Analysis of chemotactic bacterial distributions in population migration assays using a mathematical model applicable to steep or shallow attractant gradients

The mathematical model developed by Riveroet al. (1989,Chem. Engng Sci. 44, 2881–2897) is applied to literature data measuring chemotactic bacterial population distributions in response to steep as well as shallow attractant gradients. This model is based on a fundamental picture of the sensing and response mechanisms of individual bacterial cells, and thus relates individual cell properties such as swimming speed and tumbling frequency to population parameters such as the random motility coefficient and the chemotactic sensitivity coefficient. Numerical solution of the model equations generates predicted bacterial density and attractant concentration profiles for any given experimental assay. We have previously validated the mathematical model from experimental work involving a step-change in the attractant gradient (Fordet al., 1991Biotechnol. Bioengng.37, 647–660; For and Lauffenburger, 1991,Biotechnol. Bioengng,37, 661–672). Within the context of this experimental assay, effects of attractant diffusion and consumption, random motility, and chemotactic sensitivity on the shape of the profiles are explored to enhance our understanding of this complex phenomenon. We have applied this model to various other types of gradients with successful intepretation of data reported by Dalquistet al. (1972,Nature New Biol. 236, 120–123) forSalmonella typhimurum validating the mathematical model and supportin the involvement of high and low affinity receptors for serine chemotaxis by these cells.     

Shotgun Crystallization Strategy for Structural Genomics II: Crystallization Conditions that Produce High Resolution Structures for T. maritima Proteins

Currently, 119 high resolution structures of Thermotoga maritima proteins have been determined by the Joint Center for Structural Genomics (JCSG, www.jcsg.org). Sixty-seven of these were solved using the first implementation of the multi-tiered crystallization strategy at the JCSG for the efficient crystallization of large numbers of protein targets. Previously, we reported the analysis of all proteins crystallized using this multi-tiered strategy [Lesley, S.A. et al. (2002) Proc. Natl. Acad. Sci. USA 99, 11664–11669; Page, R. et al. (2003) Acta Crystallogr. D Biol. Crystallogr. 59, 1028–1037]. Here, we extend the analysis and describe the crystallization characteristics of those proteins that produced diffraction quality crystals, ultimately resulting in high resolution structures. First, we found that over 77% (52) of the crystals used for structure determination were produced directly from high-throughput coarse screens, indicating that less than one quarter of the crystals (15) required fine screening. In addition, as observed for the proteome screen [Page, R. et al. (2003) Acta Crystallogr. D Biol. Crystallogr. 59, 1028–1037], the majority of conditions that produced crystals for natively expressed proteins, whose structures have been determined, were distinct from those of their more extensively purified and selenomethionine-labeled counterparts. Finally, 99% of the proteins whose structures were solved crystallized in conditions contained in the JCSG Minimal Core Screen [Page, R. et al. (2003) Acta Crystallogr. D Biol. Crystallogr. 59, 1028–1037; Page, R. and Stevens, R.C. (2004) Methods 34, 373–389], a set of 67 conditions previously identified as those most likely to produce crystals of a diverse set of proteins, confirming its success for rapid identification of proteins with a natural propensity to crystallize.     

Phytoplankton blooms and fish recruitment rate: Effects of spatial distribution

We consider the spatio-temporal dynamics of a spatially-structured generalization of the phytoplankton-zooplankton-fish larvae model system proposed earlier (Biktashev et al., 2003, J. Plankton Res. 5, 21–33; James et al., 2003, Ecol. Model. 160, 77–90). In contrast to Pitchford and Brindley (2001, Bull. Math. Biol. 63, 527–546), who were concerned with small scale patchiness (i.e., 1–10m), on which the (stochastic) raptorial behaviour of individual larvae is important, we address here the much larger scale ‘patchy’ problems (i.e., 10–100 km), on which both larvae and plankton may be regarded as passive tracers of the fluid motion, dispersed and mixed by the turbulent diffusion processes. In particular, we study the dependence of the fish recruitment on carrying capacities of the plankton subsystem and on spatio-temporal evolution of that subsystem with respect to the larvae hatching site(s). It is shown that the main features found both in the nonstructured and age-structured spatially uniform models are observed in the spatially structured case, but that spatial effects can significantly modify the overall quantitative outcome. Spatial patterns in the metamorphosed fish distribution are a consequence of quasi-local interaction of larvae with plankton, in which the dispersion of larvae by large scale turbulent eddies plays little part due to the relatively short timescale of the larvae development. As a result, in a strong phyto/zooplankton subsystem, with fast reproduction rate and large carrying capacity of phytoplankton and high conversion ratio of zooplankton, recruitment success depends only on the localization and timing of the hatching with respect to the plankton patches. In a weak phyto/zooplankton system, with slow reproduction rate and small carrying capacity of phytoplankton and low conversion ratio of zooplankton, the larvae may significantly influence the evolution of the plankton patches, which may lead to nontrivial cooperative effects between different patches of larvae. However, in this case, recruitment is very low.     

Capacity analysis in multi-state synaptic models: a retrieval probability perspective

We define the memory capacity of networks of binary neurons with finite-state synapses in terms of retrieval probabilities of learned patterns under standard asynchronous dynamics with a predetermined threshold. The threshold is set to control the proportion of non-selective neurons that fire. An optimal inhibition level is chosen to stabilize network behavior. For any local learning rule we provide a computationally efficient and highly accurate approximation to the retrieval probability of a pattern as a function of its age. The method is applied to the sequential models (Fusi and Abbott, Nat Neurosci 10:485–493, 2007) and meta-plasticity models (Fusi et al., Neuron 45(4):599–611, 2005; Leibold and Kempter, Cereb Cortex 18:67–77, 2008). We show that as the number of synaptic states increases, the capacity, as defined here, either plateaus or decreases. In the few cases where multi-state models exceed the capacity of binary synapse models the improvement is small.     

Two remarks on the mathematical biology of automobile driving

In connection with a series of previous papers by this author (Bulletin of Mathematical Biophysics,21, 299–308, 375–385;22, 257–262, 263–267;23, 19–29;24, 319–325) results obtained by A. Crawford (Economics 5, 417–428) on the effects of irrelevant lights on reaction times toward a given light stimulus are discussed. The conclusions from a previous paper of this author (Bulletin of Mathematical Biophysics,23, 19–29) are elaborated.     

Evolutionary tradeoff and equilibrium in an aquatic predator-prey system

Due to the conventional distinction between ecological (rapid) and evolutionary (slow) timescales, ecological and population models have typically ignored the effects of evolution. Yet the potential for rapid evolutionary change has been recently established and may be critical to understanding how populations persist in changing environments. In this paper we examine the relationship between ecological and evolutionary dynamics, focusing on a well-studied experimental aquatic predator-prey system (Fussmann et al., 2000, Science, 290, 1358–1360; Shertzer et al., 2002, J. Anim. Ecol., 71, 802–815; Yoshida et al., 2003, Nature, 424, 303–306). Major properties of predator-prey cycles in this system are determined by ongoing evolutionary dynamics in the prey population. Under some conditions, however, the populations tend to apparently stable steady-state densities. These are the subject of the present paper. We examine a previously developed model for the system, to determine how evolution shapes properties of the equilibria, in particular the number and identity of coexisting prey genotypes. We then apply these results to explore how evolutionary dynamics can shape the responses of the system to ‘management’: externally imposed alterations in conditions. Specifically, we compare the behavior of the system including evolutionary dynamics, with predictions that would be made if the potential for rapid evolutionary change is neglected. Finally, we posit some simple experiments to verify our prediction that evolution can have significant qualitative effects on observed population-level responses to changing conditions.     

On the quantitative relationship between survival rate of larvae and their food supply

The survival rate of fishes in their earlier stages of development and the influencing factors present one of the most fundamental problems of fish population dynamics. After I. Hjort's (Cons. L.'explor. Ner.,20, 3–228, 1914) work, there have been many investigators in this field and there is no doubt about the very important role of ova and larvae mortality in the fate of a given fish generation. Less clear are the ideas concerning factors determining the high mortality of fishes in their earlier stages of development; especially the factor of food supply of larvae during the period of transition to exogenic nutrition. The value of this factor has been estimated differently from different points of view. For example, R. J. H. Beverton and S. J. Holt (On the Dynamics of Exploited Fish Population, 1957) have given to the food supply factor its deserved importance. On the other hand, T. V. Dekhnik (Trudy Sevastopolskoi Biologicheskoi Stantsii,13, 216–244, 1960;Ibid.,14, 222–243, 1961) has proved in her investigations that at least for pelagic larvae of Black Sea fishes there is an excessive amount of food, and that therefore food cannot play an important role in larva survival. Not wanting to stop to review the literature of the problem (see Dekhnik,Trudy Sevastopolskoi Biologicheskoi Stantsii,13, 216–244, 1960), we will only remark that the problem as a whole needs further investigation. Not only new data are needed, but also methods for following up analysis have to be worked out.     

Effect of Novel Amino Acids and Dipeptides Substituted 3-Morpholino Arecoline Derivatives as Muscarinic Receptor 1 Agonists in Alzheimer’s Dementia Models

A series of novel, potent and selective muscarinic receptor 1 agonists (M1 receptor agonists) that employ a key N-substituted morpholine arecoline moiety has been synthesized as part of research effort for the therapy of Alzheimer’s diseases. The ester group of arecoline (which is reported as mucarinic agonist) has been replaced by N-substituted morpholine ring. The structure activity relationship reveals that the increase in lipophilic carbon chain on the nitrogen atom of the morpholine ring increases the affinity of M1 receptor. In the present study, we are reporting N-amino acid substituted 9(a–k) and dipeptides substituted 10(a–j) and 11(a–j) morpholino arecoline derivatives, along with their in vitro muscarinic binding studies by using [³H]QNB and also in vivo evaluation of memory and learning in male Wistar rats (passive avoidance test plus maze studies) as M1 receptor agonist. Some molecules from the dipeptide series (10b, 10c and 10j) showed potent M1 receptor agonist activity. Other derivatives also showed considerable M1 receptor binding affinity.     

‘Random coil’ 1H chemical shifts obtained as a function of temperature and trifluoroethanol concentration for the peptide series GGXGG

Summary Proton chemical shifts of a series of disordered linear peptides (H-Gly-Gly-X-Gly-Gly-OH, with X being one of the 20 naturally occurring amino acids) have been obtained using 1D and 2D ¹H NMR at pH 5.0 as a function of temperature and solvent composition. The use of 2D methods has allowed some ambiguities in side-chain assignments in previous studies to be resolved. An additional benefit of the temperature data is that they can be used to obtain ‘random coil’ amide proton chemical shifts at any temperature between 278 and 318 K by interpolation. Changes of chemical shift as a function of trifluoroethanol concentration have also been determined at a variety of temperatures for a subset of peptides. Significant changes are found in backbone and side-chain amide proton chemical shifts in these ‘random coil’ peptides with increasing amounts of trifluoroethanol, suggesting that caution is required when interpreting chemical shift changes as a measure of helix formation in peptides in the presence of this solvent. Comparison of the proton chemical shifts obtained here for H-Gly-Gly-X-Gly-Gly-OH with those for H-Gly-Gly-X-Ala-OH [Bundi, A. and Wüthrich, K. (1979) Biopolymers, 18, 285–297] and for Ac-Gly-Gly-X-Ala-Gly-Gly-NH₂ [Wishart, D.S., Bigam, C.G., Holm, A., Hodges, R.S. and Sykes, B.D. (1995) J. Biomol. NMR, 5, 67–81] generally shows good agreement for CH protons, but reveals significant variability for NH protons. Amide proton chemical shifts appear to be highly sensitive to local sequence variations and probably also to solution conditions. Caution must therefore be exercised in any structural interpretation based on amide proton chemical shifts.     

Comparison of surface potentials due to several singularity representations of the human heart

In electrocardiography the electrical potentials due to the heart actions can be analyzed by assuming the human body to be a conductor of homogeneous medium and the heart to be a combination of singularities within it. For a spherical conductor the “interior sphere theorem” of G. Ludford, J. Martinek, and G. Yeh (Proc. Cambridge Phil. Soc.,51, 389–93, 1955) renders potential expressions due to any singularity. For a conductor of prolate spheroidal shape the potential expressions due to a source-sink pair and a general dipole have been given by J. R. Wait (Jour. App. Physics,24, 496–97, 1953) and the authors (paper at the Conference on the Electrophysiology of the Heart, Feb. 16–17, 1956, in New York, to appear in theAnn. N. Y. Acad. Sciences) respectively. (A theorem which applies to any singularity inside a prolate or oblate spheroid will be published by the authors soon). This paper presents numerical and graphical results of potentials on the surfaces of a prolate spheroid and a sphere due to source-sink pairs and dipoles of several locations and directions and compares the various representations. A discussion on the judicious choice of heart models concludes the paper. This investigation was supported by The National Heart Institute under a research grant H-2263.     

To Rink or Not to Rink Amide Link, that is the Question to Address for More Economical and Environmentally Sound Solid-Phase Peptide Synthesis

A comparative study is presented on the solid-phase peptide synthesis (SPPS) of the acyl carrier protein (ACP 65–74) sequence on a series of Rink amide resins possessing different matrix structures: poly(vinyl alcohol)-graft-poly(ethylene glycol) (PVA-g-PEG, 4), Tentagel-S-RAM (TG, 5), NovaGel (NG, 6), ChemMatrix (CM, 7) and polystyrene-divinylbenzene (PS-DVB, 8). In this comparison, the PEG-containing resins proved significantly better suited for the synthesis of pure ACP target sequence than the conventional PS-DVB solid supports (75–90% versus 52% crude purity). Amongst themselves, the PEG resins 4-7 exhibited similar capacity for providing pure peptide. Selecting PVA-g-PEG resin for a comparison of Rink amide linker versus no linker, the ACP (65–74) sequence was synthesized directly on the PVA-g-PEG resin 1, under identical conditions as employed in the synthesis on resin 4 bearing the Fmoc Rink linker, except for the final cleavage step, which was performed under more environmentally sound conditions using ester displacement with aqueous ammonia. Relative to its Rink amide counterpart 4, PVA-g-PEG resin 1 was cheaper to produce and possessed twice as much loading capacity (0.48 vs. 0.81 mmol/g). Moreover, Rink-less resin 1 gave higher yields of isolated pure peptide (61 vs. 45%) relative to its Fmoc Rink linker counterpart 4. In light of these results, the importance of the linker has been brought into question. As the need for large scale solid-phase peptide synthesis grows with greater demand for peptide products, ideal resins should be inexpensive to produce and employable under environmentally sound conditions to provide pure products. In this light, PVA-g-PEG resin 1 has demonstrated significant promise for economic and “green” SPPS.     

Some considerations on relational equilibria

A previous study (Bull. Math. Biophysics,31, 417–427, 1969) on the definitions of stability of equilibria in organismic sets determined byQ relations is continued. An attempt is made to bring this definition into a form as similar as possible to that used in physical systems determined byF-relations. With examples taken from physics, biology and sociology, it is shown that a definition of equilibria forQ-relational systems similar to the definitions used in physics can be obtained, provided the concept of stable or unstable structures of a system determined byQ-relations is considered in a probabilistic manner. This offers an illustration of “fuzzy categories,” a notion introduced by I. Bąianu and M. Marinescu (Bull. Math. Biophysics,30, 625–635, 1968), in their paper on organismic supercategories, which is designed to provide a mathematical formalism for Rashevsky's theory of Organismic Sets (Bull. Math. Biophysics,29, 389–393, 1967;30, 163–174, 1968;31, 159–198, 1969). A suggestion is made for a method of mapping the abstract discrete space ofQ-relations on a continuum of variables ofF-relations. Problems of polymorphism and metamorphosis, both in biological and social organisms, are discussed in the light of the theory.     

On the number of balanced signed graphs

The classical enumeration theorem of Pólya (Acta Math.,68, 145–254, 1937) is applied to a modified version of Harary’s (Pacific J. Math.,8, 743–755, 1958) generating functions for counting bicolored graphs to derive a counting function for the number of balanced signed graphs. Methods for computing these counting polynomial functions are discussed.     

Kinetic study of DNA binding of cisplatin and of a bicycloalkyl-substituted (ethylenediamine)dichloroplatinum(II) complex

 Salmon sperm DNA platination has been conducted under strictly pseudo-first-order conditions with cisplatin (1) and rac-{(1S,2S,4S)-exo-2-(aminomethyl)-2-amino-7-bicyclo[2.2.1]heptane}dichloroplatinum(II) (2). An aquation step first occurs for both complexes, with the rate constants k ₁ = 1.12(0.02)×10^–4 s^–1 and 1.47(0.02)×10^–4 s^–1 respectively for 1 and 2 at 37  °C, values in agreement with those previously reported. It is followed by the actual platination step whose second-order rate constant has been determined for the first time by physicochemical techniques. The values for 1 and 2 respectively are: k ₂ = 2.08(0.07) M^–1 s^–1 and 3.9(0.4) M^–1 s^–1. These kinetic data are discussed in the context of a comparison of several biological properties of the two complexes. Received: 15 May 1998 / Accepted: 26 June 1998     

A note on epilepsies and similar disorders

This paper compares two previously published neural models for epilepsies (Bull. Math. Biophysics,33, 539–553, 1971;34, 71–78, 1972). The second model is developed in more detail and an attempt is made to bring it more in line with established neurological findings. The question of classification of some epilepsies is briefly discussed.     

Homeostatic control of thyroxin concentration expressed by a set of linear differential equations

The purpose of this work is to express current concepts on the relationship between the rates of secretion of thyroxin and of thyroid stimulating hormone (TSH) by a set of linear differential equations (two attempts have been made previously in this direction; cf. Roston,Bull. Math. Biophysics,21, 271–282, 1959; Danziger and Elmergreen,Bull. Math. Biophysics,16, 15–21, 1954), and to show that the solutions to these equations fulfill two criteria: that they correctly express the previously observed behavior of thyroxin and TSH, and that they allow certain predictions to be made which are amenable to experimental verification or disproval by currently existing techniques. This mathematical model is necessarily only an approximation of reality.     

Colicins—Exocellular lethal proteins ofEscherichia coli

Colicins are toxic exoproteins produced by bacteria of colicinogenic strains ofEscherichia coli and some related species ofEnterobacteriaceae, during the growth of their cultures. They inhibit sensitive bacteria of the same family. About 35%E. coli strains appearing in human intestinal tract are colicinogenic. Synthesis of colicins is coded by genes located on Col plasmids. Until now more than 34 types of colicins have been described, 21 of them in greater detail,viz. colicins A, B, D, E1–E9, Ia, Ib, J_S, K, M, N, U, 5, 10. In general, their interaction with sensitive bacteria includes three steps: (1) binding of the colicin molecule to a specific receptor in the bacterial outer membrane; (2) its translocation through the cell envelope; and (3) its lethal interaction with the specific molecular target in the cell. The classification of colicins is based on differences in the molecular events of these three steps. The original version of this review was published in Czech in the journal “Biologické listy”,62, 107–130 (1997).     

Analysis of tracer experiments: IV. The kinetics of generalN compartment systems

The methods of C. W. Sheppard and A. S. Householder (Jour. App. Physcis,22, 510–20, 1951), H. D. Landahl (Bull. Math. Biophysics,16, 151–54, 1954) and H. E. Hart (Bull. Math. Biophysics,17, 87–94, 1955;ibid.,19, 61–72, 1957;ibid.,20, 281–87, 1958) are employed in studying the kinetics of generalN compartment systems. It is shown that the nature of the transfer processes occurring in fluid flow systems and the chemical processes occurring in quadratic systems and in catalyzed quadratic systems can in principle be completely determined for all polynomial dependencies. Systems involving three-body and higher-order interactions can be completely solved, however, only if supplementary information is available. Research supported by the Atomic Energy Commission, Contract AT (30-1)-1551.     

Effect of varying salt and urea permeabilities along descending limbs of Henle in a model of the renal medullary urine concentrating mechanism

Modelling studies have played an important role in research on the mechanism of urine concentration and dilution by the medulla of the kidney ever since Hargitay and Kuhn (1951,Z. Elektrochem. 55, 539–558) first proposed that the parallel tubular structures in the kidney medulla must function as a “countercurrent multiplication” system. Present-day models, in keeping with our considerably improved understanding of most aspects of medullary structure-function relationships, have evolved into rather sophisticated systems of parallel tubes. In spite of this increasing complexity, it has remained the case that “model medullas” do not concentrate as well as the real kidney, especially in the inner medulla where only passive, diffusional transport occurs. Inasmuch as these models take into account the majority of contemporary ideas making up our global hypothesis about the functioning of this system, their failure to behave physiologically indicates that our understanding remains incomplete. The purpose of the present modelling study was to evaluate the implications of some recent measurements showing that permeabilities of NaCl (P _s ) and urea (P _u ) vary along the length of the descending thin limbs of Henle (Imaiet al., 1988,Am. J. Physiol. 254, F323–F328), rather than being constant throughout this segment as had been assumed earlier. It was hoped that these newly measured values might explain, by a passive, diffusional process, the net solute addition at the bend of Henle’s loop observed under some circumstances and heretofore attributed (though without any supporting experimental evidence) to active transport into the descending limb. The results of the present study show that whereas incorporation of the new values forP _s andP _u in the descending limbs of short nephrons does indeed improve the concentrating power of the model, these new values are nonetheless not sufficient to allow the model to build an osmolarity gradient that increases all the way through the inner medulla. This failing, which is common to virtually all modelling studies to date using measured values from rat kidneys, probably points to a key role for preferential exchange supposed by some to exist among certain tubule segments within vascular bundles in species whose kidneys have the highest concentrating power.