Ab initio folding of terminal segments with secondary structures reveals the fine difference between two closely related all-atom statistical energy functions

One of the common methods for assessing energy functions of proteins is selection of native or near-native structures from decoys. This is an efficient but indirect test of the energy functions because decoy structures are typically generated either by sampling procedures or by a separate energy function. As a result, these decoys may not contain the global minimum structure that reflects the true folding accuracy of the energy functions. This paper proposes to assess energy functions by ab initio refolding of fully unfolded terminal segments with secondary structures while keeping the rest of the proteins fixed in their native conformations. Global energy minimization of these short unfolded segments, a challenging yet tractable problem, is a direct test of the energy functions. As an illustrative example, refolding terminal segments is employed to assess two closely related all-atom statistical energy functions, DFIRE (distance-scaled, finite, ideal-gas reference state) and DOPE (discrete optimized protein energy). We found that a simple sequence-position dependence contained in the DOPE energy function leads to an intrinsic bias toward the formation of helical structures. Meanwhile, a finer statistical treatment of short-range interactions yields a significant improvement in the accuracy of segment refolding by DFIRE. The updated DFIRE energy function yields success rates of 100% and 67%, respectively, for its ability to sample and fold fully unfolded terminal segments of 15 proteins to within 3.5 A global root-mean-squared distance from the corresponding native structures. The updated DFIRE energy function is available as DFIRE 2.0 upon request.     

Protein decoy sets for evaluating energy functions

Energy functions are crucial ingredients of protein tertiary structure prediction methods. Assessing the quality of energy functions is therefore of prime importance. It requires the elaboration of a standard evaluation scheme, whose key elements are: i). sets that contain the native and several non-native structures of proteins (decoys) in order to test whether the energy functions display the expected quality features and ii). measures to evaluate the reliability of energy functions. We present here a survey of the recent advances in these two related fields. In a first part, we analyze and review the large number of decoy sets that are available on the web, and we summarize the characteristics of a challenging decoy set. We then discuss how to define the quality of energy functions and review the measures related to it.     

Discrimination of the native from misfolded protein models with an energy function including implicit solvation.

An essential requirement for theoretical protein structure prediction is an energy function that can discriminate the native from non-native protein conformations. To date most of the energy functions used for this purpose have been extracted from a statistical analysis of the protein structure database, without explicit reference to the physical interactions responsible for protein stability. The use of the statistical functions has been supported by the widespread belief that they are superior for such discrimination to physics-based energy functions. An effective energy function which combined the CHARMM vacuum potential with a Gaussian model for the solvation free energy is tested for its ability to discriminate the native structure of a protein from misfolded conformations; the results are compared with those obtained with the vacuum CHARMM potential. The test is performed on several sets of misfolded structures prepared by others, including sets of about 650 good decoys for six proteins, as well as on misfolded structures of chymotrypsin inhibitor 2. The vacuum CHARMM potential is successful in most cases when energy minimized conformations are considered, but fails when applied to structures relaxed by molecular dynamics. With the effective energy function the native state is always more stable than grossly misfolded conformations both in energy minimized and molecular dynamics-relaxed structures. The present results suggest that molecular mechanics (physics-based) energy functions, complemented by a simple model for the solvation free energy, should be tested for use in the inverse folding problem, and supports their use in studies of the effective energy surface of proteins in solution. Moreover, the study suggests that the belief in the superiority of statistical functions for these purposes may be ill founded.     

The dependence of all-atom statistical potentials on structural training database

An accurate statistical energy function that is suitable for the prediction of protein structures of all classes should be independent of the structural database used for energy extraction. Here, two high-resolution, low-sequence-identity structural databases of 333 alpha-proteins and 271 beta-proteins were built for examining the database dependence of three all-atom statistical energy functions. They are RAPDF (residue-specific all-atom conditional probability discriminatory function), atomic KBP (atomic knowledge-based potential), and DFIRE (statistical potential based on distance-scaled finite ideal-gas reference state). These energy functions differ in the reference states used for energy derivation. The energy functions extracted from the different structural databases are used to select native structures from multiple decoys of 64 alpha-proteins and 28 beta-proteins. The performance in native structure selections indicates that the DFIRE-based energy function is mostly independent of the structural database whereas RAPDF and KBP have a significant dependence. The construction of two additional structural databases of alpha/beta and alpha + beta-proteins further confirmed the weak dependence of DFIRE on the structural databases of various structural classes. The possible source for the difference between the three all-atom statistical energy functions is that the physical reference state of ideal gas used in the DFIRE-based energy function is least dependent on the structural database.     

Scoring function accuracy for membrane protein structure prediction

We perform a systematic examination of the ability of several different high-resolution, atomic-detail scoring functions to discriminate native conformations of loops in membrane proteins from non-native but physically reasonable, or "decoy," conformations. Decoys constructed from changing a loop conformation while keeping the remainder of the protein fixed are a challenging test of energy function accuracy. Nevertheless, the best of the energy functions we examined recognized the native structure as lowest in energy around half the time, and consistently chose it as a low-energy structure. This suggests that the best of present energy functions, even without a representation of the lipid bilayer, are of sufficient accuracy to give reasonable confidence in predictions of membrane protein structure. We also constructed homology models for each structure, using other known structures in the same protein family as templates. Homology models were constructed using several scoring functions and modeling programs, but with a comparable sampling effort for each procedure. Our results indicate that the quality of sequence alignment is probably the most important factor in model accuracy for sequence identity from 20-40%; one can expect a reasonably accurate model for membrane proteins when sequence identity is greater than 30%, in agreement with previous studies. Most errors are localized in loop regions, which tend to be found outside the lipid bilayer. For the most discriminative energy functions, it appears that errors are most likely due to lack of sufficient sampling, although it should be stressed that present energy functions are still far from perfectly reliable.     

Effective knowledge‐based potentials

Empirical or knowledge‐based potentials have many applications in structural biology such as the prediction of protein structure, protein–protein, and protein–ligand interactions and in the evaluation of stability for mutant proteins, the assessment of errors in experimentally solved structures, and the design of new proteins. Here, we describe a simple procedure to derive and use pairwise distance‐dependent potentials that rely on the definition of effective atomic interactions, which attempt to capture interactions that are more likely to be physically relevant. Based on a difficult benchmark test composed of proteins with different secondary structure composition and representing many different folds, we show that the use of effective atomic interactions significantly improves the performance of potentials at discriminating between native and near‐native conformations. We also found that, in agreement with previous reports, the potentials derived from the observed effective atomic interactions in native protein structures contain a larger amount of mutual information. A detailed analysis of the effective energy functions shows that atom connectivity effects, which mostly arise when deriving the potential by the incorporation of those indirect atomic interactions occurring beyond the first atomic shell, are clearly filtered out. The shape of the energy functions for direct atomic interactions representing hydrogen bonding and disulfide and salt bridges formation is almost unaffected when effective interactions are taken into account. On the contrary, the shape of the energy functions for indirect atom interactions (i.e., those describing the interaction between two atoms bound to a direct interacting pair) is clearly different when effective interactions are considered. Effective energy functions for indirect interacting atom pairs are not influenced by the shape or the energy minimum observed for the corresponding direct interacting atom pair. Our results suggest that the dependency between the signals in different energy functions is a key aspect that need to be addressed when empirical energy functions are derived and used, and also highlight the importance of additivity assumptions in the use of potential energy functions.     

Potential energy functions for protein design

Different potential energy functions have predominated in protein dynamics simulations, protein design calculations, and protein structure prediction. Clearly, the same physics applies in all three cases. The differences in potential energy functions reflect differences in how the calculations are performed. With improvements in computer power and algorithms, the same potential energy function should be applicable to all three problems. In this review, we examine energy functions currently used for protein design, and look to the molecular mechanics field for advances that could be used in the next generation of design algorithms. In particular, we focus on improved models of the hydrophobic effect, polarization and hydrogen bonding.     

Discrimination power of knowledge-based potential dictated by the dominant energies in native protein structures

Amino Acids - Extracting a well-designed energy function is important for protein structure evaluation. Knowledge-based potential functions are one type of the energy functions which can be...     

Effective energy functions for protein structure prediction

Protein structure prediction, fold recognition, homology modeling and design rely mainly on statistical effective energy functions. Although the theoretical foundation of such functions is not clear, their usefulness has been demonstrated in many applications. Molecular mechanics force fields, particularly when augmented by implicit solvation models, provide physical effective energy functions that are beginning to play a role in this area.     

Thermodynamics of protein folding: a microscopic view

Statistical thermodynamics provides a powerful theoretical framework for analyzing, understanding and predicting the conformational properties of biomolecules. The central quantity is the potential of mean force or effective energy as a function of conformation, which consists of the intramolecular energy and the solvation free energy. The intramolecular energy can be reasonably described by molecular mechanics-type functions. While the solvation free energy is more difficult to model, useful results can be obtained with simple approximations. Such functions have been used to estimate the intramolecular energy contribution to protein stability and obtain insights into the origin of thermodynamic functions of protein folding, such as the heat capacity. With reasonable decompositions of the various energy terms, one can obtain meaningful values for the contribution of one type of interaction or one chemical group to stability. Future developments will allow the thermodynamic characterization of ever more complex biological processes.     

Critical assessment of quantum mechanics based energy restraints in protein crystal structure refinement

A critical evaluation of the performance of X-ray refinement protocols using various energy functions is presented using the bovine pancreatic trypsin inhibitor (BPTI) protein. The four potential energy functions we explored include: (1) fully quantum mechanical calculations; (2) one based on an incomplete molecular mechanics (MM) energy function employed in the Crystallography and NMR System (CNS) with empirical parameters developed by Engh and Huber (EH), which lacks electrostatic and attractive van der Waals terms; (3) one based on a complete MM energy function (AMBER ff99 parameter set); and (4) the same as 3, with the addition of a Generalized Born (GB) implicit solvation term. The R, R (free), real space R values of the refined structures and deviations from the original experimental structure were used to assess the relative performance. It was found that at 1 Angstrom resolution the physically based energy functions 1, 3, and 4 performed better than energy function 2, which we attribute to the better representation of key interactions, particularly electrostatics. The observed departures from the experimental structure were similar for the refinements with physically based energy functions and were smaller than the structure refined with EH. A test refinement was also performed with the reflections truncated at a high-resolution cutoff of 2.5 Angstrom and with random perturbations introduced into the initial coordinates, which showed that low-resolution refinements with physically based energy functions held the structure closer to the experimental structure solved at 1 Angstrom resolution than the EH-based refinements.     

A contact energy function considering residue hydrophobic environment and its application in protein fold recognition

The three-dimensional （3D） structure prediction of proteins ：is an important task in bioinformatics. Finding energy functions that can better represent residue-residue and residue-solvent interactions is a crucial way to improve the prediction accu- racy. The widely used contact energy functions mostly only consider the contact frequency between different types of residues; however, we find that the contact frequency also relates to the residue hydrophobic environment. Accordingly, we present an improved contact energy function to integrate the two factors, which can reflect the influence of hydrophobic interaction on the stabilization of protein 3D structure more effectively. Furthermore, a fold recognition （threading） approach based on this energy function is developed. The testing results obtained with 20 randomly selected proteins demonstrate that, compared with common contact energy functions, the proposed energy function can improve the accuracy of the fold template prediction from 20% to 50%, and can also improve the accuracy of the sequence-template alignment from 35% to 65%.     

Stability analysis of predator-prey models via the liapunov method

As is well-known from the classical applications in the electrical and mechanical sciences, energy is a suitable Liapunov function: thus, by analogy, all energy functions proposed in ecology are potential Liapunov functions. In this paper, a generalized Lotka-Volterra model is considered and the stability properties of its non-trivial equilibrium are studied by means of an energy function first proposed by Volterra in the context of conservative ecosystems. The advantage of this Liapunov function with respect to the one that can be induced through linearization is also illustrated.     

The energy computation paradox and ab initio protein folding

The routine prediction of three-dimensional protein structure from sequence remains a challenge in computational biochemistry. It has been intuited that calculated energies from physics-based scoring functions are able to distinguish native from nonnative folds based on previous performance with small proteins and that conformational sampling is the fundamental bottleneck to successful folding. We demonstrate that as protein size increases, errors in the computed energies become a significant problem. We show, by using error probability density functions, that physics-based scores contain significant systematic and random errors relative to accurate reference energies. These errors propagate throughout an entire protein and distort its energy landscape to such an extent that modern scoring functions should have little chance of success in finding the free energy minima of large proteins. Nonetheless, by understanding errors in physics-based score functions, they can be reduced in a post-hoc manner, improving accuracy in energy computation and fold discrimination.     

Timing of potential and metabolic brain energy

The temporal relationship between cerebral electro-physiological activities, higher brain functions and brain energy metabolism is reviewed. The duration of action potentials and transmission through glutamate and GABA are most often less than 5 ms. Subjects may perform complex psycho-physiological tasks within 50 to 200 ms, and perception of conscious experience requires 0.5 to 2 s. Activation of cerebral oxygen consumption starts after at least 100 ms and increases of local blood flow become maximal after about 1 s. Current imaging technologies are unable to detect rapid physiological brain functions. We introduce the concepts of potential and metabolic brain energy to distinguish trans-membrane gradients of ions or neurotransmitters and the capacity to generate energy from intra- or extra-cerebral substrates, respectively. Higher brain functions, such as memory retrieval, speaking, consciousness and self-consciousness are so fast that their execution depends primarily on fast neurotransmission (in the millisecond range) and action-potentials. In other words: brain functioning requires primarily maximal potential energy. Metabolic brain energy is necessary to restore and maintain the potential energy.     

Accurate ab initio calculation of the Ar–CF4 intermolecular potential energy surface

Ar–CF₄ intermolecular interaction potential is studied by ab initio calculations at the MP2 and CCSD(T) levels of theory containing the so-called bond functions ({3s3p2d1f} basis set was chosen) both with and without a correction for the basis-set superposition error. The calculations were performed with Dunning's correlation consistent basis sets (aug-cc-pVXZ, X = D, T, Q, 5) to extrapolate the Ar–CF₄ potential energy minimum and intermolecular distance to their complete basis set (CBS) limits. It is shown that the addition of bond functions results in a dramatic improvement in the convergence of the calculated interaction energies at the MP2/aug-cc-pVTZ level. The MP2/{3s3p2d1f}-aug-cc-pVTZ potential energy surface even approaches the CCSD(T)/aug-cc-pVQZ potential energy surface. The potential energy minima and the intermolecular distances are both significantly closer to the CBS limit when using the bond functions, and it implies that adding bond functions in the calculation has a great effect on the interaction energies. We also find that with bond functions included in the CCSD(T)/aug-cc-pVDZ model chemistry, the potential energy minima are extremely close to the CBS limit and are better than the CCSD(T)/aug-cc-pVQZ values. Several levels of theory described in the text were used to determine pairwise analytic potential energy surfaces for Ar+CF₄. The analytic potential energy surfaces are in very good agreement with the ab initio values.     

Evaluating CASP4 predictions with physical energy functions

Physical energy scoring functions based on implicit solvation models are tested by evaluating predictions from the most recent CASP4 competition. The best performing scoring functions are identified along with the best protocol for preparing structures before energies are evaluated. Ranking of structures with the best scoring functions is compared across CASP4 targets to establish when physical scoring functions can be expected to reliably distinguish structures that are most similar to the native fold in a set of misfolded or unfolded protein conformations. The results are used to interpret previous studies where scoring functions were tested on the standard decoy sets by Park, Levitt, and Baker. We show that the best physical scoring functions can be applied successfully in automated consensus scoring applications where a single best conformation has to be selected from a set of structures from different sources. Finally, the potential for better protein structure scoring functions is discussed with a suggestion for an empirically parameterized linear combination of energy components.     

Role of AMPK in mammals reproduction: Specific controls and whole-body energy sensing

AMP-activated protein kinase (AMPK) is a key enzyme involved in linking the energy sensing to metabolic pathways. As such, it plays a central role at the whole-body level to translate endocrine communications into adapted responses aimed either at saving energy when food is scarce or at allocating it to various functions, particularly reproduction, when food is available. AMPK also plays major roles in the energy individual cells use in order to realize their specific functions. This is of course especially true for all cells involved in the reproductive function (gonads, gametes) or in its control (hypothalamus, pituitary). In the present review, I report a survey of the various roles of AMPK functions in reproduction, either directly in reproductive organs, or indirectly in organs controlling reproduction, particularly at hypothalamus level.