We study various elements of cell processes involved in signal transduction, protein trafficking, and cell motility. Our focus is on understanding the interactions and transformations of supramolecular cellular assemblages such as protein-coated endocytic vesicles, metabolic signalling networks, and extended cytoskeletal structures. We are particularly interested in the way complex activities are coordinated in space and time and thus devote considerable effort to developing and applying novel methodologies based on mathematical and physical principles, most notably the construction of specialized fluorescence-based optical instrumentation and the use of advanced electromagnetic scattering techniques that allow examination of structures and dynamics on nanoscopic-length scales. We also develop mathematical models of specific aspects of supramolecular and cellular behavior.

Mechanics of Coated Vesicle Formation
Nossal, Boukari, Muthukumar
Much protein and lipid trafficking in eukaryotic cells involves small tubulo-vesicular entities whose generation from cell membranes is mediated by the binding of specific coat proteins. We are attempting to understand the physical mechanisms by which such structures are generated. In earlier studies, we developed methods to determine the mechanical properties of clathrin cages (the major structural elements of the protein coats of certain endocytic vesicles) and examined how the binding of accessory proteins can increase the rigidity and stability of the cages. Recently, to investigate a possible relationship between total clathrin concentration and the amount of polymerized product, we adapted an analytical model of clathrin cage assembly that accounts for the combined bending and flexing rigidities of triskelion legs and hubs, the intrinsic curvature of an isolated triskelion, and free energy changes associated with interactions between legs of neighboring triskelions. The analysis, which is similar to that used to study micelle formation by amphiphilic molecules, provides insight into the ways that intra- and intermolecular forces determine the critical concentration for polymerization (below which cages will not form) and the size distributions of the resultant structures. We found that the critical concentration can decrease below a detectable level when the interactions between the legs of neighboring triskelions in a cage are sufficiently strong. We determined that, in general, the greater the free energy change associated with cage formation (i.e., the more stable the cages), the lower is the critical concentration. We have also determined how the critical concentration explicitly depends on the various system variables (rigidity and out-of-plane curvature of clathrin triskelions and inter-leg association energies) and have begun to explore ways that such inferences may be tested by fluorescence correlation spectroscopy (FCS) and other physical techniques that might be used to measure sizes and shapes of supramolecular assemblies.

Mathematical Analysis of Cyclic Metabolic Pathways for Phosphoinositide Synthesis
Skupsky, Nossal
Considerable evidence indicates that phosphatidylinositide (PI) metabolism is involved in many cell-physiological events, including vesicle biogenesis, motility, and chemotaxis. The feedback and feed-forward character of cyclic pathways of PI metabolism suggests that PIs may act as biochemical switches. Several investigators have proposed a direct role for PIs as spatial regulators of endocytic vesicle production, as it has been known for some time that PI 3-kinase (PI3K) activity increases upon binding of growth factors or various other ligands to cell surface receptors, leading to conversion of phosphatidylinositol 4,5 bisphosphate (PIP2) to phosphatidylinositol trisphosphate (PIP3). Additional feedback loops can arise from coupling to membrane curvature. For example, PI3K activity can be enhanced if its substrate is embedded in a curved membrane while PIs may transiently induce membrane bending through their electronic charge or by attracting curvature-inducing cytosolic proteins (e.g., epsin 1). Using mathematical and computational methods, we have examined a biochemical model for regulation of PI signalling that includes the actions of PI kinases and phosphatases, phospholipase C, small g-proteins, and phosphatidic acid production. In addition to providing insight into vesicle formation in endocytosis, the model has permitted us to understand recent data implicating PI signalling in chemotaxis. In fact, a model of eukaryotic gradient sensing that is consistent with current data and based on PI cycles can be constructed in several ways. For example, our model shows qualitative differences depending on how non-linearities are included in the chemical kinetics and whether species translocating from the cytoplasm to plasma membrane exist in small or large quantities. We have begun an analysis of spatial patterns that might be triggered by an outside stimulus, depending on different values of adjustable system parameters.

Structure of Tubulin Polymers
Sackett, Watts, Nossal; in collaboration with Krueger, Steven, Chernomordik, Losert
Tubulin polymers are participants in various critical cell functions, including mitosis, intracellular transport, determination of global cell morphology, and cell motility. Polymers of purified tubulin take many forms in response to environmental factors such as pH, salt concentration, temperature, and the presence of small organic molecules that mediate tubulin-tubulin interactions. We have been investigating how several of these factors affect supramolecular tubulin structure. Specifically, we have used small-angle neutron scattering (SANS) to examine, for example, taxol-stabilized microtubules and other tubulin samples in both H2O and D2O buffers. We made measurements at pH/pD values between 6.0 and 7.8, with observed scattered intensities, I(Q), interpreted in terms of multi-component models of microtubules and related tubulin polymers. A semi-quantitative curve-fitting procedure was developed to estimate the relative amounts of the supramolecular components of the samples. We were able to show that, when pD is lowered, the samples appear to contain an appreciable amount of sheet-like structures in addition to microtubules and that the average microtubule protofilament number increases from ca. 12.5 at pD 7.0 to ca. 14 at pD 6.0. These results are of considerable interest, as several studies have demonstrated that high concentrations of D2O can be cytotoxic to mammalian cells, presumably due to inhibition of mitosis. We now have insight into the molecular changes that are involved. Moreover, extensions of our work may lead to a deeper understanding of the chemical physics generally underlying kinetically driven supramolecular assembly processes.

In a different study, we examined the mechanisms whereby cryptophycin 1 (a potential anti-cancer drug) interacts with tubulin to induce the formation of ring-like complexes. By combining results of sedimentation velocity and dynamic light scattering measurements with scanning transmission electron microscopy measurements, we obtained a ring mass that, in combination with cryoelectron microscopy, gives an accurate representation of ring structure. We thereby were able to infer that two well-defined, repeating bends create the necessary curvature to cause the ring structures to form. We are planning extensions of these investigations to understand why microtubules self-organize to form large-scale arrays; the latter are amenable to study by optical methods such as polarization and confocal microscopy, optical birefringence, and, possibly, laser tweezers.

Fluorescence Correlation Spectroscopy and Fourier Imaging (Scattering) Techniques
Boukari, Nossal, Sackett; in collaboration with Szu
We recently built a microscope-based, dual-channel fluorescence correlation spectrometer that allows us to determine the translational diffusion coefficients (and therefore the sizes) of macromolecular complexes when in solution at very low concentrations. We have used the device to characterize the unusual polymers resulting from the interaction of tubulin with cryptophycin, hemiasterlin, and dolastatin; the last is among a group of antimitotic marine natural products undergoing evaluation as potential anti-tumor drugs. These peptides inhibit tubulin polymerization into microtubules and instead induce the formation of single-walled tubulin rings of 27-nm diameter in the case of cryptophycin and 44-nm diameter in the case of hemiasterlin and dolastatin, as revealed by electron microscopy of drug-tubulin samples of relatively high (micromolar) concentration. However, by using FCS to monitor the hydrodynamic diameter and the apparent number of fluorescent entities as the samples are diluted, we have determined that cryptophycin-tubulin rings are stable even with tubulin concentrations as low as 1 nM, whereas hemiasterlin-tubulin rings depolymerize even at relatively high concentrations (100 nM). Dolastatin-tubulin rings demonstrate an intermediate level of stability. Comparison of cytotoxicity measure-ments taken on several cell lines shows a rough correlation between that of the drugs and the stability of the rings.

These studies, which are interesting in their own right, also serve as prototypes for applications involving different biomacromolecular systems. Thus, we have initiated studies of the structure and stability of protein-polysaccharide conjugates that are currently under development as vaccines against enteric pathogens. Our objective is to determine the sizes of vaccine conjugates when diluted to low physiological concentrations in a range not easily accessible to study by other physical techniques. We also plan to use the methodology to examine the assembly of reconstituted clathrin cages as well as parameters pertaining to interactions of cellular organelles. In a related investigation, we have used SANS to examine a variety of tubulin polymers in solution, including isolated rings and higher-order ring aggregates that appear when dolastatin acts on tubulin. Preliminary evidence indicates that the aggregates are composed of columnar stacks of elemental, individual rings.