Discrete state model and accurate estimation of loop entropy of RNA secondary structures

Conformational entropy makes important contribution to the stability and folding of RNA molecule, but it is challenging to either measure or compute conformational entropy associated with long loops. We develop optimized discrete k-state models of RNA backbone based on known RNA structures for computing entropy of loops, which are modeled as self-avoiding walks. To estimate entropy of hairpin, bulge, internal loop, and multibranch loop of long length (up to 50), we develop an efficient sampling method based on the sequential Monte Carlo principle. Our method considers excluded volume effect. It is general and can be applied to calculating entropy of loops with longer length and arbitrary complexity. For loops of short length, our results are in good agreement with a recent theoretical model and experimental measurement. For long loops, our estimated entropy of hairpin loops is in excellent agreement with the Jacobson-Stockmayer extrapolation model. However, for bulge loops and more complex secondary structures such as internal and multibranch loops, we find that the Jacobson-Stockmayer extrapolation model has large errors. Based on estimated entropy, we have developed empirical formulae for accurate calculation of entropy of long loops in different secondary structures. Our study on the effect of asymmetric size of loops suggest that loop entropy of internal loops is largely determined by the total loop length, and is only marginally affected by the asymmetric size of the two loops. Our finding suggests that the significant asymmetric effects of loop length in internal loops measured by experiments are likely to be partially enthalpic. Our method can be applied to develop improved energy parameters important for studying RNA stability and folding, and for predicting RNA secondary and tertiary structures. The discrete model and the program used to calculate loop entropy can be downloaded at http://gila.bioengr.uic.edu/resources/RNA.html.     

反相高效液相色谱中溶质保留过程的热力学研究Ⅲ: 熵变及其分量,吸附及解吸附熵变

从理论上推导出了用柱相比和不用柱相比时溶质在反相高效液相色谱保留过程中的总熵变△S~(~P~a~)的两个表达式, 后者亦为计量置换保留模型中线性参数与该过程热力学函数间的定量表达式。△S~(~P~a~)也能够被分成两个独立的分量,吸附熵变△S~(~Ⅰ~,~a~)和解吸附熵变△S~(~Z~,~D~)。依据液相色谱中溶质计量置换保留模型中容量因子与流动相中置换剂活度a~D间的定量关系, 对在不同a~D条件下熔质的总熵变、吸附及解吸附熵变进行了估算, 并用实验测定的总熵变△S~(~P~a~,~e~)与估算总熵变△S~(~P~a~,~c~)进行了比较, 获得了结果的一致性。本文对传统的"吸附伴随着熵减小, 相反, 解吸伴随着熵增大"的定性规律予以定量的说明。     

Thermodynamics of polymeric fluids – effect of volume on the configurational entropy of chain molecules

Various conformation‐dependent properties of chain molecules have been successfully treated within the rotational isomeric state approximation. The conformation entropy is one of such properties which can be readily defined by the partition function, the sum of all possible configurations of the chain. Flexible polymers often exhibit crystallization and in some cases liquid‐crystallization as well. In these first‐order transitions, changes in the spatial arrangement of polymer chains are considered to be a major factor involved. In order to explicitly determine the conformational contribution to the melting entropy, the latent entropy observed under the isobaric condition must be corrected for the volume change. The entropy separation involves a hypothetical assumption that the volume of the isotropic fluid may be compressed to that of the solid state without affecting the configurational part of the entropy of molecules. Finally thermodynamic significance of the conformation entropy in these transitions is emphasized on the basis of the critical studies of the entropy‐volume relation of chain molecules in the liquid state.     

Structural and ITC Characterization of Peptide-Protein Binding: Thermodynamic Consequences of Cyclization Constraints,a Case Study on Vascular Endothelial Growth Factor Ligands

Macrocyclization constraints are widely used in the design of protein ligands to stabilize their bioactive conformation and increase their affinities. However, the resulting changes in binding entropy can be puzzling and uncorrelated to affinity gains. Here, the thermodynamic (Isothermal Titration Calorimetry) and structural (X-ray, NMR and CD) analysis of a complete series of lactam-bridged peptide ligands of the vascular endothelial growth factor, and their unconstrained analogs are reported. It is shown that differences in thermodynamics arise mainly from the folding energy of the peptide upon binding. The systematic reduction in conformational entropy penalty due to helix pre-organization can be counterbalanced by an unfavorable vibrational entropy change if the constraints are too rigid. The gain in configurational entropy partially escapes the enthalpy/entropy compensation and leads to an improvement in affinity. The precision of the analytical ITC method makes this study a possible benchmark for constrained peptides optimization.     

Fast and accurate computation schemes for evaluating vibrational entropy of proteins

Standard normal mode analysis (NMA) method is able to calculate vibrational entropy of proteins, but it is computationally intensive, especially for large proteins. To evaluate vibrational entropy efficiently and accurately, we, here, propose computation schemes based on coarse-grained NMA methods. This can be achieved by rescaling coarse-grained results with a specific factor that is derived on the basis of the linear correlation of protein vibrational entropy between standard NMA and coarse-grained NMA. Our coarse-grained NMA computation schemes can repeat correctly and efficiently the results of standard NMA for large proteins.     

Validation of Markov state models using Shannon's entropy

Markov state models are kinetic models built from the dynamics of molecular simulation trajectories by grouping similar configurations into states and examining the transition probabilities between states. Here we present a procedure for validating the underlying Markov assumption in Markov state models based on information theory using Shannon's entropy. This entropy method is applied to a simple system and is compared with the previous eigenvalue method. The entropy method also provides a way to identify states that are least Markovian, which can then be divided into finer states to improve the model.     

Entropy effects during sorption of alkanes in zeolites

Recent developments in Configurational-Bias Monte Carlo (CBMC) techniques allow the accurate calculation of the sorption isotherms for alkanes, and their mixtures, in various zeolites. The CBMC simulations give new insights into subtle entropy effects affecting mixture adsorption. Three types of entropy effects can be distinguished. (1) Size entropy effects favour the component with the smaller number of C atoms because the smaller molecule finds it easier to fill in the 'gaps' within the zeolite matrix at high molecular loadings. (2) Configurational entropy effects come into play for mixtures of alkanes that differ in the degree of branching. For a mixture of linear and branched alkanes with the same number of C atoms, configurational entropy effects favour the linear isomer because such molecules 'pack' more efficiently within, say, the intersecting channel topology of MFI zeolite. (3) Length entropy effects comes into force for sorption of linear and branched alkanes within the cylindrical channels of say AFI and MOR zeolites; here the double branched alkane has the shortest length and can be packed more efficiently within the channels. We demonstrate that CBMC simulations allow the efficient screening of zeolite structures for a given separation duty and aid the development of novel separation processes exploiting entropy effects.     

Excess entropy scaling of transport properties in network-forming ionic melts (SiO(2) and BeF(2))

The regime of validity of Rosenfeld excess entropy scaling of diffusivity and viscosity is examined for two tetrahedral, network-forming ionic melts (BeF(2) and SiO(2)) using molecular dynamics simulations. With decrease in temperature, onset of local caging behavior in the diffusional dynamics is shown to be accompanied by a significant increase in the effect of three-body and higher-order particle correlations on the excess entropy, diffusivity, ionic conductivity, and entropy-transport relationships. The signature of caging effects on the Rosenfeld entropy scaling of transport properties is a distinctly steeper dependence of the logarithm of the diffusivity on the excess entropy in ionic melts. This is shown to be true also for a binary Lennard-Jones glassformer, based on available results in the literature. Our results suggest that the onset of a landscape-influenced regime in the dynamics is correlated with this characteristic departure from Rosenfeld scaling. The breakdown of the Nernst-Einstein relation in the ionic melts can also be correlated with the emerging cooperative dynamics.     

Entropy, diffusivity, and structural order in liquids with waterlike anomalies

The excess entropy, defined as the difference between the entropies of the liquid and the ideal gas under identical density and temperature conditions, is studied as a function of density and temperature for liquid silica and a two-scale ramp potential, both of which are known to possess waterlike liquid state anomalies. The excess entropy for both systems is evaluated using a fairly accurate pair correlation approximation. The connection between the excess entropy and the density and diffusional anomalies is demonstrated. Using the pair correlation approximation to the excess entropy, it can be shown that if the energetically favorable local geometries in the low and high density limits have different symmetries, then a structurally anomalous regime can be defined in terms of orientational and translational order parameters, as in the case of silica and the two-scale ramp system but not for the one-scale ramp liquid. Within the category of liquids with waterlike anomalies, we show that the relationship between the macroscopic entropy and internal energy is sufficient to distinguish between those with local anisotropy and consequent open packings at low densities and those with isotropic interactions but multiple length scales. Since it is straightforward to evaluate the pair correlation entropy and internal energy from simulations or experimental data, such plots should provide a convenient means to diagnose the existence as well as type of anomalous behavior in a range of liquids, including ionic and intermetallic melts and complex fluids with ultrasoft repulsions.     

Shannon entropy as a predictor of avoided crossing in confined atoms

Avoided crossing is one of the unique spectroscopic features of a confined atomic system. Shannon information entropy of the ground state and some of the excited states of confined H atom as a predictor of avoided crossing is studied in this work. This is accomplished by varying the strength of the confinement and examining structure properties like ionization energy and Shannon information entropy. Along with the energy level repulsion at the avoided crossing, Shannon information entropy is also exchanged between the involved states. This work also addresses a question: In addition to that regarding localization, what other property of the system can be extracted from Shannon entropy? Insightful connection is discovered between Shannon entropy and the average value of confinement potential, Coulomb potential, and kinetic energy.     

关于热力学自发过程及其判据的讨论

文章指出现有各种自发过程的判据都是在指定的约束条件下才能应用,缺乏普适性是自发过程定义多样化的引发原因。在无约束条件下将热力学第一定律代入总熵判据得出并分析讨论了总熵判据的另一种形式,结合自发过程的特点总结出了热力学变化过程中能量变化的本质,给出了自发过程的通用定义。进一步指出原总熵判据只能分辨可逆与不可逆,不能分辨自发与非自发。文章给出的总熵判据的另一种形式——封闭系统任意过程的做功能力判据具有分辨自发与非自发的能力。通过理论研讨和实际应用表明,做功能力判据与总熵判据完全等价,在相应约束条件下可还原为当前热力学中各类方向判据。填补了常见的变温过程和变压过程在以前的教科书中无自发和非自发判据的空白。以前教科书中由于自发过程定义和解释的混乱而出现的一些疑难问题,在通用定义和做功能力判据面前都能得到满意的解答。     

Entropy of Zeolitic Water

Zeolite-water heat-pump system is suitable for effective use of low temperature heat sources such as solar energy and waste heats from factories, that is,for energy saving. The heat exchange function of zeolite owes obviously to the nature of the zeolitic water, the state of which can be described in terms of the entropy value as an independent component of H₂O. Most entropy values of zeolitic water have been given so far to be intermediate between those of liquid water (69.9 J mol^-1 K^-1 at 298 K) and ice (41.5 J mol^-1 K^-1 at 273 K).The present calorimetric measurements proved, however, that the entropy value for Mg-exchanged A-type zeolite is so small, even at the ambient temperature, as to be compared with the residual entropy of ice at 0 K. This revised version was published online in July 2006 with corrections to the Cover Date.     

平行板间高分子链构象熵的动力学Monte Carlo模拟

用动力学Monte Carlo方法模拟了受限于两平行板之间的高分子链,并用扫描法计算了链的构象熵S,研究了构象熵相对于自由链的减小量(S0-S)与平行板间距D和高分子链长n的关系.结果证实了de Gennes的自由能标度关系,并给出了标度关系适用的范围.当D非常小时,高分子链受到强烈限制,S0-S与n成正比,表明单链节受到平行板的平均排斥作用力与链长无关.随着D增大,平行板对构象熵的影响越来越弱,单链节受到平行板的平均排斥作用力随链长的增长而增大.当D比较大时,平行板对构象熵的影响近似可以忽略,高分子链构象熵与自由空间中的结果一致.     

Efficient calculation of configurational entropy from molecular simulations by combining the mutual-information expansion and nearest-neighbor methods

Changes in the configurational entropies of molecules make important contributions to the free energies of reaction for processes such as protein-folding, noncovalent association, and conformational change. However, obtaining entropy from molecular simulations represents a long-standing computational challenge. Here, two recently introduced approaches, the nearest-neighbor (NN) method and the mutual-information expansion (MIE), are combined to furnish an efficient and accurate method of extracting the configurational entropy from a molecular simulation to a given order of correlations among the internal degrees of freedom. The resulting method takes advantage of the strengths of each approach. The NN method is entirely nonparametric (i.e., it makes no assumptions about the underlying probability distribution), its estimates are asymptotically unbiased and consistent, and it makes optimum use of a limited number of available data samples. The MIE, a systematic expansion of entropy in mutual information terms of increasing order, provides a well-characterized approximation for lowering the dimensionality of the numerical problem of calculating the entropy of a high-dimensional system. The combination of these two methods enables obtaining well-converged estimations of the configurational entropy that capture many-body correlations of higher order than is possible with the simple histogramming that was used in the MIE method originally. The combined method is tested here on two simple systems: an idealized system represented by an analytical distribution of six circular variables, where the full joint entropy and all the MIE terms are exactly known, and the R,S stereoisomer of tartaric acid, a molecule with seven internal-rotation degrees of freedom for which the full entropy of internal rotation has been already estimated by the NN method. For these two systems, all the expansion terms of the full MIE of the entropy are estimated by the NN method and, for comparison, the MIE approximations up to third order are also estimated by simple histogramming. The results indicate that the truncation of the MIE at the two-body level can be an accurate, computationally nondemanding approximation to the configurational entropy of anharmonic internal degrees of freedom. If needed, higher-order correlations can be estimated reliably by the NN method without excessive demands on the molecular-simulation sample size and computing time.     

人类Y染色体回文序列中的重复序列与碱基关联

利用信息论和统计学的方法并结合生物学的特征研究人类Y染色体回文序列的互信息、“n字”熵、条件熵,定量分析了回文序列的长程关联和短程关联,发现其中既存在长程关联也存在短程关联,并且它们主要是由序列中的重复序列引起的．研究表明重复序列含量越高碱基之间的关联越强．     

Optimization in flow-injection analysis and principles of linear nonequilibrium thermodynamics

It is proposed to describe the set of processes determining the steady-state conditions and signal formation in flow-injection analysis (FIA) in terms of the concept the rate of entropy production σ _S borrowed from nonequilibrium thermodynamics. The value of σ _S can be estimated from the ascending wing of the peak. It is shown that the steady-state mode with the zero rate of entropy production is reproduced at the moment of peak registration because of the equilibrium of entropy because of the chemical reaction and the diffusion and transfer of the sample plug through the detector. It was supposed that the accurate reproduction of this condition in registering each peak results in the high reproducibility of the results of measurements in FIA. The correlation between the rate of entropy production and the degree of reaction proceeding in the flow-injection system, which is easily determined experimentally, is studied. It is shown that a convenient criterion of the optimization of various systems can be the degree of reaction proceeding extrapolated to the zero rate of entropy production. The efficiency of the proposed criterion is demonstrated on practical examples.     

Structural information content of networks: graph entropy based on local vertex functionals

In this paper we define the structural information content of graphs as their corresponding graph entropy. This definition is based on local vertex functionals obtained by calculating j-spheres via the algorithm of Dijkstra. We prove that the graph entropy and, hence, the local vertex functionals can be computed with polynomial time complexity enabling the application of our measure for large graphs. In this paper we present numerical results for the graph entropy of chemical graphs and discuss resulting properties.     

Conformational stabilization of an engineered binding protein

We analyzed the thermodynamic basis for improvement of a binding protein by disulfide engineering. The Z(SPA)(-)(1) affibody binds to its Z domain binding partner with a dissociation constant K(d) = 1.6 microM, and previous analyses suggested that the moderate affinity is due to the conformational heterogeneity of free Z(SPA)(-)(1) rather than to a suboptimal binding interface. Studies of five stabilized Z(SPA)(-)(1) double cystein mutants show that it is possible to improve the affinity by an order of magnitude to K(d) = 130 nM, which is close to the range (20 to 70 nM) observed with natural Z domain binders, without altering the protein-protein interface obtained by phage display. Analysis of the binding thermodynamics reveals a balance between conformational entropy and desolvation entropy: the expected and favorable reduction of conformational entropy in the best-binding Z(SPA)(-)(1) mutant is completely compensated by an unfavorable loss of desolvation entropy. This is consistent with a restriction of possible conformations in the disulfide-containing mutant and a reduction of average water-exposed nonpolar surface area in the free state, resulting in a smaller conformational entropy penalty, but also a smaller change in surface area, for binding of mutant compared to wild-type Z(SPA)(-)(1). Instead, higher Z domain binding affinity in a group of eight Z(SPA)(-)(1) variants correlates with more favorable binding enthalpy and enthalpy-entropy compensation. These results suggest that protein-protein binding affinity can be improved by stabilizing conformations in which enthalpic effects can be fully explored.     

Calculation of entropy with computer simulation methods

A new approximate method for calculating entropy with computer simulation methods is proposed. The sampling probability of an equilibrium configuration is calculated by means of the frequency of occurrence of what we call here local states (related to the state of the particle and its neighbors); hence entropy can be calculated as well. The method is applied to the Metropolis' Monte Carlo simulation for the case of the square Ising lattice. The results are in very good agreement with theory even close to the critical temperature. The method is used also to calculate non-equilibrium entropy.     

A mechanical representation of entropy for a large finite system

The temporal evolution of the entropy of a mechanical system as described by a single trajectory is computed using the Clausius [Philos. Mag. 40, 122 (1868)] equality. This requires computing the maximal work that can be done by the system and comparing it to the actual work performed. A single trajectory suffices to determine the entropy when it is "typical," meaning that average values of mechanical variables will not be different when computed using trajectories with different initial conditions. The results are illustrated for small rare gas clusters heated and compressed by an impact at a hard surface.