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PubMed1995

Analytical expressions for the NMR apparent diffusion coefficients in an anisotropic system and a simplified method for determining fiber orientation.

NMR measurements of anisotropic diffusion were studied using a three-dimensional random-walk model. It was found that the apparent diffusion coefficient can be expressed in a canonical form as the product of a diagonal matrix, an orthonormal rotation matrix, and a vector representing the encoding magnetic field gradient. The diffusion coefficient can be interpreted as the sum of the corresponding coefficients measured along the principal diffusion axes, weighted by the squares of the directional cosines of the encoding direction with respect to the principal axes. The analysis revealed that determining the orientation of anisotropy, in a cylindrically symmetric system, requires a minimum of four diffusion measurements. A special pulse sequence which minimized gradient cross-terms and possible restricted diffusion effects was used to characterize diffusion anisotropy in cut chicken gizzards. Diffusion coefficients parallel to the muscle fibers were found to be approximately two to three times larger than those in the transverse direction. Furthermore, the method was successful in detecting the angular change when the sample was rotated by 30 degrees. Results indicate that the proposed approach to measure fiber orientation is valid and may be used to improve the time efficiency of diffusion anisotropy measurements.

Algorithms↗
PubMed1997

Double line scan diffusion imaging.

A new double line scan diffusion imaging sequence (DLSDI) is presented. In DLSDI, two lines from two separate slices are acquired in each shot. As its predecessor, LSDI, DLSDI is insensitive to motion artifacts and it can be used on conventional MR scanners. In addition, DLSDI is almost twice as fast as LSDI. Preliminary results from phantom and patient studies show excellent agreement between ADC trace maps obtained with DLSDI and LSDI. The technical and the theoretical aspects of DLSDI are studied, and it is shown how the conditional random walk model can be used as an analytical tool to derive the diffusion sensitivity in the DLSDI sequence.

Brain↗
PubMed2004

A structural pattern-based method for protein fold recognition.

A method (SPREK) was developed to evaluate the register of a sequence on a structure based on the matching of structural patterns against a library derived from the protein structure databank. The scores obtained were normalized against random background distributions derived from sequence shuffling and permutation methods. 'Random' structures were also used to evaluate the effectiveness of the method. These were generated by a simple random-walk and a more sophisticated structure prediction method that produced protein-like folds. For comparison with other methods, the performance of the method was assessed using collections of models including decoys and models from the CASP-5 exercise. The performance of SPREK on the decoy models was equivalent to (and sometimes better than) those obtained with more complex approaches. An exception was the two smallest proteins, for which SPREK did not perform well due to a lack of patterns. Using the best parameter combination from trials on decoy models, the CASP models of intermediate difficulty were evaluated by SPREK and the quality of the top scoring model was evaluated by its CASP ranking. Of the 14 targets in this class, half lie in the top 10% (out of around 140 models for each target). The two worst rankings resulted from the selection by our method of a well-packed model that was based on the wrong fold. Of the other poor rankings, one was the smallest protein and the others were the four largest (all over 250 residues).

Algorithms↗
PubMed2004

Modeling correlated main-chain motions in proteins for flexible molecular recognition.

We describe a new method for modeling protein and ligand main-chain flexibility, and show its ability to model flexible molecular recognition. The goal is to sample the full conformational space, including large-scale motions that typically cannot be reached in molecular dynamics simulations due to the computational intensity, as well as conformations that have not been observed yet by crystallography or NMR. A secondary goal is to assess the degree of flexibility consistent with protein-ligand recognition. Flexibility analysis of the target protein is performed using the graph-theoretic algorithm FIRST, which also identifies coupled networks of covalent and noncovalent bonds within the protein. The available conformations of the flexible regions are then explored with ROCK by random-walk sampling of the rotatable bonds. ROCK explores correlated motions by only sampling dihedral angles that preserve the coupled bond networks in the protein and generates conformers with good stereochemistry, without using a computationally expensive potential function. A representative set of the conformational ensemble generated this way can be used as targets for docking with SLIDE, which handles the flexibility of protein and ligand side-chains. The realism of this protein main-chain conformational sampling is assessed by comparison with time-resolved NMR studies of cyclophilin A motions. ROCK is also effective for modeling the flexibility of large cyclic and polycyclic ligands, as demonstrated for cyclosporin and zearalenol. The use of this combined approach to perform docking with main-chain flexibility is illustrated for the cyclophilin A-cyclosporin complex and the estrogen receptor in complex with zearalenol, while addressing the question of how much flexibility is allowed without hindering molecular recognition.

Biochemical Phenomena↗
PubMed1995

Smoothing splines for longitudinal data.

In a longitudinal data model with fixed and random effects, polynomials are used to model the fixed effects and smoothing polynomial splines are used to model the within-subject random effect curves. The splines are generated by modelling the data for each subject as observations of an integrated random walk with observational error. The initial conditions for each subject's deviation from the fixed effect curve are assumed to have zero mean and arbitrary covariance matrix which is estimated by maximum likelihood, producing an empirical Bayes estimate. This is in contrast to modelling a single curve using a diffuse prior. An example is presented using unbalanced longitudinal data from a pilot study in breast cancer patients.

Bayes Theorem↗
PubMed1997

Signaling mechanisms in growth factor-stimulated cell motility.

Most mammalian cells have the capacity to migrate. When placed into culture, cells will generally display a set rate of basal, unstimulated locomotion. The cells will begin to move in one direction and, after some time, change directions resulting in a random walk. External stimuli can influence cell motility in several ways to either enhance or retard the rate of migration (chemokinesis), to change the average amount of cell migration observed before the cell turns (persistence), or to increase the directionality of movement by limiting the number of turns made by the cells. Several factors have been identified that stimulate cell movement, but the signaling mechanisms that mediate this induced cell movement have only recently begun to be studied. In this review, we discuss the signals that support the directional movement of fibroblasts and epithelial cells in response to chemoattractant gradients. The work will emphasize studies carried out by our laboratory and others on the stimulation of cell motility by the PDGF. These results indicate that at least two sets of signaling molecules cooperate to regulate cell motility in vivo. These include phospholipase C-gamma, phosphoinositide-3' kinase and the Ras-GTPase activating protein Ras-GAP. The first set are those which bind to the intracellular domain of the receptor tyrosine kinase and bring about the phosphorylation and/or activation of intracellular effectors proximal to the receptor. The second is a set of down-stream effectors that regulate either the rate of cell movement or the directionality of that movement depending on the cell type. These include Ras and the Ras-related GTPase Rac along with free phosphoinositides and calcium ions that regulate the actin polymerization machinery. Signals that mediate nuclear changes leading to cell proliferation, such as elements of the MAP kinase pathway, do not appear to play a role in PDGF-stimulated cell migration. Current work thus suggests that a coordinated spatial regulation of signaling elements that interact with the cell membrane and cytoskeleton but not necessarily with nuclear elements is the controlling mediator of directional cell motility.

Animals↗
PubMed1998

Continuous and discrete mathematical models of tumor-induced angiogenesis.

Angiogenesis, the formation of blood vessels from a pre-existing vasculature, is a process whereby capillary sprouts are formed in response to externally supplied chemical stimuli. The sprouts then grow and develop, driven initially by endothelial-cell migration, and organize themselves into a dendritic structure. Subsequent cell proliferation near the sprout tip permits further extension of the capillary and ultimately completes the process. Angiogenesis occurs during embryogenesis, wound healing, arthritis and during the growth of solid tumors. In this paper we present both continuous and discrete mathematical models which describe the formation of the capillary sprout network in response to chemical stimuli (tumor angiogenic factors, TAF) supplied by a solid tumor. The models also take into account essential endothelial cell-extracellular matrix interactions via the inclusion of the matrix macromolecule fibronectin. The continuous model consists of a system of nonlinear partial differential equations describing the initial migratory response of endothelial cells to the TAF and the fibronectin. Numerical simulations of the system, using parameter values based on experimental data, are presented and compared qualitatively with in vivo experiments. We then use a discretized form of the partial differential equations to develop a biased random-walk model which enables us to track individual endothelial cells at the sprout tips and incorporate anastomosis, mitosis and branching explicitly into the model. The theoretical capillary networks generated by computer simulations of the discrete model are compared with the morphology of capillary networks observed in in vivo experiments.

Angiogenesis Inducing Agents↗
PubMed2000

Modelling run-and-tumble chemotaxis in a shear flow.

The biased random walk undergone by chemotactic bacteria such as Escherichia coli will be influenced at the microscopic level by flow in the ambient medium. In this paper, we model swimming bacteria being advected and rotated by a simple shear flow. Under certain scaling assumptions, we obtain an advection-diffusion equation for cell density, when the chemotactic response is small, which shows a coupling between the rotation and chemotaxis. We also present an alternative method for calculating the chemotactic flux in an unbounded region which is valid for more general chemotactic responses.

Chemotaxis↗
PubMed2001

Mathematical modeling of capillary formation and development in tumor angiogenesis: penetration into the stroma.

The purpose of this paper is to present a mathematical model for the tumor vascularization theory of tumor growth proposed by Judah Folkman in the early 1970s and subsequently established experimentally by him and his coworkers [Ausprunk, D. H. and J. Folkman (1977) Migration and proliferation of endothelial cells in performed and newly formed blood vessels during tumor angiogenesis, Microvasc Res., 14, 53-65; Brem, S., B. A. Preis, ScD. Langer, B. A. Brem and J. Folkman (1997) Inhibition of neovascularization by an extract derived from vitreous Am. J. Opthalmol., 84, 323-328; Folkman, J. (1976) The vascularization of tumors, Sci. Am., 234, 58-64; Gimbrone, M. A. Jr, R. S. Cotran, S. B. Leapman and J. Folkman (1974) Tumor growth and neovascularization: an experimental model using the rabbit cornea, J. Nat. Cancer Inst., 52, 413-419]. In the simplest version of this model, an avascular tumor secretes a tumor growth factor (TGF) which is transported across an extracellular matrix (ECM) to a neighboring vasculature where it stimulates endothelial cells to produce a protease that acts as a catalyst to degrade the fibronectin of the capillary wall and the ECM. The endothelial cells then move up the TGF gradient back to the tumor, proliferating and forming a new capillary network. In the model presented here, we include two mechanisms for the action of angiostatin. In the first mechanism, substantiated experimentally, the angiostatin acts as a protease inhibitor. A second mechanism for the production of protease inhibitor from angiostatin by endothelial cells is proposed to be of Michaelis-Menten type. Mathematically, this mechanism includes the former as a subcase. Our model is different from other attempts to model the process of tumor angiogenesis in that it focuses (1) on the biochemistry of the process at the level of the cell; (2) the movement of the cells is based on the theory of reinforced random walks; (3) standard transport equations for the diffusion of molecular species in porous media. One consequence of our numerical simulations is that we obtain very good computational agreement with the time of the onset of vascularization and the rate of capillary tip growth observed in rabbit cornea experiments [Ausprunk, D. H. and J. Folkman (1977) Migration and proliferation of endothelial cells in performed and newly formed blood vessels during tumor angiogenesis, Microvasc Res., 14, 73-65; Brem, S., B. A. Preis, ScD. Langer, B. A. Brem and J. Folkman (1997) Inhibition of neovascularization by an extract derived from vitreous Am. J. Opthalmol., 84, 323-328; Folkman, J. (1976) The vascularization of tumors, Sci. Am., 234, 58-64; Gimbrone, M. A. Jr, R. S. Cotran, S. B. Leapman and J. Folkman (1974) Tumor growth and neovascularization: An experimental model using the rabbit cornea. J. Nat. Cancer Inst., 52, 413-419]. Furthermore, our numerical experiments agree with the observation that the tip of a growing capillary accelerates as it approaches the tumor [Folkman, J. (1976) The vascularization of tumors, Sci. Am., 234, 58-64].

Angiogenesis Inducing Agents↗
PubMed2001

Two-category model of task allocation with application to ant societies.

In many network models of interacting units such as cells or insects, the coupling coefficients between units are independent of the state of the units. Here we analyze the temporal behavior of units that can switch between two 'category' states according to rules that involve category-dependent coupling coefficients. The behaviors of the category populations resulting from the asynchronous random updating of units are first classified according to the signs of the coupling coefficients using numerical simulations. They range from isolated fixed points to lines of fixed points and stochastic attractors. These behaviors are then explained analytically using iterated function systems and birth-death jump processes. The main inspiration for our work comes from studies of non-hierarchical task allocation in, e.g., harvester ant colonies where temporal fluctuations in the numbers of ants engaged in various tasks occur as circumstances require and depend on interactions between ants. We identify interaction types that produce quick recovery from perturbations to an asymptotic behavior whose characteristics are function of the coupling coefficients between ants as well as between ants and their environment. We also compute analytically the probability density of the population numbers, and show that perturbations in our model decay twice as fast as in a model with random switching dynamics. A subset of the interaction types between ants yields intrinsic stochastic asymptotic behaviors which could account for some of the experimentally observed fluctuations. Such noisy trajectories are shown to be random walks with state-dependent biases in the 'category population' phase space. With an external stimulus, the parameters of the category-switching rules become time-dependent. Depending on the growth rate of the stimulus in comparison to its population-dependent decay rate, the dynamics may qualitatively differ from the case without stimulus. Our simple two-category model provides a framework for understanding the rich variety of behaviors in network dynamics with state-dependent coupling coefficients, and especially in task allocation processes with many tasks.

Animals↗
PubMed1994

Primary mesenchyme cell migration in the sea urchin embryo: distribution of directional cues.

The directional migration of the primary mesenchyme cells (PMCs) of the sea urchin embryo is a critical step in the process of gastrulation. Although interactions between the migrating cells and the blastocoel environment are necessary for guiding the PMCs to their subequatorial target site, the nature of these interactions and the localization of guidance cues involved in directing the cells are not yet known. Previous studies have suggested that PMC migration is the result of random exploration and selective trapping at the target site by a pattern of adhesiveness in the ectoderm or basal lamina. To better characterize the distribution of guidance cues in the blastocoel we used a combination of time-lapse microscopy, microsurgery, and fluorescence photoablation to study the behavior of the migrating cells. By using fluorescence time-lapse microscopy, and a two-dimensional random-walk analysis of cell trajectories, we demonstrated that fluorescently labeled PMCs injected near the animal pole move in a directed fashion over a relatively long distance to reach the target site. This suggests that guidance cues are distributed globally throughout the embryo and are not restricted to the immediate ring area. To further test this hypothesis we investigated the migratory behavior of PMCs that were prevented from interacting directly with the target site. First, we examined the behavior of PMCs injected into animal embryo fragments lacking the target site. We found that PMCs move to the vegetal-most area of such embryo fragments, regardless of their size. Second, we studied the effects of photoablating a stripe of ectoderm between PMCs injected at the animal pole region (APR) and the target site. PMCs were found to accumulate along the ablated stripe and were unable to cross it for up to 6 hr after ablation. We also examined the migratory behavior of endogenous PMCs in embryos treated with lithium, a vegetalizing agent which shifts the position of the PMC ring toward the animal pole. We found that PMCs accumulated along an ablated stripe of ectoderm positioned below the shifted target site, suggesting that endogenous PMCs follow a set of directional cues to the target site which may be similar to those used by PMCs injected into the APR. As a whole, these results suggest that migrating PMCs follow a set of directional cues that are widely distributed throughout the blastocoel and that may be arranged in a gradient.

Animals↗
PubMed1999

Asymmetric Multiblock Copolymers at the Gas-Liquid Interface: Phase Diagram and Surface Pressure.

A theoretical model of copolymers made of N blocks is studied at the air-water interface. Each block is made of a sequence A of ZA hydrophobic and of a sequence B of ZB hydrophilic monomers. The A and B sequences cannot cross the interface. The conformation of an adsorbed polymer is determined as a random walk of N elements whose size is the Flory radius of a single sequence. The structure of the interfacial layer is determined as a function of alpha = ZA/ZB and of the surface concentration using scaling law arguments. Only three different regions are found in the phase diagram to describe the change of surface regime as a function of the total surface concentration. The energy of flower-like micelles of polymers is calculated and compared with the energy of adsorbed macromolecules in order to determine the surface concentration at saturation. The surface pressure is also calculated as a function of the surface concentration in the three different regions of the phase diagram. It is found that these surface pressure isotherms are not affected by the solvent quality except when the properties of the interfacial layer are dominated by a purely two-dimensional behavior (semidiluted regime of the whole polymer or of the A sequences on the air side of the interface). Finally the properties of this model are compared with experimental data obtained with protein adsorbed layers and encouraging agreement is found although proteins are much more complicated polymers than this crude model. Copyright 1999 Academic Press.

Journal Article↗
PubMed1999

Adsorption of Telechelic Poly(ethylene oxide) on Colloids: Influence on Colloid Stability.

Adsorption of telechelic polymers (hydrophobically end-capped poly(ethylene oxide)) on charged colloidal particles modifies the characteristics of the electrical double layer. These interfaces were found to be able to protect colloids against aggregation or to induce colloid destabilization depending on the molecular weight and type of surface coating. Colloid stability is observed when the polymer layer is thinner than the electrical double layer and when the adsorbed polymer adopts a tethered brush structure. Aggregation of coated latex particles induced by formation of interparticle polymer links is observed when (i) the distance of the random walk of the dangling paraffinic end group is larger than the thickness of the electrical double layer and (ii) the adsorbed polymer brush adopts a loopy structure. Aggregation is attributed to adsorption of hydrophobic end chains on two different latex particles. Copyright 1999 Academic Press.

Journal Article↗
PubMed1995

Algorithms for evaluating the long-range accessibility of protein surfaces.

Algorithms are presented for characterizing the long-range accessibilities of protein surfaces. First, we describe an analytical method for determining the maximum contact radius for each atom in a structure. The problem is simplified greatly by geometric inversion in a sphere, a type of conformal mapping. Second, we introduce the concept of diffusion accessibility of a protein surface, which we evaluate either by random-walk simulations or by numerical solution of the equations of diffusion with the protein acting as an adsorber. These two measures of exposure are compared to each other as well as to the more common notion of solvent accessibility. These new procedures provide longer-range descriptions of surface geometry which may be useful in docking studies and other areas where surface comparison is required.

Algorithms↗
PubMed1997

Triple-helical DNA as a reversible block of the branch point in a partially symmetrical DNA four-arm junction.

DNA branch migration is a fundamental process in genetic recombination. A new model system has been developed for studying branch migration in a small synthetic four-arm junction. A mathematical method for describing branch-point movement by discrete steps in such junctions is also presented. The key to our experimental system is the ability to fix the location of the branch point during the assembly of the junction with a reversible block. The block is provided by a short oligonucleotide that forms triplex DNA adjacent to the initial location branch point at low pH. Raising the pH causes the triplex strand to dissociate, making the branch point free to migrate. Once mobile, the branch point can run off the end of the junction. The time-course for this runoff is consistent with a random walk of the branch point. If it is assumed that one migration step moves the branch point one base-pair, the time-course gives a rate constant for one step of 1.4 second-1 at 37 degrees C in 10 mM MgCl2, 50 mM NaCl. These values are consistent with other measurements of non-enzymatic branch migration. We have also monitored the spread of the branch points directly with T4 endonuclease VII. Using EcoRI restriction endonuclease, we have shown that the binding of this protein to the arms of the junction essentially blocks branch migration through the binding site. In these experiments Ca2+ replaces Mg2+, and the enzyme does not cleave the DNA. In vivo there must be a special process to get branch points to migrate past bound proteins.

DNA↗
PubMed1998

Branch migration through DNA sequence heterology.

Branch migration of a DNA Holliday junction is a key step in genetic recombination. Previously, it was shown that a single base-pair heterology between two otherwise identical DNA sequences is a substantial barrier to passage of a Holliday junction during spontaneous branch migration. Here, we exploit this inhibitory effect of sequence heterology to estimate the step size of branch migration. We also devise a simulation of branch migration through mismatched base-pairs to arrive at the underlying molecular basis for the block to branch migration imposed by sequence heterology. Based on the observation that two adjacent sequence heterologies exert their effects on branch migration more or less independently, we conclude that the step size of branch migration is quite small, of the order of one or two base-pairs per migratory step. Comparison of branch migration experiments through a single base-pair heterology with simulations of a random walk through sequence heterology suggests that the inhibition of branch migration is largely attributable to a thermodynamic barrier arising from the formation of unpaired or mispaired bases in heteroduplex DNAs.

Bacteriophages↗
PubMed1999

Proteins can adopt totally different folded conformations.

The three-dimensional structure of a protein is determined by interactions between its amino acids and by interactions of the amino acids with molecules of the environment. The great influence of the latter interactions is demonstrated for the enzyme phosphoglycerate kinase from yeast (PGK). In the native state, PGK is a compact, bilobal molecule; 35% and 13% of its amino acids are organised in the form of alpha-helices and beta-sheets, respectively. The molecules unfold at acidic pH and low ionic strength forming random-walk structures with a persistence length of 3 nm. More than 90% of the amino acid residues of the ensemble have phi,psi-angles corresponding to those of a straight beta-chain. Upon addition of 50% (v/v) trifluoroethanol to the acid-unfolded protein, the entire molecule is transformed into a rod-like, flexible alpha-helix. Addition of anions, such as chloride or trichloroacetate, to the acid-unfolded protein leads to the formation of amyloid-like fibres over a period of many hours when the anion concentration exceeds a critical limit. Half of the amino acid residues are then organised in beta-sheets. Both of the non-natively folded states of PGK contain more regular secondary structure than the native one. The misfolding starts in both cases from the acid-unfolded state, in which the molecules are essentially more expanded than in other denatured states, e.g. those effected by temperature or guanidine hydrochloride.

Anions↗
PubMed1994

Mapping of turbulent intensity by magnetic resonance imaging.

The signal produced by turbulent flow in NMR pulse sequences has been analyzed by considering the effects of variations in fluid velocity on the net signal phase variance. It is shown that in a bipolar field gradient, the signal is dependent on the gradient amplitude and the precise time dependence of the velocity fluctuations. This dependence is described using an autocorrelation function, whose characteristic width is the correlation time. When the correlation time is short, the signal from fluid elements decreases as though they are diffusing in a random walk, whereas when the time is long, the signal falls more rapidly with an increase in the duration of the gradient waveform. However, many situations fall into an intermediate regime. A general expression appropriate for all correlation times has been derived and used to characterize the turbulent flow distal to a stenosis in a tube. By use of multiple images of the spatial distribution of NMR signal obtained with differing readout gradients and analyzed by application of the general expression for signal loss, the spatial variations in the correlation time and turbulent intensity have been obtained. The measured variations correlate well with computer calculations obtained by numerical simulation of the flow. This method permits turbulent and complex flows to be characterized without disturbing the flow and may have general applications.

Arterial Occlusive Diseases↗