A sequence alignment algorithm with an arbitrary gap penalty function.

An algorithm for aligning biological sequences is presented that is an adaptation of the sequence generating function approach used in the statistical mechanics of biopolymers. This algorithm uses recursion relationships developed from a partition function formalism of alignment probabilities. It is implemented within a dynamic programming format that closely resembles the forward algorithm used in hidden Markov models (HMM). The algorithm aligns sequences or structures according to the statistically dominant alignment path and will be referred to as the SDP algorithm. An advantage of this method over previous ones is that it allows more complicated and physically realistic gap penalty functions to be incorporated into the algorithm in a facile manner. The performance of this algorithm in a case study of aligning the heavy and light chain from the variable region of an immunoglobulin is investigated.

Dynamic models of gene expression and classification.

Powerful new methods, like expression profiles using cDNA arrays, have been used to monitor changes in gene expression levels as a result of a variety of metabolic, xenobiotic or pathogenic challenges. This potentially vast quantity of data enables, in principle, the dissection of the complex genetic networks that control the patterns and rhythms of gene expression in the cell. Here we present a general approach to developing dynamic models for analyzing time series of whole genome expression. In this approach, a self-consistent calculation is performed that involves both linear and non-linear response terms for interrelating gene expression levels. This calculation uses singular value decomposition (SVD) not as a statistical tool but as a means of inverting noisy and near-singular matrices. The linear transition matrix that is determined from this calculation can be used to calculate the underlying network reflected in the data. This suggests a direct method of classifying genes according to their place in the resulting network. In addition to providing a means to model such a large multivariate system this approach can be used to reduce the dimensionality of the problem in a rational and consistent way, and suppress the strong noise amplification effects often encountered with expression profile data. Non-linear and higher-order Markov behavior of the network are also determined in this self-consistent method. In data sets from yeast, we calculate the Markov matrix and the gene classes based on the linear-Markov network. These results compare favorably with previously used methods like cluster analysis. Our dynamic method appears to give a broad and general framework for data analysis and modeling of gene expression arrays.

Information dynamics of in vitro selection-amplification systems.

Selection-amplifications systems provide a means of engineering biomacromolecules with new properties. The combination of stringent functional selection with the ability to amplify single molecules confers great specificity on the evolving population. Yet such systems like many complicated chemical kinetic mechanisms can show a range of unstable and metastable behavior. These instabilities can be investigated using the Shannon entropy of the evolving population. It is shown that the Shannon entropy provides a Lyapounov function for exploring dynamic stability. A simple model of in vitro evolution is presented and stability conditions are established. It is seen that fairly simple directed evolution models can exhibit a range of dynamical behavior.

The genotypic landscape during in vitro evolution of a catalytic RNA: implications for phenotypic buffering.

The Tetrahymena group I ribozyme catalyzes the cleavage of a phosphodiester linkage in specific sequences of RNA. This phenotype can be used in an in vitro selection-amplification process to evolve variants that are capable of RNA catalysis in the presence of Ca(2+) as the sole available cation. With sufficient genotypic characterization of the population as it evolves, we have a rare opportunity of observing how the information stored in an evolving population responds to selective pressures, such as the requisite of catalyzing RNA cleavage in the absence of Mg(2+) or Mn(2+). In the present work, we examine the population dynamics of this system using sequence information from previous experimental work. We focus on two issues: How does the information content of the population evolve? and Is the system evolving as an adaptive walk on a rugged landscape? To investigate these questions, information theoretical parameters are examined. The evolution of the population is visualized by mapping the genotypic frequency distribution onto a two-dimensional projection of sequence space. The projection was generated using Hamming distances from the wild-type, starting sequence and a catalytically successful, evolved sequence. The evolution of the information content of the system was measured by calculating the grammar complexity of the observed sequences, which showed a very slight increase over 12 generations. This result is consistent with the system performing a search for a local optimum. The dynamics of the population in this sequence space is consistent with an adaptive walk on an uncorrelated, or "rugged," genotypic landscape, despite the observation that the phenotypic progress of the population appears smooth. The relative insensitivity of the phenotypic landscape to the variegation of the genotypic landscape suggests that the former is buffered against variation in the latter through various epigenetic-like mechanisms.

MDL and the statistical mechanics of protein potentials.

The combination of a wealth of structural data and impressive computational power provides detailed information pertaining to the structure and dynamics of biomacromolecules. A natural inclination is to incorporate this information into models to gain added predictive power on protein folding and stability. There has been considerable recent interest in developing "knowledge-based" potentials to describe internal interactions in proteins. In these approaches, probability distribution functions are inferred from existing knowledge. A common assumption has been the "quasi-chemical approximation" or "Boltzmann device". This method relates statistical mechanical probabilities to observed frequencies. The validity of this approach is discussed in detail from a statistical mechanics perspective. Because statistical mechanics is a form of statistical inference based on a lack of knowledge of the system, the "Boltzmann device" does not have a rigorous theoretical justification. In the present work, a statistical mechanics based on partial knowledge of the system is employed. This statistical mechanical scheme uses the minimum description length (MDL) of phase space as its main tool. With this approach, "knowledge-based" potentials can be derived in a rigorous fashion. In practical calculations, these potentials are best obtained using Bayesian inference methods similar to those used in image reconstruction.

Statistical mechanics of protein sequences.

A statistical mechanical treatment of biological macromolecules is presented that includes the sequence information as an internal coordinate. Using a path integral representation, the canonical partition function can be represented as a product of a polymer configurational path integral and a sequence walk path integral. In most biological instances, the sequence composition influences the potential energy of intersubunit interaction. Consequently, the two path integrals are not separable, but rather "interact" via a sequence-dependent configurational potential. In proteins and RNA, the sequence walk occurs in dimensions greater than three and, therefore, will be an ideal "polymer." The Markovian nature of this walk can be exploited to show that all the structural information is contained in the sequence. This latter effect is a result of the dimensionality of the sequence walk and is not necessarily a result of biological optimization of the system.

Non-equilibrium thermodynamics of molecular evolution.

The evolution of the information complexity of a large database of protein sequences is investigated. The information entropy for protein sequences is determined from their algorithmic complexity and is found to change with evolutionary time at a constant rate. The information content of changed residues is always lower than the content of conserved residues. This indicates that sequences are becoming less random throughout evolution. It also shows that the system is being driven toward minimal complexity production. The change in information content per amino acid substitution is virtually identical for all the protein sequences studied. These results are interpreted with a statistical mechanical theory that ties sequence information to the thermodynamics of protein structure. Sequence evolution is viewed as a means to drive the system to minimum thermodynamic entropy production in a stable, non-equilibrium state. This theory provides a physical framework for understanding molecular evolution and incorporates features of both the neutralist and selectionist models.

The Shannon information entropy of protein sequences.

A comprehensive data base is analyzed to determine the Shannon information content of a protein sequence. This information entropy is estimated by three methods: a k-tuplet analysis, a generalized Zipf analysis, and a "Chou-Fasman gambler." The k-tuplet analysis is a "letter" analysis, based on conditional sequence probabilities. The generalized Zipf analysis demonstrates the statistical linguistic qualities of protein sequences and uses the "word" frequency to determine the Shannon entropy. The Zipf analysis and k-tuplet analysis give Shannon entropies of approximately 2.5 bits/amino acid. This entropy is much smaller than the value of 4.18 bits/amino acid obtained from the nonuniform composition of amino acids in proteins. The "Chou-Fasman" gambler is an algorithm based on the Chou-Fasman rules for protein structure. It uses both sequence and secondary structure information to guess at the number of possible amino acids that could appropriately substitute into a sequence. As in the case for the English language, the gambler algorithm gives significantly lower entropies than the k-tuplet analysis. Using these entropies, the number of most probable protein sequences can be calculated. The number of most probable protein sequences is much less than the number of possible sequences but is still much larger than the number of sequences thought to have existed throughout evolution. Implications of these results for mutagenesis experiments are discussed.

Multifractals, encoded walks and the ergodicity of protein sequences.

A variety of statistical methods have been developed to explore correlations in protein and nucleic acid sequences. Such correlations have important implications for the evolution and stability of these macromolecules. Recently, a number of fractal analyses of sequence data have been developed. These analyses have considerable appeal as they are extremely sensitive to long range correlations and to hierarchical structures. One such analysis decodes sequence information into a random walk and the statistics of the resulting random walk is investigated. Anomalous scaling of such walks has been interpreted as indicative of a fractal structure. Alternatively, a generalized box counting analysis of decoded sequences can be used to establish multifractal properties. In this work, the connection between these two seemingly disparate approaches is established. This connection is exploited to investigate correlations in protein sequences. An ensemble consisting of a comprehensive data set of representative protein sequences is analyzed to establish the ergodicity of protein sequences. The implications of this ergodicity for information theoretical approaches to protein structure prediction is explored.

Fractal analysis of proton exchange kinetics in lysozyme.

Experimental data for the exchange of protons in tritiated lysozyme is reexamined by using a fractal model. The fraction of protons unexchanged, f, is seen to follow a stretched exponential, f infinity exp[(-t/tau)alpha], in the long time limit. Data over a range of temperatures are considered, and accurate fits are obtained with a single, unadjusted scaling exponent, alpha. The time constant, tau, follows an Arrhenius law and gives an activation energy comparable to that obtained for free peptide exchange. A model is proposed where proton exchange occurs as a result of solvent reacting with protein side groups in a restricted volume surrounding the protein. Dynamic fluctuations of the protein allow the protonated groups to enter this volume. Solvent also penetrates this volume, allowing proton exchange to occur. The fluctuations of reactants in this restricted volume dominate the kinetics and result in anomalous behavior. The topology of this reaction volume can be characterized by its fractal dimension. The fractal dimension of the space excluded by the protein is equal to 3-ds, where ds is the fractal dimension of the protein surface. The dimensionality of this "reaction space" can be used to predict the value of the exponent alpha. When the problem is treated as a reaction of the type A + B-->C in a confined region, the exponent is given by alpha = (3-ds)/4. By using the value of 2.17 previously established for the surface dimension of lysozyme [Pfeifer, P., Welz, U. & Wippermann, H. (1985) Chem. Phys. Lett. 113, 535-540], a corresponding alpha of 0.21 is calculated. Data for lysozyme at six different temperatures could be accurately fit by using this unadjusted value for alpha. These results show how the surface morphology of a protein influences diffusional processes of small molecules associating with the protein.

Fluorescence labeling of the palmitoylation sites of rhodopsin.

Two tandem cysteine residues in the carboxyl-terminal region of rhodopsin have been shown to be covalently linked to palmitate via thioester bonds (Ovchinnikov, Y. A., et al. (1988) FEBS Lett. 230, 1-5). We have synthesized a fluorescent analogue of palmitoyl coenzyme A (16-(9-anthroyloxy)hexadecanoyl coenzyme A ester) and incorporated the fluorescent derivative of palmitate into the protein in high yield (> 40%) through pretreatment of bovine rod outer segments with 1 M hydroxylamine and subsequent incubation with the fluorescent label. Covalent incorporation of label into protein was demonstrated by SDS-polyacrylamide gel electrophoresis. Proteolytic digestion of labeled rhodopsin in the disc membrane with papain and thermolysin verified the C-terminal location of the label. Treatment of SDS-solubilized, labeled rod outer segments with 10% beta-mercaptoethanol provided evidence that partial depalmitoylation may induce the formation of rhodopsin aggregates. Labeled, unbleached rhodopsin was purified by chromatography over hydroxyapatite and concanavalin A-agarose and reconstituted into dimyristoylphosphatidylcholine vesicles. SDS gels of the rhodopsin vesicle preparation verified that all unbound fluorescent label had been removed and that the thioester bond linking probe to protein was not labile.

Fluorescence studies of the location and membrane accessibility of the palmitoylation sites of rhodopsin.

Fluorescent fatty acid labels have been incorporated into the palmitoylation sites of rhodopsin and used to probe the membrane accessibility and location of these sites. The fluorescence properties of anthroyloxy and pyrenyl fatty acids bound to rhodopsin were investigated in a reconstituted vesicle system. Collisional quenching of fluorescence by stearic acid (DSA) labeled with doxyls in the 16, 12, and 5 positions was used to determine the membrane accessibility and disposition of the modifying fatty acids. To properly determine the membrane concentration of these quenchers, the dependence of the Stern-Volmer parameters on both quencher and vesicle concentration was determined. An analysis of these dependences provided a correction for partitioning of the quencher between the aqueous phase and the membrane. After this correction, the relative effectiveness of doxyl quenchers was 16-DSA > 12-DSA > 5-DSA. Parallel studies on free anthroyloxy and pyrenyl fatty acids incorporated into the reconstituted system showed the same dependence on quencher position. These results indicate that the labels at the palmitoylation sites of rhodopsin are situated in the membrane much as a free fatty acid. This anchoring of the palmitates in the membrane results in the formation of a fourth cytoplasmic loop.

Room temperature trapping of rhodopsin photointermediates.

By suspending bovine rhodopsin in trehalose-water glass films, it is possible to trap photostates in the light-activation process. Because of the unusually high vitrification temperature of trehalose-water mixtures, this trapping can be accomplished at room temperature. This allows for a facile investigation of the spectroscopic properties of rhodopsin's photointermediates. Depending on experimental conditions, it is possible to trap photolysis products that have visible absorbance spectra closely resembling the two different photointermediates, metarhodopsin I and metarhodopsin II. When rhodopsin is maintained in the native rod outer segment membrane, the photolysis product has the spectral properties of metarhodopsin I. Upon detergent solubilization, the photolysis product closely resembles metarhodopsin II. Ultraviolet circular dichroism spectra show that the metarhodopsin I product had no change in secondary structure compared with unbleached rhodopsin. The metarhodopsin II product did show a significant decrease in alpha-helical content. Resonance energy transfer was measured from extrinsic probes located on each of the cytoplasmic cysteine residues to the retinal in the trapped photoproducts. It is seen that these distances are the same for rhodopsin and metarhodopsin I while metarhodopsin II shows considerably shorter distances. Metarhodopsin II is intimately associated with the signal transduction process, and the present results suggest that large structural changes have occurred in the transition to this state. These results demonstrate the utility of room temperature trapping of photostates in trehalose-water glasses.

Effects of depalmitoylation on physicochemical properties of rhodopsin.

In an effort to determine the functionality of palmitoylation in rhodopsin, a number of physicochemical properties of depalmitoylated rhodopsin were monitored. Approximately 70% of the rhodopsin was depalmitoylated in rod outer segments by a mild hydroxylamine treatment that resulted in minimal bleaching of rhodopsin. Subsequent purification by affinity chromatography could be used to remove hydroxylamine-bleached rhodopsin. Parallel physical studies were performed on both purified, detergent-solubilized rhodopsin and rhodopsin in rod outer segments. No effect was seen on the rate of metarhodopsin II formation for depalmitoylated rhodopsin. A small effect was seen in the biphasic behavior of the rate of retinal regeneration. The circular dichroism spectrum of depalmitoylated, purified rhodopsin was virtually identical to that of the native protein. These results suggest that depalmitoylation does not greatly affect the conformational structure of rhodopsin. Circular dichroism at 222 nm was used to monitor the thermal denaturation of depalmitoylated and native rhodopsin. A small but significant decrease in the in rod outer segments. In both cases, the van't Hoff parameters showed an increase in positive enthalpy for denaturation relative to the native state. This is largely counterbalanced by an increase in positive entropy relative to the native states. The circular dichroism of the "denatured" state showed a high alpha-helix content. Depalmitoylated rhodopsin had a lower helix content than native protein in this high-temperature state. The changes in the thermodynamics upon depalmitoylation were attributed to structural changes in the denatured state.

Protein dynamics and 1/f noise.

It has long been recognized that protein dynamical processes occur over a wide temporal range. However, the functionality of this spectrum of events remains unclear. In this work, a generalized noise function analysis is applied to a collection of diverse protein dynamical systems. It is shown that a power law model with an oscillatory component can adequately describe the time course of a variety of processes. These results suggest that under the appropriate conditions, proteins are in a metastable state. A microscopic, chemical kinetic model based on a Poisson distribution of activation energies is presented. From this model specific functional forms for the parameters of the generalized noise model can be derived. Additionally, a model is presented to described kinetic hole burning effects observed at low temperatures. Scaling laws are derived for these models that provide a connection with the generalized noise analysis.

Activation parameters for the halorhodopsin photocycle: a phase lifetime spectroscopic study of the 520- and 640-nanometer intermediates.

Phase lifetime spectroscopy is used to investigate the kinetics of the 520- and 640-nm intermediates in the halorhodopsin photocycle. These intermediates decay on the millisecond time scale and are strongly implicated in the chloride transport steps. The temperature dependence of the 520 and 640 relaxations was measured for chloride and nitrate buffers at pH 6, 7, and 8 and for iodide buffer at pH 6. The 640 relaxations have small activation energies but large entropy barriers. The two relaxation times observed for the 640 intermediate were interpreted by using a mechanism in which two 640 species exist in equilibrium. The second 640 species is not along the main decay path for the photocycle. A quantitative analysis of the data allowed rate constants and activation parameters to be calculated for the elementary steps of this isomerization process. These parameters are similar for both chloride and nitrate buffers but differ somewhat in iodide. The derived calculated rate constants were consistent with the relaxation times observed for the 520 intermediate. These results indicate that the 520 and two 640 intermediates have very similar free energies as well as similar free energies of activation for the various interconversion processes.

Determination of the fractal dimension of membrane protein aggregates using fluorescence energy transfer.

It is demonstrated that fluorescence resonance energy transfer may be used to determine the fractal dimension of aggregates of membrane-bound proteins. Theoretical and experimental results are presented for two different experimental designs: energy transfer between proteins and energy transfer from lipids to proteins. For energy transfer between proteins the lattice spacing must be known independently for a fractal dimension to be uniquely determined, and this represents a disadvantage to this experimental design. Results are presented for the calcium ATPase and a fractal dimension of 1.9 is estimated for ATPase aggregates by assuming a lattice spacing of 50 A. Energy transfer from lipids to protein provides a means of estimating the length of the "coast-line" of the aggregate. In this case the fractal dimension is uniquely determined from a log-log plot. An analysis of data for bacteriohodopsin reconstituted in phospholipid vesicles gives a fractal dimension of 1.6. The structural basis of the value for the fractal dimension is discussed for these two systems. These techniques provide a means of assessing the nature of protein-protein interactions in membranous systems.

Circular dichroism of halorhodopsin: comparison with bacteriorhodopsin and sensory rhodopsin I.

Circular dichroic (CD) spectra of three related protein pigments from Halobacterium halobium, halorhodopsin (HR), bacteriorhodopsin (BR), and sensory rhodopsin I (SR-I), are compared. In native membranes the two light-driven ion pumps, HR and BR, exhibit bilobe circular dichroism spectra characteristic of exciton splitting in the region of retinal absorption, while the phototaxis receptor, SR-I, exhibits a single positive band centered at the SR-I absorbance maximum. This indicates specific aggregation of protein monomers of HR, as previously noted [Sugiyama, Y., & Mukohata, Y. (1984) J. Biochem. (Tokyo) 96, 413-420], similar to the well-characterized retinal/retinal exciton interaction in the purple membrane. The absence of this interaction in SR-I indicates SR-I is present in the native membrane as monomers or that interactions between the retinal chromophores are weak due to chromophore orientation or separation. Solubilization of HR and BR with nondenaturing detergents eliminates the exciton coupling, and the resulting CD spectra share similar features in all spectral regions from 250 to 700 nm. Schiff-base deprotonation of both BR and HR yields positive CD bands near 410 nm and shows similar fine structure in both pigments. Removal of detergent restores the HR native spectrum. HR differs from BR in that circular dichroic bands corresponding to both amino acid and retinal environments are much more sensitive to external salt concentration and pH. A theoretical analysis of the exciton spectra of HR and BR that provides a range of interchromophore distances and orientations is performed.(ABSTRACT TRUNCATED AT 250 WORDS)

Phase-lifetime spectroscopy of photocycle processes: proton release and uptake kinetics of purple membrane.

Phase-lifetime spectroscopy was used to measure the kinetics of light-driven proton transients and M intermediate decay of purple membrane. The kinetics were measured in distilled water and in 0.4 M KCl over the pH range of 5.3-8.8. The low ionic strength data showed results that were comparable to previous data. The lifetimes for the decay of the proton transient were intermediate between the two M decay lifetimes. Deviations from this behavior occurred at pH 5.3. The high-salt data showed two proton transients: a release process with a lifetime similar to the fast M decay, and an uptake process with a lifetime similar to the slow M decay. The release process is not seen in flash experiments. This difference is due to differences in the amplitudes of the relaxation processes for the two types of experiments. The high-salt data showed little pH dependence below pH 7.8. Both sets of data were shown to be consistent with a single mechanism that correlates the M kinetics with the proton kinetics. The proposed mechanism consists of the parallel formation and decay of two M intermediates that can interconvert. Only one of the intermediates results in a proton being released from the bacteriorhodopsin. Thus, this mechanism has a nonproductive pathway as well as a productive pathway in which a single proton is pumped. Detailed kinetic expressions are derived for this mechanism for the phase-lifetime and flash kinetic experiment. These results could explain the qualitative features of the distilled water data and provide a quantitative description of the high-salt data. Expressions for the quantum yield and the H(+) to M ratio were derived in terms of the individual rate constants of the mechanism. Conditions are established in which the quantum yield for proton release can vary with only small changes in the M/H(+). This reconciles an apparent conflict in previous evidence from flash experiments and demonstrates the difficulty of determining stoichiometry from kinetic amplitude information.

Chemical relaxation in a chemiosmotic-coupled system: driving the calcium adenosinetriphosphatase with bacteriorhodopsin.

Phase-lifetime spectroscopy has been used to measure chemical relaxation processes in a chemiosmotic-coupled system. In this experiment the calcium ATPase and bacteriorhodopsin were coreconstituted into phospholipid vesicles. Upon illumination the bacteriorhodopsin pumps protons into the vesicles and forms a membrane potential. This membrane potential alters the activity of the internal calcium and thus perturbs the equilibrium of ATP hydrolysis/synthesis coupled to calcium transport. Mechanically chopping the actinic light provides a periodic perturbation to the system, and small response signals can be observed by using phase-sensitive detection. It is shown that this periodic perturbation occurs about a steady-state membrane potential that is independent of chopping frequency. The amplitude dispersion curve for the fluorescence of a calcium indicator was observed and analyzed in terms of the relaxation time for the ATPase-catalyzed calcium transport. Thus, this technique provides a method of measuring ion transport kinetics against a constant chemiosmotic potential. The calcium ATPase showed a single relaxation time on this time scale. The dependence of this relaxation time on ADP and phosphate concentration was measured and analyzed with a random sequential mechanism. This analysis gave dissociation constants for ADP and phosphate of 3.2 mM and 1.4 mM, respectively. These binding steps are followed by slow isomerization steps with forward and reverse rate constants (in the direction of ATP synthesis) of 67 s-1 and 227 s-1, respectively. These results demonstrate that highly accurate kinetic data can be obtained with this modulation relaxation technique.