In search of the protein native state with a probabilistic sampling approach

The three-dimensional structure of a protein is a key determinant of its biological function. Given the cost and time required to acquire this structure through experimental means, computational models are necessary to complement wet-lab efforts. Many computational techniques exist for navigating the high-dimensional protein conformational search space, which is explored for low-energy conformations that comprise a protein's native states. This work proposes two strategies to enhance the sampling of conformations near the native state. An enhanced fragment library with greater structural diversity is used to expand the search space in the context of fragment-based assembly. To manage the increased complexity of the search space, only a representative subset of the sampled conformations is retained to further guide the search towards the native state. Our results make the case that these two strategies greatly enhance the sampling of the conformational space near the native state. A detailed comparative analysis shows that our approach performs as well as state-of-the-art ab initio structure prediction protocols.     

Complete suboptimal folding of RNA and the stability of secondary structures

An algorithm is presented for generating rigorously all suboptimal secondary structures between the minimum free energy and an arbitrary upper limit. The algorithm is particularly fast in the vicinity of the minimum free energy. This enables the efficient approximation of statistical quantities, such as the partition function or measures for structural diversity. The density of states at low energies and its associated structures are crucial in assessing from a thermodynamic point of view how well-defined the ground state is. We demonstrate this by exploring the role of base modification in tRNA secondary structures, both at the level of individual sequences from Escherichia coli and by comparing artificially generated ensembles of modified and unmodified sequences with the same tRNA structure. The two major conclusions are that (1) base modification considerably sharpens the definition of the ground state structure by constraining energetically adjacent structures to be similar to the ground state, and (2) sequences whose ground state structure is thermodynamically well defined show a significant tendency to buffer single point mutations. This can have evolutionary implications, since selection pressure to improve the definition of ground states with biological function may result in increased neutrality.     

Metabolic Basis for the Self-Referential Genetic Code

An investigation of the biosynthesis pathways producing glycine and serine was necessary to clarify an apparent inconsistency between the self-referential model (SRM) for the formation of the genetic code and the model of coevolution of encodings and of amino acid biosynthesis routes. According to the SRM proposal, glycine was the first amino acid encoded, followed by serine. The coevolution model does not state precisely which the first encodings were, only presenting a list of about ten early assignments including the derivation of glycine from serine—this being derived from the glycolysis intermediate glycerate, which reverses the order proposed by the self-referential model. Our search identified the glycine-serine pathway of syntheses based on one-carbon sources, involving activities of the glycine decarboxylase complex and its associated serine hydroxymethyltransferase, which is consistent with the order proposed by the self-referential model and supports its rationale for the origin of the genetic code: protein synthesis was developed inside an early metabolic system, serving the function of a sink of amino acids; the first peptides were glycine-rich and fit for the function of building the early ribonucleoproteins; glycine consumption in proteins drove the fixation of the glycine-serine pathway.     

Structure optimization in a three-dimensional off-lattice protein model

We studied a three-dimensional off-lattice AB model with two species of monomers, hydrophobic (A) and hydrophilic (B), and present two optimization algorithms: face-centered-cubic (FCC)-lattice pruned-enriched-Rosenbluth method (PERM) and subsequent conjugate gradient (PERM++) minimization and heuristic conjugate gradient (HCG) simulation based on "off-trap" strategy. In PERM++, we apply the PERM to the FCC-lattice to produce the initial conformation, and conjugate gradient minimization is then used to reach the minimum energy state. Both algorithms have been tested in the three-dimensional AB model for all sequences with lengths 13 < or = n < or = 55. The numerical results show that the proposed methods are very promising for finding the ground states of proteins. In several cases, we renew the putative ground states energy values.     

Substrate ground state binding energy concentration is realized as transition state stabilization in physiological enzyme catalysis

Previously published kinetic data on the interactions of seventeen different enzymes with their physiological substrates are re-examined in order to understand the connection between ground state binding energy and transition state stabilization of the enzyme-catalyzed reactions. When the substrate ground state binding energies are normalized by the substrate molar volumes, binding of the substrate to the enzyme active site may be thought of as an energy concentration interaction; that is, binding of the substrate ground state brings in a certain concentration of energy. When kinetic data of the enzyme/substrate interactions are analyzed from this point of view, the following relationships are discovered: 1) smaller substrates possess more binding energy concentrations than do larger substrates with the effect dropping off exponentially, 2) larger enzymes (relative to substrate size) bind both the ground and transition states more tightly than smaller enzymes, and 3) high substrate ground state binding energy concentration is associated with greater reaction transition state stabilization. It is proposed that these observations are inconsistent with the conventional (Haldane) view of enzyme catalysis and are better reconciled with the shifting specificity model for enzyme catalysis.     

Selected-fit versus induced-fit protein binding: kinetic differences and mutational analysis

The binding of a ligand molecule to a protein is often accompanied by conformational changes of the protein. A central question is whether the ligand induces the conformational change (induced-fit), or rather selects and stabilizes a complementary conformation from a pre-existing equilibrium of ground and excited states of the protein (selected-fit). We consider here the binding kinetics in a simple four-state model of ligand-protein binding. In this model, the protein has two conformations, which can both bind the ligand. The first conformation is the ground state of the protein when the ligand is off, and the second conformation is the ground state when the ligand is bound. The induced-fit mechanism corresponds to ligand binding in the unbound ground state, and the selected-fit mechanism to ligand binding in the excited state. We find a simple, characteristic difference between the on- and off-rates in the two mechanisms if the conformational relaxation into the ground states is fast. In the case of selected-fit binding, the on-rate depends on the conformational equilibrium constant, whereas the off-rate is independent. In the case of induced-fit binding, in contrast, the off-rate depends on the conformational equilibrium, while the on-rate is independent. Whether a protein binds a ligand via selected-fit or induced-fit thus may be revealed by mutations far from the protein's binding pocket, or other "perturbations" that only affect the conformational equilibrium. In the case of selected-fit, such mutations will only change the on-rate, and in the case of induced-fit, only the off-rate.     

Valence tautomerism and macrocycle ruffling in nickel(III) porphyrins

Nonlocal density functional calculations with full geometry optimization have been carried out on the low-lying electronic states of oxidized nickel porphyrins. For [NiIII(P)(Py)2]+, the ground state corresponds to a t2g6(z2)1 configuration and the t2g6(x2-y2)1 configuration is 0.43 eV higher in energy. In contrast, the ground state of [NiIII(P)(CN)2]- corresponds to a t2g6(x2-y2)1 configuration, the t2g6(z2)1 configuration being 0.96 eV higher in energy. The results are consistent with EPR spectroscopic results on the TPP analogs of these complexes. For [Ni(P)(Py)2]+, the a2u- and a1u-type Ni(II) porphyrin cation radical states are higher in energy by 0.63 and 1.23 eV, respectively, relative to the t2g6(z2)1 Ni(III) ground state. The Ni-N(Porphyrin) distance is significantly shorter in [NiIII(P)(Py)2]+ (196 pm) than in [NiIII(P)(CN)2]- (206 pm), which is consistent with the ruffled and planar macrocycle conformations, respectively, in the two complexes.     

Protonation states and pH titration in the photocycle of photoactive yellow protein

Photoactive yellow protein (PYP) undergoes a light-driven cycle of color and protonation states that is part of a mechanism of bacterial phototaxis. This article concerns functionally important protonation states of PYP and the interactions that stabilize them, and changes in the protonation state during the photocycle. In particular, the chromophore pK(a) is known to be shifted down so that the chromophore is negatively charged in the ground state (dark state) even though it is buried in the protein, while nearby Glu46 has an unusually high pK(a). The photocycle involves changes of one or both of these protonation states. Calculations of pK(a) values and protonation states using a semi-macroscopic electrostatic model are presented for the wild-type and three mutants, in both the ground state and the bleached (I(2)) intermediate state. Calculations allowing multiple H-bonding arrangements around the chromophore also have been carried out. In addition, ground-state pK(a) values of the chromophore have been measured by UV-visible spectroscopy for the wild-type and the same three mutants. Because of the unusual protonation states and strong electrostatic interactions, PYP represents a severe test of the ability of theoretical models to yield correct calculations of electrostatic interactions in proteins. Good agreement between experiment and theory can be obtained for the ground state provided the protein interior is assumed to have a relatively low dielectric constant, but only partial agreement between theory and experiment is obtained for the bleached state. We also present a reinterpretation of previously published data on the pH-dependence of the recovery of the ground state from the bleached state. The new analysis implies a pK(a) value of 6.37 for Glu46 in the bleached state, which is consistent with other available experimental data, including data that only became available after this analysis. The new analysis suggests that signal transduction is modulated by the titration properties of the bleached state, which are in turn determined by electrostatic interactions. Overall, the results of this study provide a quantitative picture of the interactions responsible for the unusual protonation states of the chromophore and Glu46, and of protonation changes upon bleaching.     

On the photophysics of metallophthalocyanine-based photothermal sensitizers: synergism between theory and experiment

A comprehensive understanding of the factors governing the efficiency of metallophthalocyanine-based photothermal sensitizers requires the knowledge of their excited-state dynamics. This can only be properly gained when the nature and energy of the excited states (often spectroscopically silent) lying between the photogenerated state and the ground state are known. Here the excited state deactivation mechanism of two very promising metallophthalocyanine-based photothermal sensitizers, NiPc(OBu)(8) and NiNc(OBu)(8), is reviewed. It is shown that time dependent density functional theory (TDDFT) methods are capable to provide reliable information on the nature and energies of the low-lying excited states along the relaxation pathways. TDDFT calculations and ultrafast experiments consistently show that benzoannulation of the Pc ring modifies the photodeactivation mechanism of the photogenerated S(1)(pi,pi*) state by inducing substantial changes in the relative energies of the excited states lying between the S(1)(pi,pi*) state and the ground state.     

Measurement of the signs of methyl 13C chemical shift differences between interconverting ground and excited protein states by R 1ρ : an application to αB-crystallin

Carr-Purcell-Meiboom-Gill relaxation dispersion (CPMG RD) NMR spectroscopy has emerged as a powerful tool for quantifying the kinetics and thermodynamics of millisecond time-scale exchange processes involving the interconversion between a visible ground state and one or more minor, sparsely populated invisible 'excited' conformational states. Recently it has also become possible to determine atomic resolution structural models of excited states using a wide array of CPMG RD approaches. Analysis of CPMG RD datasets provides the magnitudes of the chemical shift differences between the ground and excited states, Δ?, but not the sign. In order to obtain detailed structural insights from, for example, excited state chemical shifts and residual dipolar coupling measurements, these signs are required. Here we present an NMR experiment for obtaining signs of (13)C chemical shift differences of (13)CH(3) methyl groups using weak field off-resonance R(1ρ) relaxation measurements. The accuracy of the method is established by using an exchanging system where the invisible, excited state can be converted to the visible, ground state by altering sample conditions so that the signs of Δ? values obtained from the spin-lock approach can be validated against those measured directly. Further, the spin-lock experiments are compared with the established H(S/M)QC approach for measuring signs of chemical shift differences and the relative strengths of each method are discussed. In the case of the 650 kDa human αB-crystallin complex where there are large transverse relaxation differences between ground and excited state spins the R(1ρ) method is shown to be superior to more 'traditional' experiments for sign determination.     

Characterization of low-lying excited states of proteins by high-pressure NMR

Hydrostatic pressure alters the free energy of proteins by a few kJ mol⁻¹, with the amount depending on their partial molar volumes. Because the folded ground state of a protein contains cavities, it is always a state of large partial molar volume. Therefore pressure always destabilises the ground state and increases the population of partially and completely unfolded states. This is a mild and reversible conformational change, which allows the study of excited states under thermodynamic equilibrium conditions. Many of the excited states studied in this way are functionally relevant; they also seem to be very similar to kinetic folding intermediates, thus suggesting that evolution has made use of the ‘natural’ dynamic energy landscape of the protein fold and sculpted it to optimise function. This includes features such as ligand binding, structural change during the catalytic cycle, and dynamic allostery.     

Experimental analyses of the chemical dynamics of ribozyme catalysis

Most ribozymes in Nature catalyze alcoholysis or hydrolysis of RNA phosphodiester bonds. Studies of the corresponding non-enzymatic reactions reveal a complex mechanistic landscape allowing for a variety of transition states and both concerted and stepwise mechanisms. High-resolution structures, incisive biochemical studies and computer simulations are providing glimpses into how ribozyme catalyzed reactions traverse this landscape. However, direct experimental tests of mechanistic detail at the chemical level are not easily achieved. Kinetic isotope effects (KIEs) probe directly the differences in the vibrational 'environment' of the atoms undergoing chemical transformation on going from the ground state to the transition state. Thus, KIEs can in principle provide direct information about transition state bonding and so may be instrumental in evaluating possible transition states for ribozyme catalyzed reactions. Understanding charge distribution in the transition state may help resolve how rate acceleration is accomplished and perhaps the similarities and differences in how RNA and protein active sites operate. Several barriers to successful application of KIE analysis to ribozymes have recently been overcome, and new chemical details are beginning to emerge.     

Defining Seaward Boundaries in a Domestic Context: Norway and the Svalbard Archipelago

Abstract

This article analyzes a little-noticed aspect of the international legal controversy pertaining to Svalbard’s maritime zones. It concerns where and by which method Norway should draw the boundaries between Svalbard’s continental shelf and the 200-mile zone, on the one hand, and other maritime zones subject to Norwegian jurisdiction, on the other. The assumption upon which the discussion rests is that the Spitsbergen Treaty signatories enjoy treaty rights in the maritime zones beyond Svalbard’s territorial waters. The law of the sea does not contain rules on the drawing of maritime boundaries between different parts of a state’s territory, but the rules on delimitation between states offer a strong analogy. In the search for an equitable solution, primacy should be given to geographical factors. The article argues that Norway could do more to enhance the openness regarding the Svalbard delimitation question since its outcome will be of significant interest to other states.     

Beyond the EX1 limit: probing the structure of high-energy states in protein unfolding

Hydrogen exchange kinetics in native solvent conditions have been used to explore the conformational fluctuations of an immunoglobulin domain (CD2.domain1). The global folding/unfolding kinetics of the protein are unaltered between pH 4.5 and pH 9.5, allowing us to use the pH-dependence of amide hydrogen/deuterium exchange to characterise conformational states with energies up to 7.2kcal/mol higher than the folded ground state. The study was intended to search for discreet unfolding intermediates in this region of the energy spectrum, their presence being revealed by the concerted exchange behaviour of subsets of amide groups that become accessible at a given free energy, i.e. the spectrum would contain discreet groupings. Protection factors for 58 amide groups were measured across the pH range and the hydrogen-exchange energy profile is described.More interestingly, exchange behaviour could be grouped into three categories; the first two unremarkable, the third unexpected. (1) In 33 cases, amide exchange was dominated by rapid fluctuation, i.e. the free energy difference between the ground state and the rapidly accessed open state is sufficiently low that the contribution from crossing the unfolding barrier is negligible. (2) In 18 cases exchange is dominated by the global folding transition barrier across the whole pH range measured. The relationship between hydroxyl ion concentration and observed exchange rate is hyperbolic, with the limiting rate being that for global unfolding; the so-called EX1 limit. For these, the free energy difference between the folded ground state and any rapidly-accessed open state is too great for the proton to be exchanged through such fluctuations, even at the highest pH employed in this study. (3) For the third group, comprising five cases, we observe a behaviour that has not been described. In this group, as in category 2, the rate of exchange reaches a plateau; the EX1 limit. However, as the intrinsic exchange rate (k(int)) is increased, this limit is breached and the rate begins to rise again. This unintuitive behaviour does not result from pH instability, rather it is a consequence of amide groups experiencing two processes; rapid fluctuation of structure and crossing the global barrier for unfolding. The boundary at which the EX1 limit is overcome is determined by the equilibrium distribution of the fluctuating open and closed states (K(O/C)) and the rate constant for unfolding (k(u)). This critical boundary is reached when k(int)K(O/C)=k(u). Given that, in a simple transition state formalism: k(u)=K(#)k' (where K(#) describes the equilibrium distribution between the transition and ground state and k' describes the rate of a barrierless rearrangement), it follows that if the pH is raised to a level where k(int)=k', then the entire free energy spectrum from ground state to transition state could be sampled.     

Simulated Evolutionary Optimization and Local Search: Introduction and Application to Tree Search

Tree search and its more complicated variant, tree search and simultaneous multiple DNA sequence alignment, are difficult NP-complete optimization problems, which require the application of advanced computational techniques, if large data sets are to be solved within reasonable computation times. Traditionally tree search has been attacked with a search strategy that is best described as multistart hill-climbing; local search by branch swapping has been performed on several different starting trees. Recently a different tree search strategy was tested in the Parsigal parsimony program, which used a combination of evolutionary optimization and local search. Evolutionary optimization algorithms use principles adopted from biological evolution to solve technical optimization tasks. Evolutionary optimization is a stochastic global search method, which means that the method is able to escape local optima, and is in principle able to produce any solution in the search space (although this may take a long time). Local search techniques, such as branch swapping, employ a completely different search strategy; they exploit local information maximally in order to achieve quick improvement in the value of the objective function. However, local search algorithms lack the ability to escape from local optima, which is a fundamental requirement for any search algorithm that aims to be able to discover the global optimum of a multimodal optimization problem. Hence it seems that an optimization strategy combining the good properties of both evolutionary algorithms and local search would be ideal. In this study, aspects of global optimization and local search are discussed, and the method of simulated evolutionary optimization is reviewed in detail. The application of simulated evolutionary optimization to tree search in Parsigal is then reviewed briefly.     

Catalytic efficiency, kinetic co-operativity of oligomeric enzymes and evolution

The catalytic performance of an enzyme, whether it is monomeric or oligomeric, depends on extra costs of energy in passing from the initial ground state to the various transition states, along the reaction co-ordinate. The improvement, during evolution, of the catalytic performance of individual subunits implies that three structural requirements are met in the course of an enzyme reaction: the unstrained enzyme subunits exist in the ground states under two conformations, one corresponding to the non-liganded state and the other to the liganded state; the inter-subunit strain is relieved in the various transition states; the subunits bound to the various transition states S not equal to, X not equal to and P not equal to have the same conformation. These structural requirements are precisely those which have been used to derive structural rate equations for polymeric enzymes. When subunits are loosely coupled, their arrangement controls the various rate constants, but not the extra costs of energy required to reach the various transition states. Moreover, one cannot expect the rate curve to display any sigmoidicity under these conditions. If subunits are tightly coupled and if the strained non-liganded and half-liganded states are destabilized with respect to the corresponding unstrained states, that is if they contain more conformational energy, the oligomeric enzyme is more catalytically efficient than the ideally isolated subunits. Moreover, if the available conformational energy of the half-liganded state is more than twice that of the non-liganded state, kinetic co-operativity is positive and the rate curve is sigmoidal. It is therefore the extent of inter-subunit strain in the half-liganded state which controls the appearance of sigmoidal kinetic behaviour. If subunits are tightly coupled but if inter-subunit strain is relieved in both the non-liganded and fully-liganded states, the half-liganded state controls both the catalytic efficiency of the enzyme and the sigmoidicity of the rate curve. Sigmoidicity and high catalytic efficiency are to be observed when this half-liganded state is destabilized relative to the corresponding unstrained state.(ABSTRACT TRUNCATED AT 400 WORDS)     

Ground state isomerization of a model green fluorescent protein chromophore

The relationship between ground state cis-trans isomerization and protonation state is explored for a model green fluorescent protein chromophore, 4-hydroxybenzylidene-1,2-dimethylimidazolinone (HBDI). We find that the protonation state has only a modest effect on the free energy differences between cis and trans isomers and on the activation energies for isomerization. Specifically, the experimental free energy differences are 3.3, 8.8, and 9.6 kJ/mol for cationic, neutral, and anionic forms of HBDI, respectively, and the activation energies are 48.9, 54.8, and 54.8 kJ/mol for cationic, neutral, and anionic forms, respectively. Furthermore, these activation energies are much smaller than might be expected based on comparison with similar systems. These results suggest that there may be a sub-population of the chromophore, which is nearly equally accessible to all three protonation states, through which thermal isomerization may proceed.     

Phenotypic plasticity in fluctuating environments: consequences of the lack of individual optimization

I consider the problem of characterizing the optimal plasticresponse when there are large-scale fluctuations in the environmentaffecting all population members. Individuals differ in theirstate, and each makes a reproductive decision before the environmentalconditions are known. An individual's state, its decision, andenvironmental conditions together determine the number of descendantsleft at the next decision epoch. I restrict attention to thesimple problem in which the state of the descendants left atthis epoch does not depend on these three factors. Because theenvironment is fluctuating, there is no individual optimization;instead the best action in one state implicitly depends on thebest action in other states. I characterize an optimal state-dependentstrategy, give a method of computation, and show how behaviorof each individual following the optimal strategy may be reinterpretedas a form of "individual optimization." Concepts are illustratedwith an example of optimal dutch size as a function of territoryquality.     

Structure optimization of the two-dimensional off-lattice hydrophobic–hydrophilic model

A two-dimensional off-lattice protein model with two species of monomers, hydrophobic and hydrophilic, was studied. Low-energy configurations in the model were optimized using the improved energy landscape paving (ELP+) method. In ELP+, the energy landscape paving (ELP) was first applied to search for the low-energy states. After the ELP led to the basins of the local energy minima, the additional degree-of-freedom of bond length was introduced, and the gradient descent method was then used to search for lower energy states near the local minima. Numerical results show that the proposed methods are quite effective for finding the ground states of proteins. A comparison between ELP+ and other methods is made.