The structure of the acto-myosin subfragment 1 complex: Results of searches using data from electron microscopy and x-ray crystallography

Surmises of how myosin subfragment 1 (S1) interacts with actin filaments in muscle contraction rest upon knowing the relative arrangement of the two proteins. Although there exist crystallographic structures for both S1 and actin, as well as electron microscopy data for the acto–S1 complex (AS1), modeling of this arrangement has so far only been done “by eye.” Here we report fitted AS1 structures obtained using a quantitative method that is both more objective and makes more complete use of the data. Using undistorted crystallographic results, the best-fit AS1 structure shows significant differences from that obtained by visual fitting. The best fit is produced using the F-actin model of Holmes et al. [Holmes, K. C., Popp, D., Gebhard, W. & Kabsch, W. (1990) Nature (London) 347, 44–49]. S1 residues at the AS1 interface are now found at a higher radius as well as being translated axially and rotated azimuthally. Fits using S1 plus loops missing from the crystal structure were achieved using a homology search method to predict loop structures. These improved fits favor an arrangement in which the loop at the 50- to 20-kDa domain junction of S1 is located near the N terminus of actin. Rigid-body movements of the lower 50-kDa domain, which further improve the fit, produce closure of the large 50-kDa domain cleft and bring conserved residues in the lower 50-kDa domain into an apparently appropriate orientation for close interaction with actin. This finding supports the idea that binding of ATP to AS1 at the end of the ATPase cycle disrupts the actin binding site by changing the conformation of the 50-kDa cleft of S1.

The problem of the spreading of a liquid film along a solid surface: A new mathematical���formulation

A new mathematical model is proposed for the spreading of a liquid film on a solid surface. The model is based on the standard lubrication approximation for gently sloping films (with the no-slip condition for the fluid at the solid surface) in the major part of the film where it is not too thin. In the remaining and relatively small regions near the contact lines it is assumed that the so-called autonomy principle holds—i.e., given the material components, the external conditions, and the velocity of the contact lines along the surface, the behavior of the fluid is identical for all films. The resulting mathematical model is formulated as a free boundary problem for the classical fourth-order equation for the film thickness. A class of self-similar solutions to this free boundary problem is considered.

Structural features of the γ subunit of the Escherichia coli F1 ATPase revealed by a 4.4-Å resolution map obtained by x-ray crystallography

The F1 part of the F1FO ATP synthase from Escherichia coli has been crystallized and its structure determined to 4.4-Å resolution by using molecular replacement based on the structure of the beef-heart mitochondrial enzyme. The bacterial F1 consists of five subunits with stoichiometry α3, β3, γ, δ, and ɛ. δ was removed before crystallization. In agreement with the structure of the beef-heart mitochondrial enzyme, although not that from rat liver, the present study suggests that the α and β subunits are arranged in a hexagonal barrel but depart from exact 3-fold symmetry. In the structures of both beef heart and rat-liver mitochondrial F1, less than half of the structure of the γ subunit was seen because of presumed disorder in the crystals. The present electron-density map includes a number of rod-shaped features which appear to correspond to additional α-helical regions within the γ subunit. These suggest that the γ subunit traverses the full length of the stalk that links the F1 and FO parts and makes significant contacts with the c subunit ring of FO.

Dissociation of HIV-1 from follicular dendritic cells during HAART: Mathematical analysis

Follicular dendritic cells (FDC) provide a reservoir for HIV type 1 (HIV-1) that may reignite infection if highly active antiretroviral therapy (HAART) is withdrawn before virus on FDC is cleared. To estimate the treatment time required to eliminate HIV-1 on FDC, we develop deterministic and stochastic models for the reversible binding of HIV-1 to FDC via ligand–receptor interactions and examine the consequences of reducing the virus available for binding to FDC. Analysis of these models shows that the rate at which HIV-1 dissociates from FDC during HAART is biphasic, with an initial period of rapid decay followed by a period of slower exponential decay. The speed of the slower second stage of dissociation and the treatment time required to eradicate the FDC reservoir of HIV-1 are insensitive to the number of virions bound and their degree of attachment to FDC before treatment. In contrast, the expected time required for dissociation of an individual virion from FDC varies sensitively with the number of ligands attached to the virion that are available to interact with receptors on FDC. Although most virions may dissociate from FDC on the time scale of days to weeks, virions coupled to a higher-than-average number of ligands may persist on FDC for years. This result suggests that HAART may not be able to clear all HIV-1 trapped on FDC and that, even if clearance is possible, years of treatment will be required.

Visualization of a water-selective pore by electron crystallography in vitreous ice

The water-selective pathway through the aquaporin-1 membrane channel has been visualized by fitting an atomic model to a 3.7-Å resolution three-dimensional density map. This map was determined by analyzing images and electron diffraction patterns of lipid-reconstituted two-dimensional crystals of aquaporin-1 preserved in vitrified buffer in the absence of any additive. The aqueous pathway is characterized by a size-selective pore that is ≈4.0 ± 0.5Å in diameter, spans a length of ≈18Å, and bends by ≈25° as it traverses the bilayer. This narrow pore is connected by wide, funnel-shaped openings at the extracellular and cytoplasmic faces. The size-selective pore is outlined mostly by hydrophobic residues, resulting in a relatively inert pathway conducive to diffusion-limited water flow. The apex of the curved pore is close to the locations of the in-plane pseudo-2-fold symmetry axis that relates the N- and C-terminal halves and the conserved, functionally important N76 and N192 residues.

From cell death to embryo arrest: Mathematical models of human preimplantation embryo development

Human preimplantation embryos exhibit high levels of apoptotic cells and high rates of developmental arrest during the first week in vitro. The relation between the two is unclear and difficult to determine by conventional experimental approaches, partly because of limited numbers of embryos. We apply a mixture of experiment and mathematical modeling to show that observed levels of cell death can be reconciled with the high levels of embryo arrest seen in the human only if the developmental competence of embryos is already established at the zygote stage, and environmental factors merely modulate this. This suggests that research on improving in vitro fertilization success rates should move from its current concentration on optimizing culture media to focus more on the generation of a healthy zygote and on understanding the mechanisms that cause chromosomal and other abnormalities during early cleavage stages.

Mathematical approaches to comparative linguistics‡

The inference of the evolutionary history of a set of languages is a complex problem. Although some languages are known to be related through descent from common ancestral languages, for other languages determining whether such a relationship holds is itself a difficult problem. In this paper we report on new methods, developed by linguists Johanna Nichols (University of California, Berkeley), Donald Ringe and Ann Taylor (University of Pennsylvania, Philadelphia), and me, for answering some of the most difficult questions in this domain. These methods and the results of the analyses based on these methods were presented in November 1995 at the Symposium on the Frontiers of Science held by the National Academy of Sciences.

Location of a potential transport binding site in a sigma class glutathione transferase by x-ray crystallography.

The crystal structure of the sigma class glutathione transferase from squid digestive gland in complex with S-(3-iodobenzyl)glutathione reveals a third binding site for the glutathione conjugate besides the two in the active sites of the dimer. The additional binding site is near the crystallographic two-fold axis between the two alpha 4-turn-alpha 5 motifs. The principal binding interactions with the conjugate include specific electrostatic interactions between the peptide and the two subunits and a hydrophobic cavity found across the two-fold axis that accommodates the 3-iodobenzyl group. Thus, two identical, symmetry-related but mutually exclusive binding modes for the third conjugate are observed. The hydrophobic pocket is about 14 A from the hydroxyl group of Tyr-7 in the active site. This site is a potential transport binding site for hydrophobic molecules or their glutathione conjugates.

Failure of programmed cell death and differentiation as causes of tumors: some simple mathematical models.

Most models of tumorigenesis assume that the tumor grows by increased cell division. In these models, it is generally supposed that daughter cells behave as do their parents, and cell numbers have clear potential for exponential growth. We have constructed simple mathematical models of tumorigenesis through failure of programmed cell death (PCD) or differentiation. These models do not assume that descendant cells behave as their parents do. The models predict that exponential growth in cell numbers does sometimes occur, usually when stem cells fail to die or differentiate. At other times, exponential growth does not occur: instead, the number of cells in the population reaches a new, higher equilibrium. This behavior is predicted when fully differentiated cells fail to undergo PCD. When cells of intermediate differentiation fail to die or to differentiate further, the values of growth parameters determine whether growth is exponential or leads to a new equilibrium. The predictions of the model are sensitive to small differences in growth parameters. Failure of PCD and differentiation, leading to a new equilibrium number of cells, may explain many aspects of tumor behavior--for example, early premalignant lesions such as cervical intraepithelial neoplasia, the fact that some tumors very rarely become malignant, the observation of plateaux in the growth of some solid tumors, and, finally, long lag phases of growth until mutations arise that eventually result in exponential growth.

Direct structure analysis in protein electron crystallography: crystallographic phases for halorhodopsin to 6-A resolution.

The crystal structure of halorhodopsin was determined in (centrosymmetric) projection to 6-A resolution by direct methods that use only the amplitudes of the electron diffraction pattern. A multisolution technique was used to generate initial 15-A-resolution basis sets, and after selection of the best phase set (by the closest match of magnitude of Eobs and magnitude of Ecalc), annealing of individual reflections was used to improve its accuracy. The Sayre equation was then used to expand the phase terms to 10 A, followed again by phase annealing. A final expansion with the Sayre equation enlarged this corrected phase set to 6 A. When the condition of density flatness was used to locate the best phase solution after each extension, a final structure could be observed that was quite similar to the one found earlier by analysis of electron micrographs.

Mathematical analysis of activation thresholds in enzyme-catalyzed positive feedbacks: application to the feedbacks of blood coagulation.

A hierarchy of enzyme-catalyzed positive feedback loops is examined by mathematical and numerical analysis. Four systems are described, from the simplest, in which an enzyme catalyzes its own formation from an inactive precursor, to the most complex, in which two sequential feedback loops act in a cascade. In the latter we also examine the function of a long-range feedback, in which the final enzyme produced in the second loop activates the initial step in the first loop. When the enzymes generated are subject to inhibition or inactivation, all four systems exhibit threshold properties akin to excitable systems like neuron firing. For those that are amenable to mathematical analysis, expressions are derived that relate the excitation threshold to the kinetics of enzyme generation and inhibition and the initial conditions. For the most complex system, it was expedient to employ numerical simulation to demonstrate threshold behavior, and in this case long-range feedback was seen to have two distinct effects. At sufficiently high catalytic rates, this feedback is capable of exciting an otherwise subthreshold system. At lower catalytic rates, where the long-range feedback does not significantly affect the threshold, it nonetheless has a major effect in potentiating the response above the threshold. In particular, oscillatory behavior observed in simulations of sequential feedback loops is abolished when a long-range feedback is present.