Stochastic models for subpopulation emergence in heterogeneous tumors

A stochastic analog to a deterministic model describing subpopulation emergence in heterogeneous tumors is developed. The resulting system is described by the Fokker-Planck or forward Kolmogorov equation. A finite element approach for the numerical solution to this equation is described. Four biological and clinical scenarios are simulated (emergence of heterogeneity, exclusion of a subpopulation, and induction of drug resistance in both pure and heterogeneous tumors). The results of the simulations show that the stochastic model describes the same basic dynamics as its deterministic counterpart via a convective component, but that for each simulation a distribution of tumor sizes and mixes can also be derived from a diffusive component in the model. These distributions yield estimates for subpopulation extinction probabilities. The biological and clinical relevance of these results are discussed. Research support provided in part by ACS Grant IN45-Z and ACS PDT 243 B. Research support provided in part by the National Science Foundation under NSF Grant MCS-8504316, and by the Air Force Office of Scientific Research under Contract F49620-86-C-0111.     

Diffusion-driven instability in a predator-prey system with time-varying diffusivities

We consider an ecological model by Levin and Segel (1976) for predator-prey planktonic species, which consists of two reaction-diffusion equations, and extend it to plankton populations with time-varying diffusivities. The local stability of uniform equilibria is examined both analytically and numerically. It is found that diffusive instability is less likely to occur in systems with time-varying diffusivity than those with constant diffusivity. Contribution No. 803 of the Marine Sciences Research Center, State University of New York, Stony Brook Supported by the Danish Science Research Council (Grant nos. 11-7128, 11-8321), the Danish Research Academy (Grant nos. E-880011, V-890085) and a Travel Grant for Mathematicians (Rejselegat for Matematikere) Supported by Hudson River Foundation, Grant no. 01488AO37     

An asymptotic solution for traveling waves of a nonlinear-diffusion Fisher's equation

We examine traveling-wave solutions for a generalized nonlinear-diffusion Fisher equation studied by Hayes [J. Math. Biol. 29, 531–537 (1991)]. The density-dependent diffusion coefficient used is motivated by certain polymer diffusion and population dispersal problems. Approximate solutions are constructed using asymptotic expansions. We find that the solution will have a corner layer (a shock in the derivative) as the diffusion coefficient approaches a step function. The corner layer at z = 0 is matched to an outer solution for z < 0 and a boundary layer for z > 0 to produce a complete solution. We show that this model also admits a new class of nonphysical solutions and obtain conditions that restrict the set of valid traveling-wave solutions. Supported by a National Science Foundation graduate fellowship. This work was performed under National Science Foundation grant DMS-9024963 and Air Force Office of Scientific Research grant AFOSR-F49620-94-1-0044.     

Differential contributions of circular and elongated spatial filters to the Café Wall illusion

The Café Wall illusion is induced in part by the spatial filtering properties of circular-symmetric receptive fields at the retina, and in part by cortical processes, presumably that have elongated receptive fields. Two experiments are reported that address these two contributions. The first shows a reversal effect that is consistent with brightness-induction and related theories. The second shows extinction of the effect in a manner predicted by recent observations of gating nonlinearity in cortical cells. Supported by Air Force Office of Scientific Research Grant AFSR-85-0359. The authors gratefully acknowledge the insightful suggestions provided by Allen Brookes     

An analysis of the kinematics of swarming ofAnarete pritchardi kim (Diptera: Cecidomyiidae)

Summary Movies of swarming of a cecidomyiid midgeAnarete pritchardi Kim, are analysed to provide kinematic data for swarming. Statistical characteristics calculated from the basic data are the mean, variance, standard deviation, skewness, and kurtosis of the insect coordinates; insect number-density distribution in space, frequency distribution of insect velocities; velocity autocorrelation coefficients; effective advective velocity and diffusivity for swarming, among others. Analyses show that the motion inside the swarm is more or less random although we occasionally notice some kind of interaction between individuals. An inward velocity is detected for insects flying at the edge of the swarm. Mathematical models based on a combined process of diffusion and of nondiffusion are compared with the kinetic data to evaluate the parameters involved in the model. The result indicates that the mathematical model is realistic and useful. A few features of midge behavior in swarming flight are also noted. The result obtained for midge kinetics should be applicable to swarms of some other insects and animals. Contribution No. 211 from Chesapeake Bay Institute, Department of Earth and Planetary Sciences, The Johns Hopkins University, and Paper No. 8791 Scientific Journal Series, Agricultural Experimental Station, University of Minnesota. Supported by the Office of Naval Research under research contract N00014-67-A-0163-0006, Research Project NR083-016, and National Science Foundation Grant, GA-16603 to The Johns Hopkins University, and National Science Foundation Grant GB-1755 to the University of Minnesota.     

Multi-species interactions with hereditary effects and spatial diffusion

Summary We study the effect of spatial diffusion on oscillatory states in arbitrary multi-species growth models having hereditary terms. We show that it is a general principle that the addition of spatial diffusion to a stable oscillatory ecological community induces a periodic diffusion wave in which the original wavenumber (or phase) evolves according to a nonlinear evolution equation of generalized Burgers' type.Supported in part by the U.S. Army Research Office (Durham) under Contract DAHC-04-68-C-0006 and by the National Science Foundation under Grant GP-32157X2     

Consensus weak hierarchies

An axiomatic characterization is presented for consensus functions defined on weak hierarchies. These functions are generalizations of the majority rule consensus. Research supported by Grant N00014-89-J-1643 from the Office of Naval Research.     

Transient behavior of population density with competition for resources

In this paper we derive a perturbation expansion of some nonlinear diffusion equations proposed by J. G. Skellam and M. Kimura. The solution is derived by converting the differential equations into integral equations by means of the Green's function for the diffusion equation. This research was supported in part by the Office of Naval Research under Contract Nonr-595(17).     

Global asymptotic stability of a periodic solution to an epidemic model

In this paper a periodic delay differential equation with spatial spread is investigated. This equation can be used to model the growth of malaria which is transmitted by a mosquito. Using monotone techniques, it is shown that the following bifurcation holds: either the disease dies out or the density of infectious people tends to a spatially homogeneous, time periodic and positive solution.Research partially supported by NSF Grant MCS 810-4837     

An avoidance learning situation. A neural net model

A simple avoidance situation is considered in terms of a neural net learning model. Data for the control situation can be represented by an expression having three parameters which determine the initial and the steady state activities together with the transient aspects. The introduction of a learning parameter then allows one to calculate satisfactorily the results obtained in the experimental situation in which shock is applied. This research was supported in part by the United States Air Force through the Air Force Office of Scientific Research of the Air Research Development Command under Grant No. AF AFOSR 370-63 and in part by the United States Public Health Service Grant RCA GM K6 18,420.     

Transmitter contents of cells and fibers in the cephalic sensory organs of the gastropod mollusc<Emphasis Type="Italic"> Phestilla sibogae</Emphasis>

While the central ganglia of gastropod molluscs have been studied extensively, relatively little is known about the organization and functions of the peripheral nervous system in these animals. In the present study, we used immunohistochemical procedures to examine the innervation of the rhinophores, oral tentacles and region around the mouth of the aeolid nudibranch, Phestilla sibogae. Serotonin-like immunoreactivity was found in an extensive network of efferent projections apparently originating from central neurons, but was not detected within any peripheral cell bodies. In contrast, large numbers of peripheral, and presumably sensory, somata exhibited reactivity to an antibody raised against tyrosine hydroxylase (the enzyme catalyzing the initial step in the conversion of tyrosine into the catecholamines). Additional tyrosine hydroxylase-like immunoreactivity was detected in afferent fibers of the peripheral cells and in several cells within the rhinophoral ganglia. The presence of serotonin, dopamine and norepinephrine in the rhinophores, tentacles and central ganglia was confirmed using high-performance liquid chromatography. Finally, FMRFamide-like immunoreactivity was detected in cells and tangles of fibers found within the rhinophore, possibly revealing glomerulus-like structures along olfactory pathways. FMRFamide-like immunoreactivity was also found in somata of the rhinophoral ganglia, in a small number of cells located in the body wall lateral to the tentacles and in what appeared to be varicose terminals of efferent projections to the periphery. Together, these results indicate several new features of the gastropod peripheral nervous system and suggest future experiments that will elucidate the function of the novel cells and innervation patterns described here.This research was supported by Natural Sciences and Research Council of Canada Grant #OPG38863 to R.P.C. and Office of Naval Research Grant #N00014-94-1-0524 to M.G.H.     

Decision equation for binary systems. Application to neuronal behavior

Summary The exact mathematical treatment is given for a non linear equation describing the delayed yes-or-no response to a binary system to external stimulations, in some typical cases of interest. Comparison is made with neurophysiological data on the frequency rate of firings of stimulated neurons; the same equation, however, can be conceivably applied to a vast variety of phenomena. The procedures followed to solve the problems that arise in connection with this equation could be extended to more general types of non linear equations.The research reported in this document has been sponsored in part by the Air Force Office of Scientific Research under Contract no. AF EOAR 65-44 through the European Office of Aerospace Research, OAR, United States Air Force.     

Structure and function of the photosynthetic reaction center fromRhodobacter sphaeroides

The three-dimensional structure of the photosynthetic reaction center fromRhodobacter sphaeroides is described. The reaction center is a transmembrane protein that converts light into chemical energy. The protein has three subunits: L, M, and H. The mostly helical L and M subunits provide the scaffolding and the finely tuned environment in which the chromophores carry out electron transfer. The details of the protein-chromophore interactions are from studies of a trigonal crystal form that diffracted to 2.65-Å resolution. Functional studies of the multi-subunit complex by site-specific replacement of key amino acid residues are summarized in the context of the molecular structure.This work was supported in part by the U.S. Department of Energy, Office of Health and Environmental Research, under Contract No. W-31-109-ENG-38 and by Public Health Service Grant GM36598.     

Monoaminergic mechanisms in the nervous system of Lumbricus terrestris (L.)

Summary The localization and intraneuronal distribution of the monoaminergic transmitters in the nervous system of the earthworm, Lumbricus terrestris, have been investigated in detail with the aid of the histochemical fluorescence method of Falck and Hillarp.In the ventral nerve cord, many yellow fluorescent, 5-hydroxytryptamine containing neurons are found, but only few green fluorescent noradrenaline containing cell bodies, which, however, are numerous in the peripheral nervous system. There is an abundance of both fibre types in the neuropile.The 5-hydroxytryptaminergic neurons probably have a motor (possibly inhibitor) function; the adrenergic neurons in the body segments are supposed to have a receptor (exteroceptive and possibly proprioceptive) function.In the cerebral ganglion, both 5-hydroxytryptamine and noradrenaline containing neurons are found in large numbers, and there are closely packed numerous fibres of both types in the neuropile. Their function is more obscure, though an associative function can be presumed for some adrenergic neurons; smaller 5-hydroxytryptaminergic neurons might have a motor (perhaps inhibitor) function.Adrenergic sensory cells are found in the body integument, most frequently in the clitellum segments, in the prostomium, and in the roof of the buccal cavity. These cells give off varicose fibres that form a basi-epithelial network which is in communication with the green fluorescent sensory fascicles in the ventral nerve cord via the epidermal nerves, the ring nerves, and the segmental nerves. No direct adrenergic sensory-effector innervation of either circular and/or longitudinal musculature or gland cells seems to exist. No adrenergic free nerve endings in the body integument have been observed. Instead, there must be a synaptic contact with the motoneurons, either directly in the neuropile or via an interjacent neuron.No synaptic contacts have been observed in the ventral nerve cord between adrenergic or 5-hydroxytryptaminergic fibres and either the giant fibres or fluorescent or nonfluorescent perikarya.An adrenergic innervation of the pharynx musculature has been found, and sensory cells of a different type are present in and below the epithelium; here, a direct senso-motoric innervation of the pharyngeal musculature cannot be excluded. It is established that the adrenergic neurons in the stomatogastric nervous system have an exciting function on the pharynx, whereas a direct monoaminergic influence of the muscular movements of the intestine probably does not exist.Abbreviations Used A adrenaline - CA catecholamine - DA dopamine - 5-HT 5-hydroxytryptamine - MA monoamine - NA noradrenaline The research reported in this document has been sponsored by the Air Force Office of Scientific Research under Grant AF EOAR 67-15 through the European Office of Aerospace Research (OAR), United States Air Force, by the Swedish Natural Science Research Council (99-34, 6627), and by the Swedish Medical Research Council (B67-12X-712-02A).     

Discussion: McCulloch-Pitts and related neural nets from 1943 to 1989

The McCulloch-Pitts paper “A Logical Calculus of the Ideas Immanent in Nervous Activity” was published in theBulletin of Mathematical Biophysics in 1943, a decade before the work of Hodgkin, Huxley, Katz and Eccles. The McCulloch-Pitts neuron is an extremely simplified representation of neural properties, based simply on the existence of a threshold for the activation of an action potential. This work has been supported in part by Grants from the University of Chicago Brain Research Foundation, and the U.S. Department of the Navy, Office of Naval Research (Grant No. N 00014-89J-1099).     

The Danziger-Elmergreen theory of periodic catatonic schizophrenia

Properties of the solutions of the mathematical model introduced by Danziger and Elmergreen for the study of periodic catatonic schizophrenia are studied, and it is shown that the properties established suggest that the model can be used to study other forms of catatonic schizophrenia besides periodic catatonic schizophrenia. The research in this paper was supported by the U.S. Army Research Office (Durham) (Grant No. DA-ARO-D-31-124-G1098) and by a Rutgers Research Council Faculty Fellowship.     

Mathematical aspects of peridic catatonic schizophrenia

The mathematical study of periodic catatonic schizophrenia initiated by Danziger and Elmergreen is generalized by considering the class of differential equations that could be used to describe the periodic behavioral symptoms and periodic variation of biochemical levels in periodic catatonic schizophrenia. The existence of asymptotically stable periodic solutions is studied mathematically and the physical significance of such periodic solutions is discussed. The occurrence of relaxation oscillations is briefly considered. The research in this paper was supported partly by the U.S. Army Research Office (Durham) (Grant No. DA-ARO-D-31-124-72-C69) and partly by a Rutgers Research Council Faculty Fellowship.     

Expression of enzymatically active and correctly targeted strictosidine synthase in transgenic tobacco plants

Strictosidine, a precursor to over 1000 indole alkaloids including the anti-tumor drugs vinblastine, vincristine, and camptothecin, is produced by the condensation of tryptamine and secologanin. Strictosidine synthase, the enzyme responsible for this condensation, is the first committed step in the indole-alkaloid pathway. We have introduced a modified cDNA encoding Strictosidine synthase from Catharanthus roseus (L.) Don. (McKnight et al. 1990, Nucl. Acids Res. 18, 4939) driven by the CaMV 35S promoter into tobacco (Nicotiana tabacum L.). Transgenic tobacco plants expressing this construct had from 3 to 22 times greater strictosidinesynthase activity than C. roseus plants. Ultrastructural immunolocalization demonstrated that strictosidine synthase is a vacuolar protein in C. roseus and is correctly targeted to the vacuole in transgenic tobacco. Immunoblot analysis of strictosidine synthase showed that two distinct forms of the enzyme were produced in transgenic tobacco plants but that only a single form was made in C. roseus. This observation indicates that the second form of the protein is not simply a result of overexpression in tobacco, but may reflect differences in protein processing between tobacco and C. roseus.Abbreviations cDNA complementary DNA - TLC thin-layer chromatography We thank Dr. C.A. Roessner for providing the E. coli strain expressing strictosidine synthase, Dr. J. Balsevich for providing alkaloid standards, and Dr. L. Cloney for assisting with antibody preparation. This work was supported by a National Institutes of Health Biomedical Research Support Grant to T.D.M and by a grant from the US Department of Agriculture, Competitive Research Grants Office (90-37262-5375) to C.L.N.     

Electron microscopic observations on the reactional changes occurring in insect nerve fibers after transection

Summary An electron microscopic study of the interganglionic connectives of an orthopteran (Laplatacris dispar) demonstrated that a reaction starts in the sectioned fibers very soon after transection (30 minutes). This reaction is closely similar to that observed in sectioned nerves of vertebrates and consists in the appearance of microvesicles and the proliferation of mitochondria. Sectioned connectives were studied from 30 minutes to 88 hours after section. The reaction mentioned increases progressively during the indicated lapse of time.This research was supported by the United States Air Force under Contract No AF 49 (638) 585 and Grant Af 61-64, monitored by the Air Force Office of Scientific Research, Air Force Research Division of the Air Research and Development Command.     

On visual adaptation: I. Photochemistry

Quantitative aspects of the photochemistry of visual adaptation are considered. A simplied model is given that fits data on changes of rhodopsin concentration during and following strong illumination. A variation on Wald’s compartment hypothesis is shown to fit the quasilinear dependence of log threshold upon pigment concentration. Finally, there is a brief review of pertinent data on cone pigments. This research was supported in part by the United States Air Force through the Air Force Office of Scientific Research of the Air Research Development Command under Contract No. AF(638)-414, and in part by the United States Public Health Service Training Grant 2G-833.