Evolution of discontinuous solutions to transport equations in stellarators

It was shown earlier that, in the range of rare collisions, transport equations for stellarators allow steady discontinuous solutions for the ambipolar electric field and for the plasma density and temperature gradients. Moreover, such solutions are non-single-valued; that is, their explicit form depends on the initial values of the ambipolar electric field. The time-independent transport equations are derived under the conventional quasineutrality condition; i.e., it is assumed that the electron and ion densities, N _e and N _i , are related by the relationship N _e = ZN _i (where Z is the ion charge number). In other words, the plasma charge density is assumed to be much less than the product e _i N _i . Under typical conditions, the corresponding inequality is satisfied by a large margin. However, if the electric field E has discontinuities, then it can be seen from the equation ▿·E = 4πρ that, at the discontinuity points, the charge density becomes infinite and the relationship N _e = ZN _i fails to hold, so it is necessary to replace it with N _e = ZN _i + ρ/e _e . In the transport equations, this latter replacement produces additional terms, proportional to the second radial derivative of the field E. With these additional terms, the steady solutions are modified substantially. First, the ambipolar field and the derivatives of the density and temperatures all become continuous functions of the coordinates, a result that seems to be quite obvious. The second, not-so-obvious result is that the steady solutions become single-valued, i.e., independent of the initial values of the ambipolar electric field. It turns out that, in this case, two regimes are possible, depending on the values of the plasma parameters. In the first regime, the solution is unique and is independent of the initial conditions. In the second regime, two steady solutions can exist, depending on the initial conditions. One of the solution is similar to that obtained in the first regime, and the other differs from the first one both in the ambipolar field profile and in the dependence of the density and temperatures on the minor plasma radius. It cannot be excluded that different plasma confinement modes revealed in experiments are associated with the existence of such solutions.     

Polymerization rate difference of N-alkyl glycine NCAs: Steric hindrance or not?

Polypeptoids synthesized from N-substituted glycine N-carboxyanhydrides (NNCAs) are widely applied in biological fields. The effect of side groups in NNCA polymerizations is a key to develop novel polypeptoids with complex topologies and constituents. In this work, density functional theory (DFT) calculations are employed to investigate the propagation of a series of alkyl substituted NNCAs with solvation model. According to both computational and experimental results, carbonyl addition is confirmed as rate determining step and steric hindrance is recognized as the major factor of low reactivity in β-C branched NNCAs. However, in linear and γ-C branched case, aggregation of side groups instead of bulkiness is considered responsible for low polymerization rate.     

Permanent coexistence in general models of three interacting species

We address the question of the long term coexistence of three interacting species whose dynamics are governed by the ordinary differential equations x _i = X _i f _i (i = 1, 2, 3). In order for any theory in this area to be useful in practice, it must utilize as little information as possible concerning the forms of the f _i , in view of the great difficulty of determining these experimentally. Here we obtain, under rather general conditions on the equations, a criterion for judging whether the species will coexist in a biologically realistic manner. This criterion depends only on the behaviour near the one or two species equilibria of the two dimensional subsystems, the behaviour there being relatively easy to examine experimentally. We show that with the exception of one class of cases, which is a generalization of a classical example of May and Leonard [21], invasibility at each such equilibrium suitably interpreted is both necessary and sufficient for a strong form of coexistence to hold. In the exceptional case, a single additional condition at the equilibria is enough to ensure coexistence.     

N-methylleucine gramicidin-S and (di-N-methylleucine) gramicidin-S conformations with cis L-Orn-L-N-MeLeu peptide bonds.

Conformations containing all trans peptide bonds have previously been proposed for N-methylleucine gramicidin-S and (di-N-methylleucine) gramicidin-S based on an evaluation of proton nuclear magnetic resonance parameters in a series of solvents. These gramicidin-S derivatives exhibit full biological activity despite the fact that the proposed solution conformations differ in backbone topology and relative orientation of the Phe and Orn side chains compared to gramicidin-S. The present authors discuss conformations for N-methylleucine gramicidin-S and (di-N-methylleucine) gramicidin-S which incorporate cis peptide bonds at L -Orn-L -N-MeLeu, where the gramicidin-S backbone is essentially retained, and the relative orientation of the Pro, Orn, Val, and Phe side chains correspond to those observed for gramicidin-S. A novel hydrogen-bond arrangement involving one carbonyl group interacting with two peptide protons (1 ←4 and 1 ←5 types) is proposed to stabilize the backbone conformation in the gramicidin-S derivatives. A recent report on the cyclic heptapeptide antibiotic, Ilamycin B₁, shows the presence of cis peptide bonds at N-CH₃ amino acids, as well as the novel hydrogen-bond arrangement presented above.     

How do the ovaries count?

A model is studied that regulates follicle development through hormonal feedback. The model takes the form of a many body problem where each body (follicle) is assumed to inherit the same maturation function and interact through a common summed variable (estradiol). Numerical solutions are obtained to a system of ordinary differential equations in which the number of equations changes in time as new follicles randomly enter into the interacting population from a dormant immature pool. Spontaneous cycles of follicle development emerge in which the number of follicles that reach full maturity is well controlled and insensitive to the rate of activation. The system of N interacting follicles is transformed to a gradient system on the unit sphere. Its stability properties are analyzed and compared with numerical and experimental results. Features of both spontaneous and induced ovulators are simulated. For certain parameter values solutions are obtained that correspond to known pathology. The model also suggests that the large number of follicles that are activated but atrophy before reaching full maturity help to regulate the cycle.     

A region-based model framework for the rat urine concentrating mechanism

The highly structured organization of tubules and blood vessels in the outer medulla of the mammalian kidney is believed to result in preferential interactions among tubules and vessels; such interactions may promote solute cycling and enhance urine concentrating capability. In this study, we formulate a new model framework for the urine concentrating mechanism in the outer medulla of the rat kidney. The model simulates preferential interactions among tubules and vessels by representing two concentric regions and by specifying the fractions of tubules and vessels assigned to each of the regions. The model equations are based on standard expressions for transmural transport and on solute and water conservation. Model equations, which are derived in dynamic form, are solved to obtain steady-state solutions by means of a stable and efficient numerical method, based on the semi-Lagrangian semi-implicit method and on Newton’s method. In this application, the computational cost scales as O(N ²), where N is the number of spatial subintervals along the medulla. We present representative solutions and show that the method generates approximations that are second-order accurate in space and that exhibit mass conservation.     

A Microenvironment Based Model of Antimitotic Therapy of Gompertzian Tumor Growth

A model of tumor growth, based on two-compartment cell population dynamics, and an overall Gompertzian growth has been previously developed. The main feature of the model is an inter-compartmental transfer function that describes the net exchange between proliferating (P) and quiescent (Q) cells and yields Gompertzian growth for tumor cell population N = P + Q. Model parameters provide for cell reproduction and cell death. This model is further developed here and modified to simulate antimitotic therapy. Therapy decreases the reproduction-rate constant and increases the death-rate constant of proliferating cells with no direct effect on quiescent cells. The model results in a system of two ODE equations (in N and P/N) that has an analytical solution. Net tumor growth depends on support from the microenvironment. Indirectly, this is manifested in the transfer function, which depends on the proliferation ratio, P/N. Antimitotic therapy will change P/N, and the tumor responds by slowing the transfer rate from P to Q. While the cellular effects of therapy are modeled as dependent only on antimitotic activity of the drug, the tumor response also depends on the tumor age and any previous therapies—after therapy, it is not the same tumor. The strength of therapy is simulated by the parameter λ, which is the ratio of therapy induced net proliferation rate constant versus the original. A pharmacodynamic factor inversely proportional to tumor size is implemented. Various chemotherapy regimens are simulated and the outcomes of therapy administered at different time points in the life history of the tumor are explored. Our analysis shows: (1) for a constant total dose administered, a decreasing dose schedule is marginally superior to an increasing or constant scheme, with more pronounced benefit for faster growing tumors, (2) the minimum dose to stop tumor growth is age dependent, and (3) a dose-dense schedule is favored. Faster growing tumors respond better to dose density.     

A singular dispersion relation arising in a caricature of a model for morphogenesis

In 1983 Oster et al. proposed a model for morphogenesis consisting of a system of partial differential equations in which the dispersion relation for the problem linearised about the zero solution has a singularity. That is, the initial growth rate of a small perturbation of wave number k from the zero solution tends to positive or negative infinity as k tends to some critical value k _c from above or below respectively. We consider here as a caricature of the model a single partial differential equation with a similar dispersion relation in a bounded one-dimensional domain. The wave number, or equivalently the domain size, may be thought of as a bifurcation parameter. For the Neumann problem a phenomenon arises in which, as the domain size l increases past a critical value l _l ,the linear stability of the n-th mode jumps from one solution to a remote solution. That is, for l _n the trivial solution is unstable and a certain non-trivial solution is stable to perturbations of mode n, whereas for l>l _n the opposite is true. For the Dirichlet or the Robin problem a linear stability change in the trivial solution occurs, but no corresponding change in any other solution has been found. The corresponding initial boundary value problems are then considered. An asymptotic analysis is performed in the weakly nonlinear limit in the particular case in which only one mode is unstable and gives an asymptotic solution for two classes of nonlinearity, one symmetric and the other asymmetric about u=0. A development of the method of harmonic balance is then used to obtain approximate solutions in the strongly nonlinear case and when more than one mode may be unstable.     

Nuclear magnetic resonance study on the model compounds for poly(N-alkylglycine)s

Cis-trans isomerism was investigated with N-acetyl and N-propionyl, N-alkylglycine dimethylamides as model compounds for poly(N-alkylglycine dimethlamides as model compounds for (N-alkylglycine)s using n.m.r. spectroscopy. The population of the cis isomer measured in benzene and methylene chloride solutions did not show any marked dependence on the bulkiness of N-alkyl substituents. This contrasts with polyN-alkylglycine)s, whose cis isomer population increased with the introduction of bulky N-alkyl groups. Kinetics of the Cis-trans isomerization was also investigated with N-acetyldimethylamides of sacrosine, N-n-propylglycine, and N-isopropylglycine in tetrachloroethane solution. The δG? values for Cis-trans isomerization in these amides were 18 ～ 19 kcal/mole, which were virtually the same as that of polysacrosine.     

Kinetics of reversible reactions on linear lattices with neighbor effects

As a model for a variety of reaction processes on long chain molecules, for example, helix formation, a kinetic theory on a linear lattice is presented. Each reaction site can undergo reversible transitions between two states (0 and 1) with rates depending on the slate of its nearest neighbors. The system of coupled rate equations for the frequencies of specified runs of 0's and 1's is infinite for an infinite chain. In contrast to the case of irreversible processes, the system cannot, be written down by inspection. A procedure for the systematic derivation of the rate equations is developed which can be programmed on a computer. Explicit expressions for runs up to length four, involving runs up to length five are obtained without recourse to the computer. For the solution of the rate equations a closure must necessarily be imposed, and a possible procedure is pointed out. Furthermore the equilibrium relations following from the model are considered. The well-known equilibrium results for nearest-neighbor interactions represent a special case of these equations.     

Neocortical development and social structure in primates

The relationships between the relative size of the neocortex and differences in social structures were examined in prosimians and anthropoids. The relative size of the neocortex (RSN) of a given congeneric group in each superfamily of primates was measured based on the allometric relationships between neocortical volume and brain weight for each superfamily, to control phylogenetic affinity and the effects of brain size. In prosimians, “troop-making” congeneric groups (N=3) revealed a significantly larger RSN than solitary groups (N=6), and there was a significant, positive correlation between RSN and troop size. In the case of anthropoids, polygynous/frugivorous groups (N=5) revealed a significantly larger RSN than monogynous/frugivorous groups (N=8). Furthermore, a significant, positive correlation between RSN and troop size was found for frugivorous congeneric groups of the Ceboidea. These results suggest that neocortical development is associated with differences in social structure among primates.     

Parameter estimation techniques for interaction and redistribution models: a predator-prey example

Summary The use of parameter estimation techniques for partial differential equations is illustrated using a predatorprey model. Whereas ecologists have often estimated parameters in models, they have not previously been able to do so for models that describe interactions in heterogeneous environments. The techniques we describe for partial differential equations will be generally useful for models of interacting species in spatially complex environments and for models that include the movement of organisms. We demonstrate our methods using field data from a ladybird beetle (Coccinella septempunctata) and aphid (Uroleucon nigrotuberculatum) interaction. Our parameter estimation algorithms can be employed to identify models that explain better than 80% of the observed variance in aphid and ladybird densities. Such parameter estimation techniques can bridge the gap between detail-rich experimental studies and abstract mathematical models. By relating the particular bestfit models identified from our experimental data to other information on Coccinella behavior, we conclude that a term describing local taxis of ladybirds towards prey (aphids in this case) is needed in the model.     

Thermodynamics of solvation for methylproline peptides

We have determined thermodynamic parameters for transfer of N-acetyl,N′-methylamide derivatives of proline and methylprolines from carbon tetrachloride and from chloroform to water. The hydrophilic nature of the diamide model peptides is demonstrated by the negative free energies and enthalpies for transfer. Chloroform solvates the peptides considerably better than carbon tetrachloride. Heats of dilution in carbon tetrachloride arise from disruption of intermolecular peptide–peptide hydrogen bonds. After extrapolation to dilute solution, differences in thermodynamic parameters among the isomeric mono-methylproline peptides are correlated with the population of the intramolecularly hydrogen-bonded C₇ conformer in the nonpolar solvent. However, the thermodynamic parameters aslo reflect differences in solvation due to the proximity of the two peptide groups and the side chain.     

The biofilm matrix of Campylobacter jejuni determined by fluorescence lectin-binding analysis

Campylobacter jejuni is responsible for the most common bacterial foodborne gastroenteritis. Despite its fastidious growth, it can survive harsh conditions through biofilm formation. In this work, fluorescence lectin-binding analysis was used to determine the glycoconjugates present in the biofilm matrix of two well-described strains. Screening of 72 lectins revealed strain-specific patterns with six lectins interacting with the biofilm matrix of both strains. The most common sugar moiety contained galactose and N-acetylgalactosamine. Several lectins interacted with N-acetylglucosamine and sialic acid, probably originated from the capsular polysaccharides, lipooligosaccharides and N-glycans of C. jejuni. In addition, glycoconjugates containing mannose and fucose were detected within the biofilm, which have not previously been found in the C. jejuni envelope. Detection of thioflavin T and curcumin highlighted the presence of amyloids in the cell envelope without association with specific cell appendages. The lectins ECA, GS-I, HMA and LEA constitute a reliable cocktail to detect the biofilm matrix of C. jejuni.     

Sedimentation coefficients of DNA topoisomers: the helical wormlike chain

The sedimentation coefficient s_N of the DNA topoisomer with the linking number N is evaluated as a function of N and chain length on the basis of a (circular) twisted wormlike chain, i.e., a special case of the helical wormlike chain. Evaluation is carried out by an application of the Oseen–Burgers procedure of hydrodynamics to the cylinder model with the preaveraged Oseen tensor. The necessary mean reciprocal distance between two contour points is obtained by a Monte Carlo method. It is shown that s_N increases as |ΔN| is increased from 0 in the range of small |ΔN|, where ΔN = N ? N , with N the number of helix turns in the linear DNA chain in the undeformed state. It is found that there is semiquantitative agreement between the Monte Carlo values and the experimental data obtained by Wang for s_N.     

Contribution to the probabilistic theory of neural nets: IV. Various models for inhibition

Input-output formulas are derived for a neuron upon which converge single axones of two other neurons, which are subjected to a Poisson shower, where a number of different assumptions are made concerning the mechanism of inhibition. In one assumption so-called “bilateral pre-inhibition” is considered. That is to say, both neuronsN ₁ andN ₂ may exciteN ₃, but if the stimulus of one of them follows within a certain interval σ of the other, the second stimulus is not effective. This model is essentially no different from that involving two excitatory neurons acting upon a neuron having a refractory period. Another mechanism considered involves so-called “pre-and-post” inhibition, in which if two stimuli fromN ₁ andN ₂ fall within σ,both are ineffective. This case being mathematically much more involved than the preceding, an approximation method is used for deriving the input-output formula. Previous papers of this series are denoted by I, II, and III in this paper.     

Clustering with neural networks

Partitioning a set ofN patterns in ad-dimensional metric space intoK clusters — in a way that those in a given cluster are more similar to each other than the rest — is a problem of interest in many fields, such as, image analysis, taxonomy, astrophysics, etc. As there are approximatelyK ^N/K! possible ways of partitioning the patterns amongK clusters, finding the best solution is beyond exhaustive search whenN is large. We show that this problem, in spite of its exponential complexity, can be formulated as an optimization problem for which very good, but not necessarily optimal, solutions can be found by using a Hopfield model of neural networks. To obtain a very good solution, the network must start from many randomly selected initial states. The network is simulated on the MPP, a 128 × 128 SIMD array machine, where we use the massive parallelism not only in solving the differential equations that govern the evolution of the network, but also in starting the network from many initial states at once thus obtaining many solutions in one run. We achieve speedups of two to three orders of magnitude over serial implementations and the promise through Analog VLSI implementations of further speedups of three to six orders of magnitude.Supported by a National Research Council-NASA Research Associatship