A measure of conformational entropy change during thermal protein unfolding using neutron spectroscopy

Thermal unfolding of proteins at high temperatures is caused by a strong increase of the entropy change which lowers Gibbs free energy change of the unfolding transition (DeltaG(unf) = DeltaH - TDeltaS). The main contributions to entropy are the conformational entropy of the polypeptide chain itself and ordering of water molecules around hydrophobic side chains of the protein. To elucidate the role of conformational entropy upon thermal unfolding in more detail, conformational dynamics in the time regime of picoseconds was investigated with neutron spectroscopy. Confined internal structural fluctuations were analyzed for alpha-amylase in the folded and the unfolded state as a function of temperature. A strong difference in structural fluctuations between the folded and the unfolded state was observed at 30 degrees C, which increased even more with rising temperatures. A simple analytical model was used to quantify the differences of the conformational space explored by the observed protein dynamics for the folded and unfolded state. Conformational entropy changes, calculated on the basis of the applied model, show a significant increase upon heating. In contrast to indirect estimates, which proposed a temperature independent conformational entropy change, the measurements presented here, demonstrated that the conformational entropy change increases with rising temperature and therefore contributes to thermal unfolding.     

基于SNP数据检测染色体拷贝数结果可信度分析

利用SNP数据检测肿瘤细胞染色体拷贝数变异是癌症相关研究的一个热点,目前已有多种方法可以通过分析SNP array数据检测染色体拷贝数。然而在某些情况下,这些检测方法检测结果与真实拷贝数具有一定错误率。目前并没有方法研究预测结果发生错误的规律。本文分别分析了GPHMM,ASCAT两种检测方法结果信息熵与检测正确率的关系,发现检测正确率与信息熵存在很强的相关性。通过对比不同肿瘤细胞比例下信息熵与正确率关系,本文发现随着肿瘤细胞比例的增大,检测结果信息熵平均值增大,方差减小;同时平均检测正确率也越来越大,方差显著减小。这些结果显示信息熵的大小可以反映出检测结果正确率的高低。最后,本文以高肿瘤细胞比例下拷贝数检测结果为例,研究了在变异类型单一,信息熵小的情况下,染色体倍性检测的正确率。结果表明信息熵可以作为衡量检测结果可信度的指标：即信息熵越高,检测结果越可信。     

Structural fluctuations and conformational entropy in proteins: entropy balance in an intramolecular reaction in methemoglobin.

The reversible intramolecular binding of the distal histidine side chain to the heme iron in methemoglobin is of special interest due to the very large negative reaction entropy which overcompensates the large reaction enthalpy. It may be considered as a prominent example of the ability of proteins (including enzymes) to provide global entropy in a local process. In this work new experiments and model calculations are reported which aim at finding the structural elements contributing to the reaction entropy. Geometrical studies prove the implication of the 20 residue E-helix being shifted by more than 2 A. Vibrational entropies are calculated by a procedure derived from the method of Karplus and Kushik. It turns out that neither the histidine alone nor the complete E-helix contribute more than 15 per cent of the required entropy.     

Protein design simulations suggest that side-chain conformational entropy is not a strong determinant of amino acid environmental preferences

Loss of side-chain conformational entropy is an important force opposing protein folding and the relative preferences of the amino acids for being buried or solvent exposed may be partially determined by which amino acids lose more side-chain entropy when placed in the core of a protein. To investigate these preferences, we have incorporated explicit modeling of side-chain entropy into the protein design algorithm, RosettaDesign. In the standard version of the program, the energy of a particular sequence for a fixed backbone depends only on the lowest energy side-chain conformations that can be identified for that sequence. In the new model, the free energy of a single amino acid sequence is calculated by evaluating the average energy and entropy of an ensemble of structures generated by Monte Carlo sampling of amino acid side-chain conformations. To evaluate the impact of including explicit side-chain entropy, sequences were designed for 110 native protein backbones with and without the entropy model. In general, the differences between the two sets of sequences are modest, with the largest changes being observed for the longer amino acids: methionine and arginine. Overall, the identity between the designed sequences and the native sequences does not increase with the addition of entropy, unlike what is observed when other key terms are added to the model (hydrogen bonding, Lennard-Jones energies, and solvation energies). These results suggest that side-chain conformational entropy has a relatively small role in determining the preferred amino acid at each residue position in a protein.     

On the relationship between dissipation and the rate of spontaneous entropy production from linear irreversible thermodynamics

When systems are far from equilibrium, the temperature, the entropy and the thermodynamic entropy production are not defined and the Gibbs entropy does not provide useful information about the physical properties of a system. Furthermore, far from equilibrium, or if the dissipative field changes in time, the spontaneous entropy production of linear irreversible thermodynamics becomes irrelevant. In 2000 we introduced a definition for the dissipation function and showed that for systems of arbitrary size, arbitrarily near or far from equilibrium, the time integral of the ensemble average of this quantity can never decrease. In the low-field limit, its ensemble average becomes equal to the spontaneous entropy production of linear irreversible thermodynamics. We discuss how these quantities are related and why one should use dissipation rather than entropy or entropy production for non-equilibrium systems.     

最大信息熵原理与群体遗传平衡

建立了用最大信息熵原理推导群体遗传平衡定律的统一数学模型,并给出了模型的统一解,此解正是Hardy-Weinberg定律所给出的平衡群体的基因型频率,说明当群体信息熵达到最大时,群体基因型频率不再变化,即达到“平衡”。这证明了最大熵分布就是Hardy-Weinberg平衡分布。Hardy-Weinberg平衡定律与最大信息熵原理的内在一致性说明,杂交和随机交配是一个不可逆过程,使群体基因型信息熵增大,无序性增,是选择和近亲交配使群体的信息熵降低,有序性增加,育种过程实际就是调节群体信息熵的过程。过程信息熵的含义是表示一个概率分布的不确定性,最大熵原理意味着在一定的约束条件,选择具有最大不确定性的分布,从而其分布是最为随机的。最大熵原理在信息,工程,天文,地理,图像处理,模式识别等自然科学和社会科学领域都有广泛的成功应用,本文从群体遗传学角度证明了这一原理具有普遍适用性。熵是描述系统状态的函数,而最大熵原理则表明了系统发展变化的趋势,系统的最终状态必然是熵增加至最大值的状态,对于任何系统都是如此。因此,群体遗传系统的平衡定律可以统一用最大熵原理进行判定和描述;任意群体的基因型信息熵在随机交配世代传递时有不断增加的趋势;在一定约束条件下基因型信息熵达到最大值时,就称之为达到遗传平衡。本文将信息论原理应用于群体遗传学研究,揭示了基因信息熵的生物学意义,并表明可以用信息学和控制论的原理和方法来研究群体遗传学问题。     

一个基于熵的指数精细定位人类复杂性状位点

近来,一个基于熵的指数被提出用来对人类复杂性状位点进行连锁不平衡定位．这个熵指数比较了患病个体与正常个体或极端样本之间标记基因频率的熵和条件熵．本文基于熵理论,提出了另一个备选指数．这个新的指数比较患病个体与正常个体之间标记基因型频率的熵和条件熵．计算机模拟结果表明本文提出的新指数平行于之前的熵指数．而基于遗传性血色病（hereditary haemochromatosis,HH）数据的分析表明了这个新指数能有效对人类复杂性状位点进行精细定位．     

Accurate estimation of entropy in very short physiological time series: the problem of atrial fibrillation detection in implanted ventricular devices

Entropy estimation is useful but difficult in short time series. For example, automated detection of atrial fibrillation (AF) in very short heart beat interval time series would be useful in patients with cardiac implantable electronic devices that record only from the ventricle. Such devices require efficient algorithms, and the clinical situation demands accuracy. Toward these ends, we optimized the sample entropy measure, which reports the probability that short templates will match with others within the series. We developed general methods for the rational selection of the template length m and the tolerance matching r. The major innovation was to allow r to vary so that sufficient matches are found for confident entropy estimation, with conversion of the final probability to a density by dividing by the matching region volume, 2r(m). The optimized sample entropy estimate and the mean heart beat interval each contributed to accurate detection of AF in as few as 12 heartbeats. The final algorithm, called the coefficient of sample entropy (COSEn), was developed using the canonical MIT-BIH database and validated in a new and much larger set of consecutive Holter monitor recordings from the University of Virginia. In patients over the age of 40 yr old, COSEn has high degrees of accuracy in distinguishing AF from normal sinus rhythm in 12-beat calculations performed hourly. The most common errors are atrial or ventricular ectopy, which increase entropy despite sinus rhythm, and atrial flutter, which can have low or high entropy states depending on dynamics of atrioventricular conduction.     

Does microbial life always feed on negative entropy? Thermodynamic analysis of microbial growth.

Schr?dinger stated in his landmark book, What is Life?, that life feeds on negative entropy. In this contribution, the validity of this statement is discussed through a careful thermodynamic analysis of microbial growth processes. In principle, both feeding on negative entropy, i.e. yielding products of higher entropy than the substrates, and generating heat can be used by microorganisms to rid themselves of internal entropy production resulting from maintenance and growth processes. Literature data are reviewed in order to compare these two mechanisms. It is shown that entropy-neutral, entropy-driven, and entropy-retarded growth exist. The analysis of some particularly interesting microorganisms shows that enthalpy-retarded microbial growth may also exist, which would signify a net uptake of heat during growth. However, the existence of endothermic life has never been demonstrated in a calorimeter. The internal entropy production in live cells also reflects itself in the Gibbs energy dissipation accompanying growth, which is related quantitatively to the biomass yield. An empirical correlation of the Gibbs energy dissipation in terms of the physico-chemical nature of the growth substrate has been proposed in the literature and can be used to predict the biomass yield approximately. The ratio of enthalpy change and Gibbs energy change can also be predicted since it is shown to be approximately equal to the same ratio of the relevant catabolic process alone.     

153 Wonderful roles of the entropy in protein dynamics,binding and folding

The entropy, which is central to the second law of thermodynamics, determines that the thermal energy always flows spontaneously from regions of higher temperature to regions of lower temperature. In the protein–solvent thermodynamic system, the entropy is defined as a measure of how evenly the thermal energy would distribute over the entire system (Liu et al., 2012). Such tendency to distribute energy as evenly as possible will reduce the state of order of the initial system, and hence, the entropy can be regarded as an expression of the disorder, or randomness of the system (Yang et al., 2012). For a protein–solvent system under a constant solvent condition, the origin of entropy is the thermal energy stored in atoms, which makes atoms jostle around and bump onto one another, thus leading to vibrations of the covalent bonds connecting two atoms (occurring on the fs timescale) and the rotational and translational motions of amino acid side chain groups (occurring on ps timescale) and water molecules. These motions break the noncovalent bonds around structural regions that are weakly constrained thereby triggering the competitive interactions among residues or between residues and water molecules leading ultimately to the loop motions (occurring on ns timescale) around the protein surface. The loop motions can further transmit either through the water network around the protein surface or via specific structural components (such as the hinge-bending regions) over the entire protein molecule leading to large concerted motions (occurring on μs to s timescales) that are most relevant to protein functions (Amadei, Linssen & Berendsen, 1993; Tao, Rao & Liu, 2010). Thus, the multiple hierarchies of the protein dynamics on distinct timescales (Henzler-Wildman & Kern, 2007) are a consequence of the cascade amplification of the microscopic motions of atoms and groups for which the entropy originating from atomic thermal energy is most fundamental. In the case of protein–ligand binding, the importance of the entropy is embodied in the following aspects. (i) The release of the water molecule kinetic energy (which is a process of the solvent entropy maximization) will cause Brownian motions of individual water molecules which result in strong Brownian bombardments to solute molecules causing molecule wanders/diffusions and subsequent accident contacts/collisions between proteins and ligands. (ii) Such collisions will inevitably cause water molecule displacement and, if the contact interfaces are properly complementary, the requirement to increase the solvent entropy would further displace the water network around the binding interfaces thus leading to the formation the initial protein-ligand complex. (iii) In the initial complex, the loose association of the two partners provide the opportunity for protein to increase conformational entropy, thus triggering the conformational adjustments through competitive interaction between protein residues and ligand, leading ultimately to the formation of tightly associated complex (Liu et al., 2012). In the protein folding process, the first stage, i.e. the rapid hydrophobic collapse (Agashe, Shastry & Udgaonkar, 1995; Dill, 1985), is in fact driven by the effect of the solvent entropy maximization. Specifically, the requirement to maintain as many as possible the dynamic hydrogen bonds among the water molecules will squeeze/sequestrate the hydrophobic amino acid side chains into the interior of the folding intermediates and expose the polar/charged side chains onto the intermediate surface. This will minimize the solvent accessible surface area of the folding intermediates and as thus maximize the entropy of the solvent. The resulting molten globule states (Ohgushi & Wada, 1983) may contain a few secondary structural components and native tertiary contacts, while many native contacts, or close residue–residue interactions present in the native state have not yet formed. However, the nature to increase the protein conformational entropy can trigger a further conformational adjustment process, i.e. the conformational entropy increase breaks the transient secondary or tertiary contacts and triggers the competitive interactions among protein residues and between residues and water. This process may repeat many rounds until the negative enthalpy change resulting from the noncovalent formations can overcompensate for protein conformational entropy loss. In summary, we consider that the tendency to maximize the entropy of the protein–solvent system, which originates from the atomic thermal energy, is the most fundamental driving factor for protein folding, binding, and dynamics, whereas the enthalpy reduction, an opposing factor that tends to make the system become ordered, can compensate for the effect of entropy loss to ultimately allow the system to reach equilibrium at the free energy minima, either global or local.     

Thermal coefficients of the methyl groups within ubiquitin

Physiological processes such as protein folding and molecular recognition are intricately linked to their dynamic signature, which is reflected in their thermal coefficient. In addition, the local conformational entropy is directly related to the degrees of freedom, which each residue possesses within its conformational space. Therefore, the temperature dependence of the local conformational entropy may provide insight into understanding how local dynamics may affect the stability of proteins. Here, we analyze the temperature dependence of internal methyl group dynamics derived from the cross-correlated relaxation between dipolar couplings of two CH bonds within ubiquitin. Spanning a temperature range from 275 to 308 K, internal methyl group dynamics tend to increase with increasing temperature, which translates to a general increase in local conformational entropy. With this data measured over multiple temperatures, the thermal coefficient of the methyl group order parameter, the characteristic thermal coefficient, and the local heat capacity were obtained. By analyzing the distribution of methyl group thermal coefficients within ubiquitin, we found that the N-terminal region has relatively high thermostability. These results indicate that methyl groups contribute quite appreciably to the total heat capacity of ubiquitin through the regulation of local conformational entropy.     

Statistical sequence analyses of G-protein-coupled receptors: structural and functional characteristics viewed with periodicities of entropy, hydrophobicity, and volume

G-protein-coupled receptors (GPCRs) play a crucial role in signal transduction and receive a wide variety of ligands. GPCRs are a major target in drug design, as nearly 50% of all contemporary medicines act on GPCRs. GPCRs are membrane proteins possessing a common structural feature, seven transmembrane helices. In order to design an effective drug to act on a GPCR, knowledge of the three-dimensional (3D) structure of the target GPCR is indispensable. However, as GPCRs are membrane bound, their 3D structures are difficult to obtain. Thus we conducted statistical sequence analyses to find information about 3D structure and ligand binding using the receptors' primary sequences. We present statistical sequence analyses of 270 human GPCRs with regard to entropy (Shannon entropy in sequence alignment), hydrophobicity and volume, which are associated with the alpha-helical periodicity of the accessibility to the surrounding lipid. We found periodicity such that the phase changes once in the middle of each transmembrane region, both in the entropy plot and in the hydrophobicity plot. The phase shift in the entropy plot reflects the variety of ligands and the generality of the mechanism of signal transduction. The two periodic regions in the hydrophobicity plot indicate the regions facing the hydrophobic lipid chain and the polar phospholipid headgroup. We also found a simple periodicity in the plot of volume deviation, which suggests conservation of the stable structural packing among the transmembrane helices.     

Flow-induced endothelial surface reorganization and minimization of entropy generation rate

Effects of hydrodynamic shear on the shape of the endothelial surface are examined based on evaluations of the rate of entropy generation at the cell surface. A linear solution of the flow over a sinusoidally varying endothelial surface is used to evaluate the entropy generation rate on the cell surface for which measured cell dimensions are available. Both the local rate of entropy generation (equivalent to the rate of energy dissipation by viscous shear) at the peak of a cell and the total entropy generation rate over the cell surface are minimized under conditions of a constant cell surface area and a constant cell peak height; which yields horizontal cell dimensions that are close to those obtained experimentally.     

生物多样性的熵值结构

多样性一般定义为生物进化过程中物种表现型的丰富程度。从广义上讲,生物进化与环境归类作用的结构复杂性对生物为多样性的解释是很有益处的。我们认为区域水平与样地水平上的气候归类型与植被归类型是最基本的划分因子,本文以中国东北部黑龙江省的山地植被为例,采用多等级熵值分配的方法来证明所使用的方法。我们得出气候归类型与植被归类型所占有的熵值成分要弱于物种表现型丰富度所占有的熵值成份。值得注意的是其相对的熵值贡献率对每一个因子的最大熵值的变化规律。分析结果表明气候归类型占有较低的熵值而植被归类型占有较高的熵值。因此仅仅以在区域气候型为主要指标所划分的物种生态位是不够的,而以小区域内的植被型为指标所产生的物种生态位具有较高利用价值。     

A theoretical prediction of the normal cardiac oxygen consumption

A model is described from which the entropy production associated with the process of transporting oxygen and carbon dioxide between the lungs and the muscles of the body can be calculated. The two entropy sources which are assumed to be the dominant ones for this process are the entropy production associated with the metabolism of the heart and the entropy production associated with the diffusion of oxygen and carbon dioxide into and out of the blood. The hypothesis that the observed blood flow is the one for which a given amount of oxygen and carbon dioxide is transported between the lungs and the muscles with minimum total entropy production is used to predict the value of the slope of the cardiac oxygen consumption vs. blood flow curve. At a blood flow of 15 liters/min, the predicted value of the slope of this curve is 1.2 ml/liter.     

Side-chain conformational entropy in protein unfolded states

The largest force disfavoring the folding of a protein is the loss of conformational entropy. A large contribution to this entropy loss is due to the side-chains, which are restricted, although not immobilized, in the folded protein. In order to accurately estimate the loss of side-chain conformational entropy that occurs upon folding it is necessary to have accurate estimates of the amount of entropy possessed by side-chains in the ensemble of unfolded states. A new scale of side-chain conformational entropies is presented here. This scale was derived from Monte Carlo computer simulations of small peptide models. It is demonstrated that the entropies are independent of host peptide length. This new scale has the advantage over previous scales of being more precise with low standard errors. Better estimates are obtained for long (e.g., Arg and Lys) and rare (e.g., Trp and Met) side-chains. Excellent agreement with previous side-chain entropy scales is achieved, indicating that further advancements in accuracy are likely to be small at best. Strikingly, longer side-chains are found to possess a smaller fraction of the theoretical maximum entropy available than short side-chains. This indicates that rotations about torsions after chi(2) are significantly affected by side-chain interactions with the polypeptide backbone. This finding invalidates previous assumptions about side-chain-backbone interactions. Proteins 2000;40:443-450.     

Dynamic Approximate Entropy Electroanatomic Maps Detect Rotors in a Simulated Atrial Fibrillation Model

There is evidence that rotors could be drivers that maintain atrial fibrillation. Complex fractionated atrial electrograms have been located in rotor tip areas. However, the concept of electrogram fractionation, defined using time intervals, is still controversial as a tool for locating target sites for ablation. We hypothesize that the fractionation phenomenon is better described using non-linear dynamic measures, such as approximate entropy, and that this tool could be used for locating the rotor tip. The aim of this work has been to determine the relationship between approximate entropy and fractionated electrograms, and to develop a new tool for rotor mapping based on fractionation levels. Two episodes of chronic atrial fibrillation were simulated in a 3D human atrial model, in which rotors were observed. Dynamic approximate entropy maps were calculated using unipolar electrogram signals generated over the whole surface of the 3D atrial model. In addition, we optimized the approximate entropy calculation using two real multi-center databases of fractionated electrogram signals, labeled in 4 levels of fractionation. We found that the values of approximate entropy and the levels of fractionation are positively correlated. This allows the dynamic approximate entropy maps to localize the tips from stable and meandering rotors. Furthermore, we assessed the optimized approximate entropy using bipolar electrograms generated over a vicinity enclosing a rotor, achieving rotor detection. Our results suggest that high approximate entropy values are able to detect a high level of fractionation and to locate rotor tips in simulated atrial fibrillation episodes. We suggest that dynamic approximate entropy maps could become a tool for atrial fibrillation rotor mapping.     

异质性干细胞增殖过程中的熵变化

干细胞在多细胞生物体内广泛存在,其增殖过程在生命体的生长、发育、衰老、组织修复过程中起着重要作用。正常组织中的细胞增殖过程受到严格的控制,干细胞的异常增殖与恶性肿瘤、肥胖症、再生障碍性贫血等疾病有密切关系。生命体内异质性细胞的增殖过程是复杂的动力系统行为,干细胞异常增殖过程伴随细胞的可塑性变化和细胞间相互作用的再平衡过程,如何对这一过程进行定量描述是重要的研究课题。本文构建包含细胞的增殖分化指标和异常增殖性指标异质性的干细胞增殖模型,通过所建立的模型研究由于微环境变化引起的细胞异常增殖过程的熵变化,建立不同增殖条件下的系统熵变化与宏观动力学和系统参数之间的关系。结果表明,在细胞微环境变化引起异常增殖和恢复的过程中,系统的熵与细胞数量之间存在对应关系,而与微环境变化的路径无关。 此外,熵对细胞数量的依赖关系在异常增殖和恢复阶段表现出不同的行为,显示了生物过程的微观不可逆性。本文从物理学的角度对细胞异常增殖过程中熵变化与细胞数量变化的动力学给出定量刻画,为定量描述异质性干细胞增殖过程给出新的研究思路。     

A watershed approach for improving medical image segmentation

In this paper, a novel watershed approach based on seed region growing and image entropy is presented which could improve the medical image segmentation. The proposed algorithm enables the prior information of seed region growing and image entropy in its calculation. The algorithm starts by partitioning the image into several levels of intensity using watershed multi-degree immersion process. The levels of intensity are the input to a computationally efficient seed region segmentation process which produces the initial partitioning of the image regions. These regions are fed to entropy procedure to carry out a suitable merging which produces the final segmentation. The latter process uses a region-based similarity representation of the image regions to decide whether regions can be merged. The region is isolated from the level and the residual pixels are uploaded to the next level and so on, we recall this process as multi-level process and the watershed is called multi-level watershed. The proposed algorithm is applied to challenging applications: grey matter–white matter segmentation in magnetic resonance images (MRIs). The established methods and the proposed approach are experimented by these applications to a variety of simulating immersion, multi-degree, multi-level seed region growing and multi-level seed region growing with entropy. It is shown that the proposed method achieves more accurate results for medical image oversegmentation.     

The nature of the free energy barriers to two-state folding

We introduce a simple procedure to analyze the temperature dependence of the folding and unfolding rates of two-state proteins. We start from the simple transition-state-like rate expression: k = D(eff)exp(-DeltaG(TS)/RT), in which upper and lower bounds for the intra-chain effective diffusion coefficient (D(eff)) are obtained empirically using the timescales of elementary processes in protein folding. From the changes in DeltaG(TS) as a function of temperature, we calculate enthalpies and heat capacities of activation, together with the more elusive entropies of activation. We then estimate the conformational entropy of the transition state by extrapolation to the temperature at which the solvation entropy vanishes by cancellation between polar and apolar terms. This approach is based on the convergence temperatures for the entropy of solvating apolar (approximately 385 K) and polar groups (approximately 335 K), the assumption that the structural properties of the transition state are somewhere in between the unfolded and folded states, and the established relationship between observed heat capacity and solvent accessibility.1 To circumvent the lack of structural information about transition states, we use the empirically determined heat capacities of activation as constraints to identify the extreme values of the transition state conformational entropy that are consistent with experiment. The application of this simple approach to six two-state folding proteins for which there is temperature-dependent data available in the literature provides important clues about protein folding. For these six proteins, we obtain an average equilibrium cost in conformational entropy of -4.3 cal x mol(-1)K(-1)per residue, which is in close agreement to previous empirical and computational estimates of the same quantity. Furthermore, we find that all these proteins have a conformationally diverse transition state, with more than half of the conformational entropy of the unfolded state. In agreement with predictions from theory and computer simulations, the transition state signals the change from a regime dominated by loss in conformational entropy to one driven by the gain in stabilization free energy (i.e., including protein interactions and solvation effects). Moreover, the height of the barrier is determined by how much stabilization free energy is realized at that point, which is related to the relative contribution of local versus non-local interactions. A remarkable observation is that the fraction of conformational entropy per residue that is present in the transition state is very similar for the six proteins in this study. Based on this commonality, we propose that the observed change in thermodynamic regime is connected to a change in the pattern of structure formation: from one driven by formation of pairwise interactions to one dominated by coupling of the networks of interactions involved in forming the protein core. In this framework, the barrier to two-state folding is crossed when the folding protein reaches a "critical native density" that allows expulsion of remaining interstitial water and consolidation of the core. The principle of critical native density should be general for all two-state proteins, but can accommodate different folding mechanisms depending on the particularities of the structure and sequence.