumiferana antifreeze protein, and (ii) a hyperactive ice-binding antifreeze protein, RmAFP1, from a longhorn beetle Rhagium mordax. We have shown that RmAFP1 (but not mIBP83) definitely decreased the ice nucleation temperature of water in test tubes (where ice originates at much higher temperatures than in bulk water and thus the process is affected by some ice-nucleating surfaces) and, most importantly, that both of the studied ice-binding proteins significantly decreased the ice nucleation temperature that had been significantly raised in the presence of potent ice nucleators (CuO powder and ice-nucleating bacteria Pseudomonas syringae). Additional experiments on human cells have shown that mIBP83 is concentrated in some cell regions of the cooled cells. Thus, the ice-binding protein interacts not only with ice, but also with other sites that act or potentially may act as ice nucleators. Such ice-preventing interaction may be the crucial biological task of ice-binding proteins.

How Can Ice Emerge at 0 °C?

The classical nucleation theory shows that bulk water freezing does not occur at temperatures above ≈ -30 °C, and that at higher temperatures ice nucleation requires the presence of some ice-binding surfaces. The temperature and rate of ice nucleation depend on the size and level of complementarity between the atomic structure of these surfaces and various H-bond-rich/depleted crystal planes. In our experiments, the ice nucleation temperature was within a range from -8 °C to -15 °C for buffer and water in plastic test tubes. Upon the addition of ice-initiating substances (i.e., conventional AgI or CuO investigated here), ice appeared in a range from -3 °C to -7 °C, and in the presence of the ice-nucleating bacterium Pseudomonas syringae from -1 °C to -2 °C. The addition of an antifreeze protein inhibited the action of the tested ice-initiating agents.

Calculation of Crystal-Solution Dissociation Constants.

The calculation of dissociation constants is an important problem in molecular biophysics. For such a calculation, it is important to correctly calculate both terms of the binding free energy; that is, the enthalpy and entropy of binding. Both these terms can be computed using molecular dynamics simulations, but this approach is very computationally expensive, and entropy calculations are especially slow. We develop an alternative very fast method of calculating the binding entropy and dissociation constants. The main part of our approach is based on the evaluation of movement ranges of molecules in the bound state. Then, the range of molecular movements in the bound state (here, in molecular crystals) is used for the calculation of the binding entropies and, then (using, in addition, the experimentally measured sublimation enthalpies), the crystal-to-vapor dissociation constants. Previously, we considered the process of the reversible sublimation of small organic molecules from crystals to vapor. In this work, we extend our approach by considering the dissolution of molecules, in addition to their sublimation. Similar to the sublimation case, our method shows a good correlation with experimentally measured dissociation constants at the dissolution of crystals.

Protein folding problem: enigma, paradox, solution.

The ability of protein chains to spontaneously form their three-dimensional structures is a long-standing mystery in molecular biology. The most conceptual aspect of this mystery is how the protein chain can find its native, "working" spatial structure (which, for not too big protein chains, corresponds to the global free energy minimum) in a biologically reasonable time, without exhaustive enumeration of all possible conformations, which would take billions of years. This is the so-called "Levinthal's paradox." In this review, we discuss the key ideas and discoveries leading to the current understanding of protein folding kinetics, including folding landscapes and funnels, free energy barriers at the folding/unfolding pathways, and the solution of Levinthal's paradox. A special role here is played by the "all-or-none" phase transition occurring at protein folding and unfolding and by the point of thermodynamic (and kinetic) equilibrium between the "native" and the "unfolded" phases of the protein chain (where the theory obtains the simplest form). The modern theory provides an understanding of key features of protein folding and, in good agreement with experiments, it (i) outlines the chain length-dependent range of protein folding times, (ii) predicts the observed maximal size of "foldable" proteins and domains. Besides, it predicts the maximal size of proteins and domains that fold under solely thermodynamic (rather than kinetic) control. Complementarily, a theoretical analysis of the number of possible protein folding patterns, performed at the level of formation and assembly of secondary structures, correctly outlines the upper limit of protein folding times.

Sublimation Entropy and Dissociation Constants Prediction by Quantitative Evaluation of Molecular Mobility in Crystals.

Prediction of binding free energies (or dissociation constants) is a crucial challenge for computational biochemistry. One of the main problems here consists in fast and accurate evaluation of binding entropy, which is far more time-consuming than evaluation of binding enthalpy. Here, we offer a fast and rather accurate approach to evaluate the sublimation entropy (i.e., entropy of binding of a vapor molecule to a crystal, taken with the opposite sign) from the average range of molecular movements in the solid state. To estimate this range (and the corresponding amplitude), we considered equilibrium sublimation of small organic molecules from molecular crystals. The calculations were based on experimental data on the sublimation enthalpy, pressure of saturated vapor, and structural characteristics of the molecule in question. The resulting average amplitude (0.17 ± 0.01 Å) of molecular movements in crystals was used to predict sublimation entropies and dissociation constants for sublimation of 28 molecular crystals. The results of these predictions are in close agreement with experimental values.

There and back again: Two views on the protein folding puzzle.

The ability of protein chains to spontaneously form their spatial structures is a long-standing puzzle in molecular biology. Experimentally measured folding times of single-domain globular proteins range from microseconds to hours: the difference (10-11 orders of magnitude) is the same as that between the life span of a mosquito and the age of the universe. This review describes physical theories of rates of overcoming the free-energy barrier separating the natively folded (N) and unfolded (U) states of protein chains in both directions: "U-to-N" and "N-to-U". In the theory of protein folding rates a special role is played by the point of thermodynamic (and kinetic) equilibrium between the native and unfolded state of the chain; here, the theory obtains the simplest form. Paradoxically, a theoretical estimate of the folding time is easier to get from consideration of protein unfolding (the "N-to-U" transition) rather than folding, because it is easier to outline a good unfolding pathway of any structure than a good folding pathway that leads to the stable fold, which is yet unknown to the folding protein chain. And since the rates of direct and reverse reactions are equal at the equilibrium point (as follows from the physical "detailed balance" principle), the estimated folding time can be derived from the estimated unfolding time. Theoretical analysis of the "N-to-U" transition outlines the range of protein folding rates in a good agreement with experiment. Theoretical analysis of folding (the "U-to-N" transition), performed at the level of formation and assembly of protein secondary structures, outlines the upper limit of protein folding times (i.e., of the time of search for the most stable fold). Both theories come to essentially the same results; this is not a surprise, because they describe overcoming one and the same free-energy barrier, although the way to the top of this barrier from the side of the unfolded state is very different from the way from the side of the native state; and both theories agree with experiment. In addition, they predict the maximal size of protein domains that fold under solely thermodynamic (rather than kinetic) control and explain the observed maximal size of the "foldable" protein domains.

Reduction of the Search Space for the Folding of Proteins at the Level of Formation and Assembly of Secondary Structures: A New View on the Solution of Levinthal's Paradox.

The complete volume of the protein conformation space is, by many orders of magnitude, smaller at the level of secondary structure elements than that at the level of amino acid residues; the latter, according to Levinthal's estimate, scales approximately as 10(2 L), with L being the number of residues in the chain, whereas the former, as demonstrated in this paper, scales no faster than ∼L(N), with N being the number of the secondary structure elements, which is approximately equal to L/15. This drastic decrease in the exponent (L/15 instead of 2 L) explains why sampling of the conformation space does not contradict the ability of the protein chain to find its most stable fold.

Restrictions to protein folding determined by the protein size.

Experimentally measured rates of spontaneous folding of single-domain globular proteins range from microseconds to hours: the difference (11 orders of magnitude!) is akin to the difference between the life span of a mosquito and the age of the Universe. We show that physical theory with biological constraints outlines the possible range of folding rates for single-domain globular proteins of various size and stability, and that the experimentally measured folding rates fall within this narrow "golden triangle" built without any adjustable parameters, filling it almost completely. This "golden triangle" also successfully predicts the maximal allowed size of the "foldable" protein domains, as well as the maximal size of protein domains that fold under solely thermodynamic (rather than kinetic) control. In conclusion, we give a phenomenological formula for dependence of the folding rate on the size, shape and stability of the protein fold.

Golden triangle for folding rates of globular proteins.

The ability of protein chains to spontaneously form their spatial structures is a long-standing puzzle in molecular biology. Experimentally measured rates of spontaneous folding of single-domain globular proteins range from microseconds to hours: the difference (11 orders of magnitude) is akin to the difference between the life span of a mosquito and the age of the universe. Here, we show that physical theory with biological constraints outlines a "golden triangle" limiting the possible range of folding rates for single-domain globular proteins of various size and stability, and that the experimentally measured folding rates fall within this narrow triangle built without any adjustable parameters, filling it almost completely. In addition, the golden triangle predicts the maximal size of protein domains that fold under solely thermodynamic (rather than kinetic) control. It also predicts the maximal allowed size of the "foldable" protein domains, and the size of domains found in known protein structures is in a good agreement with this limit.

ComSin: database of protein structures in bound (complex) and unbound (single) states in relation to their intrinsic disorder.

Most of the proteins in a cell assemble into complexes to carry out their function. In this work, we have created a new database (named ComSin) of protein structures in bound (complex) and unbound (single) states to provide a researcher with exhaustive information on structures of the same or homologous proteins in bound and unbound states. From the complete Protein Data Bank (PDB), we selected 24 910 pairs of protein structures in bound and unbound states, and identified regions of intrinsic disorder. For 2448 pairs, the proteins in bound and unbound states are identical, while 7129 pairs have sequence identity 90% or larger. The developed server enables one to search for proteins in bound and unbound states with several options including sequence similarity between the corresponding proteins in bound and unbound states, and validation of interaction interfaces of protein complexes. Besides that, through our web server, one can obtain necessary information for studying disorder-to-order and order-to-disorder transitions upon complex formation, and analyze structural differences between proteins in bound and unbound states. The database is available at http://antares.protres.ru/comsin/.

FoldAmyloid: a method of prediction of amyloidogenic regions from protein sequence.

MOTIVATION: Amyloidogenic regions in polypeptide chains are very important because such regions are responsible for amyloid formation and aggregation. It is useful to be able to predict positions of amyloidogenic regions in protein chains. RESULTS: Two characteristics (expected probability of hydrogen bonds formation and expected packing density of residues) have been introduced by us to detect amyloidogenic regions in a protein sequence. We demonstrate that regions with high expected probability of the formation of backbone-backbone hydrogen bonds as well as regions with high expected packing density are mostly responsible for the formation of amyloid fibrils. Our method (FoldAmyloid) has been tested on a dataset of 407 peptides (144 amyloidogenic and 263 non-amyloidogenic peptides) and has shown good performance in predicting a peptide status: amyloidogenic or non-amyloidogenic. The prediction based on the expected packing density classified correctly 75% of amyloidogenic peptides and 74% of non-amyloidogenic ones. Two variants (averaging by donors and by acceptors) of prediction based on the probability of formation of backbone-backbone hydrogen bonds gave a comparable efficiency. With a hybrid-scale constructed by merging the above three scales, our method is correct for 80% of amyloidogenic peptides and for 72% of non-amyloidogenic ones. Prediction of amyloidogenic regions in proteins where positions of amyloidogenic regions are known from experimental data has also been done. In the proteins, our method correctly finds 10 out of 11 amyloidogenic regions. AVAILABILITY: The FoldAmyloid server is available at http://antares.protres.ru/fold-amyloid/.

Intrinsic disorder in protein interactions: insights from a comprehensive structural analysis.

We perform a large-scale study of intrinsically disordered regions in proteins and protein complexes using a non-redundant set of hundreds of different protein complexes. In accordance with the conventional view that folding and binding are coupled, in many of our cases the disorder-to-order transition occurs upon complex formation and can be localized to binding interfaces. Moreover, analysis of disorder in protein complexes depicts a significant fraction of intrinsically disordered regions, with up to one third of all residues being disordered. We find that the disorder in homodimers, especially in symmetrical homodimers, is significantly higher than in heterodimers and offer an explanation for this interesting phenomenon. We argue that the mechanisms of regulation of binding specificity through disordered regions in complexes can be as common as for unbound monomeric proteins. The fascinating diversity of roles of disordered regions in various biological processes and protein oligomeric forms shown in our study may be a subject of future endeavors in this area.

More compact protein globules exhibit slower folding rates.

We have demonstrated that, among proteins of the same size, alpha/beta proteins have on the average a greater number of contacts per residue due to their more compact (more "spherical") structure, rather than due to tighter packing. We have examined the relationship between the average number of contacts per residue and folding rates in globular proteins according to general protein structural class (all-alpha, all-beta, alpha/beta, alpha+beta). Our analysis demonstrates that alpha/beta proteins have both the greatest number of contacts and the slowest folding rates in comparison to proteins from the other structural classes. Because alpha/beta proteins are also known to be the oldest proteins, it can be suggested that proteins have evolved to pack more quickly and into looser structures.

Different packing of external residues can explain differences in the thermostability of proteins from thermophilic and mesophilic organisms.

MOTIVATION: Understanding the basis of protein stability in thermophilic organisms raises a general question: what structural properties of proteins are responsible for the higher thermostability of proteins from thermophilic organisms compared to proteins from mesophilic organisms? RESULTS: A unique database of 373 structurally well-aligned protein pairs from thermophilic and mesophilic organisms is constructed. Comparison of proteins from thermophilic and mesophilic organisms has shown that the external, water-accessible residues of the first group are more closely packed than those of the second. Packing of interior parts of proteins (residues inaccessible to water molecules) is the same in both cases. The analysis of amino acid composition of external residues of proteins from thermophilic organisms revealed an increased fraction of such amino acids as Lys, Arg and Glu, and a decreased fraction of Ala, Asp, Asn, Gln, Thr, Ser and His. Our theoretical investigation of folding/unfolding behavior confirms the experimental observations that the interactions that differ in thermophilic and mesophilic proteins form only after the passing of the transition state during folding. Thus, different packing of external residues can explain differences in thermostability of proteins from thermophilic and mesophilic organisms. AVAILABILITY: The database of 373 structurally well-aligned protein pairs is available at http://phys.protres.ru/resources/termo_meso_base.html. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

Understanding the folding rates and folding nuclei of globular proteins.

The first part of this paper contains an overview of protein structures, their spontaneous formation ("folding"), and the thermodynamic and kinetic aspects of this phenomenon, as revealed by in vitro experiments. It is stressed that universal features of folding are observed near the point of thermodynamic equilibrium between the native and denatured states of the protein. Here the "two-state" ("denatured state" <--> "native state") transition proceeds without accumulation of metastable intermediates, but includes only the unstable "transition state". This state, which is the most unstable in the folding pathway, and its structured core (a "nucleus") are distinguished by their essential influence on the folding/unfolding kinetics. In the second part of the paper, a theory of protein folding rates and related phenomena is presented. First, it is shown that the protein size determines the range of a protein's folding rates in the vicinity of the point of thermodynamic equilibrium between the native and denatured states of the protein. Then, we present methods for calculating folding and unfolding rates of globular proteins from their sizes, stabilities and either 3D structures or amino acid sequences. Finally, we show that the same theory outlines the location of the protein folding nucleus (i.e., the structured part of the transition state) in reasonable agreement with experimental data.