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.     

Characterization the Effects of Structure and Energetics of Intermolecular Interactions on the Oligomerization of Peptides in Aqueous 2, 2, 2-Trifluoroethanol via Circular Dichroism and Nuclear Magnetic Resonance Spectroscopy

Intermolecular interactions are of fundamental importance to fully comprehend a wide range of protein behaviors such as oligomerization, folding and recognition. Two peptides, NPY^[18−36] and NPY^[15−29], segmented from human neuropeptide Y (hNPY), were synthesized in this work to study the interaction between species. Information about intermolecular interactions was extracted from their oligomerizing behaviors. The results from CD and NMR showed that the addition of 2, 2, 2-trifluoroethanol (TFE) induces a stable helix in each peptides and an extended helix in NPY^[18−36], formed between residues 30-36. Pulsed field gradient NMR data revealed that NPY^[15−29] forms a larger oligomer at lower temperatures and continuously dissociates into the monomeric form with increasing temperature. NPY^[18−36] was also found to undergo an enhanced interaction with TFE and a more favorable self-association at higher temperatures. We characterized the changes of oligomerized states with respect to temperature to infer the effects of entropy and interaction energy on the association-dissociation equilibrium. As shown by NPY^[15−29], deletion of helical secondary structure or residues from the C-terminal segment may disrupt the solvation by TFE and results in entropy increase as the oligomer dissociates. Unlike that in NPY^[15−29], the extended helix in NPY^[18−36] improves the binding of TFE, and as a result, entropy is gained via the transfer of the TFE cluster from the interface between monomeric peptides into the bulk solvent. This observation suggests that the oligomerized state may be modulated by the entropy and energetics contributed by helical segments in the oligomerization process.     

Age Dynamics of Body Mass and Human Lifespan

Age dynamics of human body mass (0–90 years) is described as a function of periodic damped oscillations. Common regularities are found in the age changes of mass, of entropic equivalent—a parameter equivalent to thermodynamic entropy, and of intensity of natural mortality. It is shown that the mass reflects the biological system thermodynamic state and is measured oppositely directed to the entropy value. The second, third, and fourth extremes of the mass age dynamics correspond to the mean (70–75 years), the commonly accepted maximal (100–110 years), and the maximal known (140–150 years) human lifespan, while the mass oscillations cease at the age associated with the maximal known lifespan—about 145 years.     

Protein folding ‐ seeing is deceiving

This Perspective is intended to raise questions about the conventional interpretation of protein folding. According to the conventional interpretation, developed over many decades, a protein population can visit a vast number of conformations under unfolding conditions, but a single dominant native population emerges under folding conditions. Accordingly, folding comes with a substantial loss of conformational entropy. How is this price paid? The conventional answer is that favorable interactions between and among the side chains can compensate for entropy loss, and moreover, these interactions are responsible for the structural particulars of the native conformation. Challenging this interpretation, the Perspective introduces a proposal that high energy (i.e., unfavorable) excluding interactions winnow the accessible population substantially under physical–chemical conditions that favor folding. Both steric clash and unsatisfied hydrogen bond donors and acceptors are classified as excluding interactions, so called because conformers with such disfavored interactions will be largely excluded from the thermodynamic population. Both excluding interactions and solvent factors that induce compactness are somewhat nonspecific, yet together they promote substantial chain organization. Moreover, proteins are built on a backbone scaffold consisting of α‐helices and strands of β‐sheet, where the number of hydrogen bond donors and acceptors is exactly balanced. These repetitive secondary structural elements are the only two conformers that can be both completely hydrogen‐bond satisfied and extended indefinitely without encountering a steric clash. Consequently, the number of fundamental folds is limited to no more than ~10,000 for a protein domain. Once excluding interactions are taken into account, the issue of “frustration” is largely eliminated and the Levinthal paradox is resolved. Putting the “bottom line” at the top: it is likely that hydrogen‐bond satisfaction represents a largely under‐appreciated parameter in protein folding models.     

Application of ITC in foods: A powerful tool for understanding the gastrointestinal fate of lipophilic compounds

BackgroundIsothermal titration calorimetry (ITC) is a biophysical technique widely used to study molecular interactions in biological and non-biological systems. It can provide important information about molecular interactions (such as binding constant, number of binding sites, free energy, enthalpy, and entropy) simply by measuring the heat absorbed or released during an interaction between two liquid solutions.Scope of the reviewIn this review, we present an overview of ITC applications in food science, with particular focus on understanding the fate of lipids within the human gastrointestinal tract. In this area, ITC can be used to study micellization of bile salts, inclusion complex formation, the interaction of surface-active molecules with proteins, carbohydrates and lipids, and the interactions of lipid droplets.Major conclusionsITC is an extremely powerful tool for measuring molecular interactions in food systems, and can provide valuable information about many types of interactions involving food components such as proteins, carbohydrates, lipids, surfactants, and minerals. For systems at equilibrium, ITC can provide fundamental thermodynamic parameters that can be used to establish the physiochemical origin of molecular interactions.General significanceIt is expected that ITC will continue to be utilized as a means of providing fundamental information about complex materials such as those found in foods. This knowledge may be used to create functional foods designed to behave in the gastrointestinal tract in a manner that will improve human health and well-being. This article is part of a Special Issue entitled Microcalorimetry in the BioSciences — Principles and Applications, edited by Fadi Bou-Abdallah.     

Free energy calculations shed light on the nuclear pore complex’s selective barrier nature

The nuclear pore complex (NPC) is the exclusive gateway for traffic control across the nuclear envelope. Although smaller cargoes (less than 5–9 nm in size) can freely diffuse through the NPC, the passage of larger cargoes is restricted to those accompanied by nuclear transport receptors (NTRs). This selective barrier nature of the NPC is putatively associated with the intrinsically disordered, phenylalanine-glycine repeat-domains containing nucleoporins, termed FG-Nups. The precise mechanism underlying how FG-Nups carry out such an exquisite task at high throughputs has, however, remained elusive and the subject of various hypotheses. From the thermodynamics perspective, free energy analysis can be a way to determine cargo’s transportability because the traffic through the NPC must be in the direction of reducing the free energy. In this study, we developed a computational model to evaluate the free energy composed of the conformational entropy of FG-Nups and the energetic gain associated with binding interactions between FG-Nups and NTRs and investigated whether these physical features can be the basis of NPC’s selectivity. Our results showed that the reduction in conformational entropy by inserting a cargo into the NPC increased the free energy by an amount substantially greater than the thermal energy (≫k_BT), whereas the free energy change was negligible (<k_BT) for small cargoes (less than ~6 nm in size), indicating the size-dependent selectivity emerges from the entropic effect. Our models suggested that the entropy-induced selectivity of the NPC depends sensitively upon the physical parameters such as the flexibility and the length of FG-Nups. On the other hand, the energetic gain via binding interactions effectively counteracted the entropic reduction, increasing the size limit of transportable cargoes up to the nuclear pore size. We further investigated the geometric effect of the binding spot spatial distribution and found that the clustered binding spot distribution decreased the free energy more efficiently as compared to the scattered distribution.     

Ion exchange of amino acids at a natural organic ion exchanger: Thermodynamics and energetics

The thermodynamics and energetics of the ion exchange of four amino acids at a cellulosic ion exchanger have been studied. Experimental work included determination of ion exchange isotherms and the use of high-sensitivity titration microcalorimetry. A rigorous thermodynamic analysis of the data was developed allowing calculation of the standard free energy, the standard enthalpy, and standard entropy of exchange, and also the differential free energy, incremental enthalpy, and incremental entropy of exchange. The results show that the relative contributions of the enthalpy and entropy to the overall free energy differ markedly for the chosen amino acids. The reasons for these differences are analyzed and discussed. A knowledge of these fundamental thermodynamic properties indicates the solution conditions likely to give enhanced affinity of the ion exchanger for selected amino acids. The experimental techniques and analysis procedures developed are generally applicable to ion exchange separations of biomolecules. (c) 1995 John Wiley & Sons, Inc.     

Free energy calculations shed light on the nuclear pore complex’s selective barrier nature

The nuclear pore complex (NPC) is the exclusive gateway for traffic control across the nuclear envelope. Although smaller cargoes (less than 5–9 nm in size) can freely diffuse through the NPC, the passage of larger cargoes is restricted to those accompanied by nuclear transport receptors (NTRs). This selective barrier nature of the NPC is putatively associated with the intrinsically disordered, phenylalanine-glycine repeat-domains containing nucleoporins, termed FG-Nups. The precise mechanism underlying how FG-Nups carry out such an exquisite task at high throughputs has, however, remained elusive and the subject of various hypotheses. From the thermodynamics perspective, free energy analysis can be a way to determine cargo’s transportability because the traffic through the NPC must be in the direction of reducing the free energy. In this study, we developed a computational model to evaluate the free energy composed of the conformational entropy of FG-Nups and the energetic gain associated with binding interactions between FG-Nups and NTRs and investigated whether these physical features can be the basis of NPC’s selectivity. Our results showed that the reduction in conformational entropy by inserting a cargo into the NPC increased the free energy by an amount substantially greater than the thermal energy (≫k_BT), whereas the free energy change was negligible (<k_BT) for small cargoes (less than ~6 nm in size), indicating the size-dependent selectivity emerges from the entropic effect. Our models suggested that the entropy-induced selectivity of the NPC depends sensitively upon the physical parameters such as the flexibility and the length of FG-Nups. On the other hand, the energetic gain via binding interactions effectively counteracted the entropic reduction, increasing the size limit of transportable cargoes up to the nuclear pore size. We further investigated the geometric effect of the binding spot spatial distribution and found that the clustered binding spot distribution decreased the free energy more efficiently as compared to the scattered distribution.     

Conformational Entropy of the RNA Phosphate Backbone and Its Contribution to the Folding Free Energy

While major contributors to the free energy of RNA tertiary structures such as basepairing, base-stacking, and charge and counterion interactions have been studied extensively, little is known about the intrinsic free energy of the backbone. To assess the magnitude of the entropic strains along the phosphate backbone and their impact on the folding free energy, we have formulated a mathematical treatment for computing the volume of the main-chain torsion-angle conformation space between every pair of nucleobases along any sequence to compute the corresponding backbone entropy. To validate this method, we have compared the computed conformational entropies against a statistical free energy analysis of structures in the crystallographic database from several-thousand backbone conformations between nearest-neighbor nucleobases as well as against extensive computer simulations. Using this calculation, we analyzed the backbone entropy of several ribozymes and riboswitches and found that their entropic strains are highly localized along their sequences. The total entropic penalty due to steric congestions in the backbone for the native fold can be as high as 2.4 cal/K/mol per nucleotide for these medium and large RNAs, producing a contribution to the overall free energy of up to 0.72 kcal/mol per nucleotide. For these RNAs, we found that low-entropy high-strain residues are predominantly located at loops with tight turns and at tertiary interaction platforms with unusual structural motifs.     

Thermodynamic perspectives on genetic instructions, the laws of biology and diseased states

This article examines in a broad perspective entropy and some examples of its relationship to evolution, genetic instructions and how we view diseases. Living organisms are programmed by functional genetic instructions (FGI), through cellular communication pathways, to grow and reproduce by maintaining a variety of hemistable, ordered structures (low entropy). Living organisms are far from equilibrium with their surrounding environmental systems, which tends towards increasing disorder (increasing entropy). Organisms free themselves from high entropy (high disorder) to maintain their cellular structures for a period of time sufficient to allow reproduction and the resultant offspring to reach reproductive ages. This time interval varies for different species. Bacteria, for example need no sexual parents; dividing cells are nearly identical to the previous generation of cells, and can begin a new cell cycle without delay under appropriate conditions. By contrast, human infants require years of care before they can reproduce. Living organisms maintain order in spite of their changing surrounding environment that decreases order according to the second law of thermodynamics. These events actually work together since living organisms create ordered biological structures by increasing local entropy. From a disease perspective, viruses and other disease agents interrupt the normal functioning of cells. The pressure for survival may result in mechanisms that allow organisms to resist attacks by viruses, other pathogens, destructive chemicals and physical agents such as radiation. However, when the attack is successful, the organism can be damaged until the cell, tissue, organ or entire organism is no longer functional and entropy increases.     

Conformational Entropy of the RNA Phosphate Backbone and Its Contribution to the Folding Free Energy

While major contributors to the free energy of RNA tertiary structures such as basepairing, base-stacking, and charge and counterion interactions have been studied extensively, little is known about the intrinsic free energy of the backbone. To assess the magnitude of the entropic strains along the phosphate backbone and their impact on the folding free energy, we have formulated a mathematical treatment for computing the volume of the main-chain torsion-angle conformation space between every pair of nucleobases along any sequence to compute the corresponding backbone entropy. To validate this method, we have compared the computed conformational entropies against a statistical free energy analysis of structures in the crystallographic database from several-thousand backbone conformations between nearest-neighbor nucleobases as well as against extensive computer simulations. Using this calculation, we analyzed the backbone entropy of several ribozymes and riboswitches and found that their entropic strains are highly localized along their sequences. The total entropic penalty due to steric congestions in the backbone for the native fold can be as high as 2.4 cal/K/mol per nucleotide for these medium and large RNAs, producing a contribution to the overall free energy of up to 0.72 kcal/mol per nucleotide. For these RNAs, we found that low-entropy high-strain residues are predominantly located at loops with tight turns and at tertiary interaction platforms with unusual structural motifs.     

Entropy budget of conifer branches

From energy budget data for a branch of ponderosa pine given by Gates, Tibbals and Kreith, entropy fluxes into or out of the branch due to solar radiation, infrared radiation, transpiration and convection are calculated. Net entropy flow into the branch is negative. Assuming that the entropy in the branch is at steady state, the entropy production in the branch of ponderosa pine is calculated and shown to be positive. A positive entropy production indicates that the Second Law of Thermodynamics is certainly valid in the branch. Entropy productions in other conifers, blue spruce and white fir, and in a single pine needle in a horizontal position are also calculated. The entropy production (S_prod) increases linearly with the solar energy absorbed by the plant surface (E_solar); S_prod≈(30.6 E_solar)×10⁻⁴. The ratio (S_prod/E_solar) does not differ between deciduous leaves reported earlier and conifer branches. The theorem of oscillating entropy production proposed earlier holds also for conifer branches and will be of universal nature, applicable to all plant leaves.     

Comment on the Letter by A. Ben-Shaul: “Entropy,Energy, and Bending of DNA in Viral Capsids”

The conformational entropic penalty associated with packaging double-stranded DNA into viral capsids remains an issue of contention. So far, models based on a continuum approximation for DNA have either left the question unexamined, or they have assumed that the entropic penalty is negligible, following an early analysis by Riemer and Bloomfield. In contrast, molecular-dynamics (MD) simulations using bead-and-spring models consistently show a large penalty. A recent letter from Ben-Shaul attempts to reconcile the differences. While the letter makes some valid points, the issue of how to include conformational entropy in the continuum models remains unresolved. In this Comment, I show that the free energy decomposition from continuum models could be brought into line with the decomposition from the MD simulations with two adjustments. First, the entropy from Flory-Huggins theory should be replaced by the estimate of the entropic penalty given in Ben-Shaul’s letter, which corresponds closely to that from the MD simulations. Second, the DNA-DNA repulsions are well described by the empirical relationship given by the Cal Tech group, but the strength of these should be reduced by about half, using parameters based on the Rau-Parsegian experiments, rather than treating them as “fitting parameters (tuned) to fit the data from (single molecule pulling) experiments.”     

154 Protein folding and binding funnels: a common driving force and a common mechanism

Under the free energy landscape theory, both the protein-folding and protein–ligand binding processes are driven by the decrease in total Gibbs free energy of the protein-solvent or protein–ligand-solvent system, which involves the non-complementary changes between the entropy and enthalpy, ultimately leading to a global free energy minimization of these thermodynamic systems (Ji & Liu, 2011; Liu et al., 2012; Yang, Ji & Liu, 2012). In the case of protein folding, the lowering of the system free energy coupled with the gradual reduction in conformational degree of freedom of the folding intermediates determines that the shape of the free energy landscape for protein folding must be funnel-like (Dill & Chan, 1997), rather than non-funneled shapes (Ben-Naim, 2012). In the funnel-like free energy landscape, protein folding can be viewed as going down the hill via multiple parallel routes from a vast majority of individual non-native states on surface outside the funnel to the native states located around the bottom of the funnel. The first stage of folding, i.e. the rapid hydrophobic collapse process, is driven by the solvent entropy maximization. Concretely, the water molecules squeeze and sequestrate the hydrophobic amino acid side chains within the interior of the folding intermediates while exposing the polar and electrostatically charged side chains on the intermediate surface so as to minimize the solvent-accessible surface area of the solute and thus, the minimal contacts between the folding intermediates and the water molecules. This will maximize the entropy of the solvent, thus contributing substantially to lowering of the system free energy due to an absolute advantage of the solvent in both quantity and mass (Yang, Ji & Liu, 2012). The resulting molten globule states (Ohgushi & Wada, 1983), within which a few transient secondary structural components and tertiary contacts have been formed but many native contacts or close residue–residue interactions has yet to form, need to be further sculptured into the native states. This is a relatively slow “bottleneck” process because the competitive interactions between protein residues within the folding intermediates and between residues and water molecules may repeat many rounds to accumulate a large enough number of stable noncovalent bonds capable of counteracting the conformational entropy loss of the intermediates, thus putting this bottleneck stage under the enthalpy control (i.e. negative enthalpy change), contributing further to the lowering of the system free energy. Although the protein–ligand association occurs around the rugged bottom of the free energy landscape, the exclusion of water from the binding interfaces and the formation of noncovalent bonds between the two partners can still lower the system free energy. In conjunction with the loss of the rotational and translational degrees of freedom of the two partners as well as the loss of the conformational entropy of the protein, these processes could merge, downwards expand, and further narrow the free energy wells within which the protein–ligand binding process takes place, thereby making them look like a funnel, which we term the binding funnel. In this funnel, the free energy downhill process follows a similar paradigm to the protein-folding process. For example, if the initial collisions/contacts occur between the properly complementary interfaces of the protein and ligand, a large amount of water molecules (which usually form a water network around the solute surface) will be displaced to suit the need for maximizing the solvent entropy. This process is similar to that of the hydrophobic collapse during protein folding, resulting in a loosely associated protein–ligand complex that needs also to be further adapted into a tight complex, i.e. the second step which is mainly driven by the negative enthalpy change through intermolecular competitive interactions to gradually accumulate the noncovalent bonds and ultimately, to stabilize the complex at a tightly bound state. Taken together, we conclude that whether in the protein-folding or in the protein–ligand binding process, both the entropy-driven first step and the enthalpy-driven second step contribute to the lowering of the system free energy, resulting in the funnel-like folding or binding free energy landscape.     

Quantifying the Entropy of Binding for Water Molecules in Protein Cavities by Computing Correlations

Protein structural analysis demonstrates that water molecules are commonly found in the internal cavities of proteins. Analysis of experimental data on the entropies of inorganic crystals suggests that the entropic cost of transferring such a water molecule to a protein cavity will not typically be greater than 7.0 cal/mol/K per water molecule, corresponding to a contribution of approximately +2.0 kcal/mol to the free energy. In this study, we employ the statistical mechanical method of inhomogeneous fluid solvation theory to quantify the enthalpic and entropic contributions of individual water molecules in 19 protein cavities across five different proteins. We utilize information theory to develop a rigorous estimate of the total two-particle entropy, yielding a complete framework to calculate hydration free energies. We show that predictions from inhomogeneous fluid solvation theory are in excellent agreement with predictions from free energy perturbation (FEP) and that these predictions are consistent with experimental estimates. However, the results suggest that water molecules in protein cavities containing charged residues may be subject to entropy changes that contribute more than +2.0 kcal/mol to the free energy. In all cases, these unfavorable entropy changes are predicted to be dominated by highly favorable enthalpy changes. These findings are relevant to the study of bridging water molecules at protein-protein interfaces as well as in complexes with cognate ligands and small-molecule inhibitors.     

Reply to the Comment by S. Harvey on “Entropy,Energy, and Bending of DNA in Viral Capsids”

The comment by Stephen Harvey in this issue of the Biophysical Journal concludes with two statements regarding my recent letter about DNA packaging into viral capsids. Harvey agrees with my interpretation of the origin of the large confinement entropy predicted by the molecular-dynamics simulations of his group, and its sensitive dependence on the molecular parameters of their wormlike chain model of double-stranded DNA. On the other hand, he doubts my assertion that the confinement entropy is already included in the interstrand repulsion free energy derived from osmotic stress measurements, which constitutes the major contribution to the packaging free energy used in recent continuum theories of this process. Harvey suggests instead that the confinement entropy should be added to this free energy as a separate term (using, for instance, the method described in my letter). I will argue that this addition is redundant, and, in a brief discussion of continuum theories, will also discuss his comments as relates to the work of other researchers.     

Reply to the Comment by S. Harvey on “Entropy,Energy, and Bending of DNA in Viral Capsids”

The comment by Stephen Harvey in this issue of the Biophysical Journal concludes with two statements regarding my recent letter about DNA packaging into viral capsids. Harvey agrees with my interpretation of the origin of the large confinement entropy predicted by the molecular-dynamics simulations of his group, and its sensitive dependence on the molecular parameters of their wormlike chain model of double-stranded DNA. On the other hand, he doubts my assertion that the confinement entropy is already included in the interstrand repulsion free energy derived from osmotic stress measurements, which constitutes the major contribution to the packaging free energy used in recent continuum theories of this process. Harvey suggests instead that the confinement entropy should be added to this free energy as a separate term (using, for instance, the method described in my letter). I will argue that this addition is redundant, and, in a brief discussion of continuum theories, will also discuss his comments as relates to the work of other researchers.     

Maximum entropy,logistic regression,and species abundance

There is considerable debate about the utility of statistical mechanics in predicting diversity patterns in terms of life history traits. Here, I reflect on this debate and show that a community is controlled by the balance of two opposite forces: the entropic part (the natural tendency of the system to be in the configuration with the highest possible entropy) and environmental, ecological and evolutionary constraints maintaining order (reducing entropy). The Boltzmann distribution law that can be derived from the maximum entropy formalism provides a fundamental model for linking species abundance to life history traits and environmental constraining factors. This model predicts a global pattern of diversity evenness along a latitudinal gradient. Although the Boltzmann distribution and the logistic regression models represent two fundamentally different approaches, the two models have an identical mathematical form. Their identical formalisms facilitate the interpretation of logistic regression models with statistical mechanics, and reveal several limitations of the maximum entropy formalism. I argued that although maximum entropy formalism is a promising tool for modeling species abundances and for linking microscopic quantities of individual life history traits to macroscopic patterns of diversity, it is necessary to revise the Boltzmann distribution law for successful prediction of species abundance.