Studies on protein folding,unfolding, and fluctuations by computer simulation. II. A. Three-dimensional lattice model of lysozyme

A three-dimensional lattice model of protein designed to assimilate lysozyme is introduced. An attractive interaction is assumed to work between preassigned specific pairs of units, when they occupy the nearest-nighbor lattice points. The behavior of this lattice lysozyme is studied by a Monte Carlo simulation method. Because of the specific interunit interactions,“native state” of the lattice lysozyme is stable at low temperatures. Conformational fluctuations in the native state are observed to occur at both termini and loop regions of the main chain existing on the surface. The process of unfolding and denatured states of this model are discussed. Complete refolding from a denatured state was not observed. However, by starting from partially folded structures, the native conformation could be attained. From these observation it is concluded that, in the process of folding of proteins as simplified in a lattice model, nulceation is a rate-limiting factor. The artificial character of this model and possible improvement are discussed.     

Development states associated with the floral transition.

Floral initiation can be analyzed from a developmental perspective by focusing upon how developmental fates are imprinted, remembered, and expressed. This is not an altogether new perspective, since people studying flowering have been concerned for a long time with the commitment of meristems to form flowers and the morphological, cellular, and molecular changes associated with this commitment. What is novel is the emphasis on developmental states as opposed to physiological processes. This developmental focus indicates that there appear to be at least three major developmental states that are acquired and expressed in the process of a meristem initiating floral morphogenesis. The meristem cells must first become competent to respond to a developmental signal that evokes them into a florally determined state. The leaves are the usual source of this signal and a specific leaf may or may not have the capacity to be inductively active. When a leaf does develop the capacity for inductive activity, this capacity is usually correlated with the ontogeny of the leaf. Inductive activity, however, may be continually expressed as in some day-neutral plants or may be latent as in plants where the photoperiod is the external cue for activity. Competent shoot apical meristems respond to inductive leaf signal by being evoked into a florally determined state. Under permissive conditions this florally determined state is expressed as the initiation of floral morphogenesis. Many mechanisms have evolved to regulate entry into and expression of these developmental states. As we learn more about the developmental states associated with flowering and how they are acquired and expressed, we will understand better how the various patterns of flowering are related to one another as well as which developmental processes are common to all angiosperms.     

Electron-conformational model of ryanodine receptor lattice dynamics

We propose a simple, physically reasonable electron-conformational model for the ryanodine receptor (RyR) and, on that basis, present a theory to describe RyR lattice responses to L-type channel triggering as an induced non-equilibrium phase transition. Each RyR is modelled with a single open and a single closed (electronic) state only, described utilizing a s=12 pseudospin approach. In addition to the fast electronic degree of freedom, the RyR channel is characterized by a slow classical conformational coordinate, Q, which specifies the RyR channel calcium conductance and provides a multimodal continuum of possible RyR states. The cooperativity in the RyR lattice is assumed to be determined by inter-channel conformational coupling. Given a threshold sarcoplasmic reticulum (SR) calcium load, the RyR lattice fires due to a nucleation process with a step-by-step domino-like opening of a fraction of lattice channels, providing for a sufficient release to generate calcium sparks. The optimal mode of RyR lattice functioning during calcium-induced calcium release implies a fractional release with a robust termination due to a decrease in SR calcium load, accompanied by a respective change in effective conformational strain of the lattice. SR calcium overload is shown to result in excitation of RyR lattice auto-oscillations with spontaneous RyR channel opening and closure.     

A proposed mechanism for the sol-gel transformation

The transformation from gel to sol in cell cytoplasm is treated as the transition from a lattice of macromolecules linked by Ca⁺⁺ ions to a random distribution of the macromolecules. The transition is a cooperative process, whose probability is expressed in terms of the theory of runs. The process is related to cell metabolism by the assumption that available Ca⁺⁺ concentration is regulated by metabolically produced endogenous chelating agents. Contribution 681 from the Division of Basic Health Sciences, Emory University.     

A triangle lattice model that predicts transmembrane helix configuration using a polar jigsaw puzzle

We developed a method of predicting the tertiary structures of seven transmembrane helical proteins in triangle lattice models, assuming that the configuration of helices is stabilized by polar interactions. Triangle lattice models having 12 or 11 nearest neighbor pairs were used as general templates of a seven-helix system, then the orientation angles of all helices were varied at intervals of 15 degrees. The polar interaction energy for all possible positions of each helix was estimated using the calculated polar indices of transmembrane helices. An automated system was constructed and applied to bacteriorhodopsin, a typical membrane protein with seven transmembrane helices. The predicted optimal and actual structures were similar. The top 100 predicted helical configurations indicated that the helix-triangle, CFG, occurred at the highest frequency. In fact, this helix-triangle of bacteriorhodopsin forms an active proton-pumping site, suggesting that the present method can identify functionally important helices in membrane proteins. The possibility of studying the structure change of bacteriorhodopsin during the functional process by this method is discussed, and may serve to explain the experimental structures of photointermediate states.     

Studies on protein folding, unfolding and fluctuations by computer simulation. I. The effect of specific amino acid sequence represented by specific inter-unit interactions.

A lattice model of proteins is introduced. "A protein molecule" is a chain of nown-intersecting units of a given length on the two-dimensional square lattice. The copolymeric character of protein molecules is incorporated into the model in the form of specificities of inter-unit interactions. This model proved most effective for studying the statistical mechanical characteristics of protein folding, unfolding and fluctuations. The specificities of inter-unit interactions are shown to be the primary factors responsible for the all-or-none type transition from native to denatured states of globular proteins. The model has been studied by the Monte Carlo method of Metropolis et al., which is now shown applied to approximately simulating a kinetic process. In the strong limit of the specificity of the inter-unit interaction the native conformation was reached in this method by starting from an extended conformation. The possible generalization and application of this method for finding the native conformation of proteins form their amino sequence are discussed.     

Time-resolved microspectroscopy on a single crystal of bacteriorhodopsin reveals lattice-induced differences in the photocycle kinetics

The determination of the intermediate state structures of the bacteriorhodopsin photocycle has lead to an unprecedented level of understanding of the catalytic process exerted by a membrane protein. However, the crystallographic structures of the intermediate states are only relevant if the working cycle is not impaired by the crystal lattice. Therefore, we applied visible and Fourier transform infrared spectroscopy (FTIR) microspectroscopy with microsecond time resolution to compare the photoreaction of a single bacteriorhodopsin crystal to that of bacteriorhodopsin residing in the native purple membrane. The analysis of the FTIR difference spectra of the resolved intermediate states reveals great similarity in structural changes taking place in the crystal and in PM. However, the kinetics of the photocycle are significantly altered in the three-dimensional crystal as compared to PM. Strikingly, the L state decay is accelerated in the crystal, whereas the M decay is delayed. The physical origin of this deviation and the implications for trapping of intermediate states are discussed. As a methodological advance, time-resolved step-scan FTIR spectroscopy on a single protein crystal is demonstrated for the first time which may be used in the future to gauge the functionality of other crystallized proteins with the molecular resolution of vibrational spectroscopy.     

Development of an equation for potentiometric titration of polyelectrolytes using a periodic lattice model. A numerical analysis of interactions among ionizable groups

A numerical method is presented for analysing the potentiometric titration behavior of linear polyelectrolytes. A polyelectrolyte molecule is treated as a one-dimensional lattice containing a large number of lattice points, each of which has an identical ionizable group. In this method, the polyelectrolyte model lattice is divided into identical repeating unit cell systems with a finite number of ionizable groups to calculate the thermodynamic partition function of the system of polyelectrolyte solution. The electrostatic interaction between ionized groups adopted in the present study is given by the Debye-Hückel type screened Coulombic potential. The titration behavior as well as several thermodynamic quantities is derived by making a canonical ensemble summation of all states in a unit cell system under an appropriate periodic boundary condition. This method serves as a model of the two-step ionization often appearing in polyions with strong neighboring interactions such as homo- and copolymers of maleic acid. Several characteristics of the titration behavior of these polyelectrolytes are well reproduced by using a lattice model with pairwise intervals, including the effects of hydrogen bond formation and change in dielectric constant of the medium around an ionizable group. In addition, this method is valid for a more detailed analysis of the titration behavior of polyelectrolytes with various kinds of arrangements of charged groups.     

Spatial pattern analysis in forest dynamics: deviation from power law and direction of regeneration waves

Grid-based models have been used to understand spatial heterogeneity of the vegetation height in forests and to analyze spatio-temporal dynamics of the forest regeneration process. In this report, we present two methods of identifying lattice models when spatio-temporal data are given. The first method detects directionality of regeneration waves based on the timing of local disturbance events. The second evaluates the forest pattern by recording the fraction of high and low vegetation areas at multiple spatial scales. We illustrate these methods by applying them to patterns generated using three simple stochastic lattice models: (1) two-state model, distinguishing sites with high and low vegetation, (2) three-state model, in which sites can be in an additional disturbed state, and (3) Shimagare model, which considers a continuous range of states. The combination of the two methods provides efficient means of distinguishing the models. The first method has a more direct ecological meaning, while the second is useful when forest data are limited in time.     

Pair potentials for protein folding: choice of reference states and sensitivity of predicted native states to variations in the interaction schemes

We examine the similarities and differences between two widely used knowledge-based potentials, which are expressed as contact matrices (consisting of 210 elements) that gives a scale for interaction energies between the naturally occurring amino acid residues. These are the Miyazawa-Jernigan contact interaction matrix M and the potential matrix S derived by Skolnick J et al., 1997, Protein Sci 6:676-688. Although the correlation between the two matrices is good, there is a relatively large dispersion between the elements. We show that when Thr is chosen as a reference solvent within the Miyazawa and Jernigan scheme, the dispersion between the M and S matrices is reduced. The resulting interaction matrix B gives hydrophobicities that are in very good agreement with experiment. The small dispersion between the S and B matrices, which arises due to differing reference states, is shown to have dramatic effect on the predicted native states of lattice models of proteins. These findings and other arguments are used to suggest that for reliable predictions of protein structures, pairwise additive potentials are not sufficient. We also establish that optimized protein sequences can tolerate relatively large random errors in the pair potentials. We conjecture that three body interaction may be needed to predict the folds of proteins in a reliable manner.     

Rates of evolution on the time scale of the evolutionary process

A generational time scale, involving change from one generation to the next, is the time scale of evolution by natural selection. Microevolutionary and macroevolutionary patterns reflect this process on longer time scales. Rates of evolution are most efficiently expressed in haldane units, H, in standard deviations per generation, indexed by the log of the time interval. Rates from replicated selection experiments and simulations have rate-interval [RI] and log rate-log interval [LRI] scaling relations enabling directional, stationary, and random time series to be distinguished. Empirical microevolutionary and macroevolutionary data exhibit stationary scaling, but point to generational rates of evolution (H ₀) conservatively on the order of 0.2 standard deviations per generation on the time scale of the evolutionary process. This paradox of long-term stationary scaling and short-term high rates of change can be explained by considering the shape of an heuristic time-form evolutionary lattice. Cenozoic mammals occupy a lattice that is about four orders of magnitude longer in time than it has ever been wide in form. The evolutionary process is dynamic but operates within relatively narrow morphological constraints compared to the time available for change.     

The role of protein homochirality in shaping the energy landscape of folding

The homochirality, or isotacticity, of the natural amino acids facilitates the formation of regular secondary structures such as alpha-helices and beta-sheets. However, many examples exist in nature where novel polypeptide topologies use both l- and d-amino acids. In this study, we explore how stereochemistry of the polypeptide backbone influences basic properties such as compactness and the size of fold space by simulating both lattice and all-atom polypeptide chains. We formulate a rectangular lattice chain model in both two and three dimensions, where monomers are chiral, having the effect of restricting local conformation. Syndiotactic chains with alternating chirality of adjacent monomers have a very large ensemble of accessible conformations characterized predominantly by extended structures. Isotactic chains on the other hand, have far fewer possible conformations and a significant fraction of these are compact. Syndiotactic chains are often unable to access maximally compact states available to their isotactic counterparts of the same length. Similar features are observed in all-atom models of isotactic versus syndiotactic polyalanine. Our results suggest that protein isotacticity has evolved to increase the enthalpy of chain collapse by facilitating compact helical states and to reduce the entropic cost of folding by restricting the size of the unfolded ensemble of competing states.     

Cell locomotion forces versus cell contraction forces for collagen lattice contraction: an in vitro model of wound contraction

Cultured human dermal fibroblasts suspended in a rapidly polymerizing collagen matrix produce a fibroblast-populated collagen lattice. With time, this lattice will undergo a reduction in size referred to as lattice contraction. During this process, two distinct cell populations develop. At the periphery of the lattice, highly oriented sheets of cells, morphologically identifiable as myofibroblasts, show cell-to-cell contacts and thick, actin-rich staining cytoplasmic stress fibers. It is proposed that these cells undergoing cell contraction produce a multicellular contractile unit which reorients the collagen fibrils associated with them. The cells in the central region, referred to as fibroblasts, are randomly oriented, with few cell-to-cell contacts and faintly staining actin cytoplasmic filaments. In contrast it is proposed that cells working as single units use cell locomotion forces to reorient the collagen fibrils associated with them. Using this model, we sought to determine which of these two mechanisms, cell contraction or cell locomotion, is responsible for the force that contracts collagen lattices. Our experiments showed that fibroblasts produce this contractile force, and that the mechanism for lattice contraction appears to be related to cell locomotion. This is in contrast to a myofibroblast; where the mechanism for contraction is based upon cell contractions. Fibroblasts attempting to move within the collagen matrix reorganize the surrounding collagen fibrils; when these collagen fibrils can be organized no further and cell-to-cell contacts develop, which occurs at the periphery of the lattice first, these cells can no longer participate in the dynamic aspects of lattice contraction.     

Statistical mechanical model of the energetics of coated vesicle formation.

Adsorptive endocytosis is mediated by a structural transformation of a two-dimensional hexagonal lattice to a polyhedron or coated vesicle. Clathrin is the structural protein involved in this process. A theoretical model that focuses on the energetics of clathrin in coated vesicle formation is presented. Fisher's cluster model of phase transitions is applied to this problem. The equilibrium constant for the process of converting a large two-dimensional patch to a coated vesicle and a smaller patch is calculated. There are three energetic contributions to be considered. They are the surface energy, the interior or lattice energy, and a loop entropy. Features of the Ising model are introduced into this model by scaling the critical exponents. In determining the configurational partition functions required for equilibrium constant calculations, the polyhedron is represented as a planar graph with no surfaces. The equilibrium constants are extremely sensitive to changes in the lattice and surface energies. The loop entropy contributions favor bimodal distributions in vesicle size under certain conditions. Conditions can also be established where the energy required to form the vesicle is small.     

Identification of the death zone: a spatially restricted region for programmed cell death that sculpts the fly eye

Programmed cell death (PCD) sculpts many developing tissues. The final patterning step of the Drosophila retina is the elimination, through PCD, of a subset of interommatidial lattice cells during pupation. It is not understood how this process is spatially regulated to ensure that cells die in the proper positions. To address this, we observed PCD of lattice cells in the pupal retina in real time. This live-visualization method demonstrates that lattice cell apoptosis is a highly specific process. In all, 85% of lattice cells die in exclusive 'death zone' positions between adjacent ommatidia. In contrast, cells that make specific contacts with primary pigment cells are protected from death. Two signaling pathways, Drosophila epidermal growth factor receptor (dEgfr) and Notch, that are thought to be central to the regulation of lattice cell survival and death, are not sufficient to establish the death zone. Thus, application of live visualization to the fly eye gives new insight into a dynamic developmental process.     

Cooperativity and the origins of rapid, single-exponential kinetics in protein folding

The folding of naturally occurring, single-domain proteins is usually well described as a simple, single-exponential process lacking significant trapped states. Here we further explore the hypothesis that the smooth energy landscape this implies, and the rapid kinetics it engenders, arises due to the extraordinary thermodynamic cooperativity of protein folding. Studying Miyazawa-Jernigan lattice polymers, we find that, even under conditions where the folding energy landscape is relatively optimized (designed sequences folding at their temperature of maximum folding rate), the folding of protein-like heteropolymers is accelerated when their thermodynamic cooperativity is enhanced by enhancing the nonadditivity of their energy potentials. At lower temperatures, where kinetic traps presumably play a more significant role in defining folding rates, we observe still greater cooperativity-induced acceleration. Consistent with these observations, we find that the folding kinetics of our computational models more closely approximates single-exponential behavior as their cooperativity approaches optimal levels. These observations suggest that the rapid folding of naturally occurring proteins is, in part, a consequence of their remarkably cooperative folding.     

Pathogen invasion and host extinction in lattice structured populations

We examined the propagation of an infectious disease and the eventual extinction of the host population in a lattice-structured population. Both the host colonization and pathogen transmission processes are assumed to be restricted to act between the nearest neighbor sites. The model is analyzed by an improved version of pair approximation (IPA). Pair approximation is a technique to trace the dynamics of the number of nearest neighbor pairs having particular states, and IPA takes account of the clustering property of lattice models more precisely. The results are checked by computer simulations. The analysis shows: (i) in a one-dimensional lattice population, a pathogen cannot invade a host population no matter how large is the transmission rate; (ii) in a two-dimensional lattice population, pathogens will drive the host to extinction if the transmission rate is larger than a threshold. These results indicate that spatially structured population models may give qualitatively different results from conventional population models, such as Lotka-Volterra ones, without spatial structure.