Nucleotide Frequencies in Human Genome and Fibonacci Numbers

This work presents a mathematical model that establishes an interesting connection between nucleotide frequencies in human single-stranded DNA and the famous Fibonacci’s numbers. The model relies on two assumptions. First, Chargaff’s second parity rule should be valid, and second, the nucleotide frequencies should approach limit values when the number of bases is sufficiently large. Under these two hypotheses, it is possible to predict the human nucleotide frequencies with accuracy. This result may be used as evidence to the Fibonacci string model that was proposed to the sequence growth of DNA repetitive sequences. It is noteworthy that the predicted values are solutions of an optimization problem, which is commonplace in many of nature’s phenomena.     

Growth characteristics and growth modeling of <Emphasis Type="Italic">Microcystis aeruginosa</Emphasis> and <Emphasis Type="Italic">Planktothrix agardhii</Emphasis> under iron limitation

Although iron is a key nutrient for algal growth just as are nitrogen and phosphorus in aquatic systems, the effects of iron on algal growth are not well understood. The growth characteristics of two species of cyanobacteria, Microcystis aeruginosa and Planktothrix agardhii, in iron-limited continuous cultures were investigated. The relationships between dissolved iron concentration, cell quota of iron, and population growth rate were determined applying two equations, Monod’s and Droop’s equations. Both species produced hydroxamate-type siderophores, but neither species produced catechol-type siderophores. The cell quota of nitrogen for both M. aeruginosa and P. agardhii decreased with decreasing cell quota of iron. The cell quota of phosphorus for M. aeruginosa decreased with decreasing cell quota of iron, whereas those for P. agardhii did not decrease. Iron uptake rate was measured in ironlimited batch cultures under different degrees of iron starvation. The results of the iron uptake experiments suggest that iron uptake rates are independent of the cell quota of iron for M. aeruginosa and highly dependent on the cell quota for P. agardhii. A kinetic model under iron limitation was developed based on the growth characteristics determined in our study, and this model predicted accurately the algal population growth and iron consumption. The model simulation suggested that M. aeruginosa is a superior competitor under iron limitation. The differences in growth characteristics between the species would be important determinants of the dominance of these algal species.     

Growth-induced buckling of an epithelial layer

We use a proof-of-concept experiment and two mathematical models to explore growth-induced tissue buckling, as may occur in colorectal crypt formation. Our experiment reveals how growth of a cultured epithelial monolayer on a thin flexible substrate can cause out-of-plane substrate deflections. We describe this system theoretically using a ‘bilayer’ model in which a growing cell layer adheres to a thin compressible elastic beam. We compare this with the ‘supported-monolayer’ model due to Edwards and Chapman (Bull Math Biol 69:1927–1942, 2007) for an incompressible expanding beam (representing crypt epithelium), which incorporates viscoelastic tethering to underlying stroma. We show that the bilayer model can exhibit buckling via parametric growth (in which the system passes through a sequence of equilibrium states, parameterised by the total beam length); in this case, non-uniformities in cell growth and variations in cell–substrate adhesion are predicted to have minimal effect on the shape of resulting buckled states. The supported-monolayer model reveals how competition between lateral supports and stromal adhesion influences the wavelength of buckled states (in parametric growth), and how non-equilibrium relaxation of tethering forces influences post-buckled shapes. This model also predicts that non-uniformities in growth patterns have a much weaker influence on buckled shapes than non-uniformities in material properties. Together, the experiment and models support the concept of patterning by growth-induced buckling and suggest that targeted softening of a growing cell layer provides greater control in shaping tissues than non-uniform growth.     

Disparity estimation through Green’s functions of matching equations

Binocular disparities arise from positional differences of scene features projected in the two retinae, and constitute the primary sensory cue for stereo vision. Here we introduce a new computational model for disparity estimation, based on the Green’s function of an image matching equation. When filtering a Gabor-function-modulated signal, the considered Green’s function yields a similarly modulated but shifted version of the original signal. Since a Gabor function models the receptive field of a cortical simple cell, the Green’s kernel thus allows the simulation of relative shifts between the cell’s left and right binocular inputs. A measure of the local degree of matching of such shifted inputs can then be introduced which affords disparity estimation in a similar manner to the energy model of the complex cortical cells. We have therefore effectively reformulated, in physiologically plausible terms, an image matching approach to disparity estimation. Our experiments show that the Green’s function method allows the detection of disparities both from random-dot and real-world stereograms. Partially supported by CNPq-Brazil.     

Waddington redux: models and explanation in stem cell and systems biology

Stem cell biology and systems biology are two prominent new approaches to studying cell development. In stem cell biology, the predominant method is experimental manipulation of concrete cells and tissues. Systems biology, in contrast, emphasizes mathematical modeling of cellular systems. For scientists and philosophers interested in development, an important question arises: how should the two approaches relate? This essay proposes an answer, using the model of Waddington’s landscape to triangulate between stem cell and systems approaches. This simple abstract model represents development as an undulating surface of hills and valleys. Originally constructed by C. H. Waddington to visually explicate an integrated theory of genetics, development and evolution, the landscape model can play an updated unificatory role. I examine this model’s structure, representational assumptions, and uses in all three contexts, and argue that explanations of cell development require both mathematical models and concrete experiments. On this view, the two approaches are interdependent, with mathematical models playing a crucial but circumscribed role in explanations of cell development.     

Function of attention in learning process in the olfactory bulb

It has been suggested that in the olfactory bulb, odor information is processed through parallel channels and learning depends on the cognitive environment. The synapse’s spike effective time is defined as the effective time for a spike from pre-synapse to post-synapse, which varies with the type of synapse. A learning model of the olfactory bulb was constructed for synapses with varying spike effective times. The simulation results showed that such a model can realize the multi-channel processing of information in the bulb. Furthermore, the effect of the cognitive environment on the learning process was also studied. Different feedback frequencies were used to express different attention states. Considering the information’s multi-channel processing requirement for learning, a learning rule considering both spike timing and average spike frequency is proposed. Simulation results showed that habituation and anti-habituation of an odor in the olfactory bulb might be the result of learning guided by a common local learning rule but at different attention states.     

Energetics and dynamics of global integrals modeling interaction between stiff filaments

The attractive and spacing interaction between pairs of filaments via cross-linkers, e.g. myosin oligomers connecting actin filaments, is modeled by global integral kernels for negative binding energies between two intersecting stiff and long rods in a (projected) two-dimensional situation, for simplicity. Whereas maxima of the global energy functional represent intersection angles of ‘minimal contact’ between the filaments, minima are approached for energy values tending to −∞, representing the two degenerate states of parallel and anti-parallel filament alignment. Standard differential equations of negative gradient flow for such energy functionals show convergence of solutions to one of these degenerate equilibria in finite time, thus called ‘super-stable’ states. By considering energy variations under virtual rotation or translation of one filament with respect to the other, integral kernels for the resulting local forces parallel and orthogonal to the filament are obtained. For the special modeling situation that these variations only activate ‘spring forces’ in direction of the cross-links, explicit formulas for total torque and translational forces are given and calculated for typical examples. Again, the two degenerate alignment states are locally ‘super-stable’ equilibria of the assumed over-damped dynamics, but also other stable states of orthogonal arrangement and different asymptotic behavior can occur. These phenomena become apparent if stochastic perturbations of the local force kernels are implemented as additive Gaussian noise induced by the cross-link binding process with appropriate scaling. Then global filament dynamics is described by a certain type of degenerate stochastic differential equations yielding asymptotic stationary processes around the alignment states, which have generalized, namely bimodal Gaussian distributions. Moreover, stochastic simulations reveal characteristic sliding behavior as it is observed for myosin-mediated interaction between actin filaments. Finally, the forgoing explicit and asymptotic analysis as well as numerical simulations are extended to the more realistic modeling situation with filaments of finite length.

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Electronic supplementary material  The online version of this article (doi:) contains supplementary material, which is available to authorized users.      

EEG simulation: variation of spectral envelope, pulse synchrony and ≈40 Hz oscillation

 Macroscopic EEG travelling wave phenomena and cortical pulse synchronisation effects are related within a single simple simulation. Non-specific activation acts to control the transfer function of the simulated cortex, and thus determines the relative amplitude of macroscopic EEG waves generated by rhythmic inputs. When concurrent asynchronous excitatory inputs to separate, local, cortical sites are introduced, the simulation reproduces both gamma-band (40 Hz) electrocorticogram (ECoG) activity and synchronous oscillation of action potential pulse density at the separate sites. The gamma-band ECoG and pulse synchrony effects depend on different mechanisms: the former upon local excitatory/inhibitory interactions, and the latter on cortico-cortical interactions. The pattern of synchronous activity depends upon both structural and dynamic aspects of gain, and is sustained by linearised versions of the simulation’s state equations. Dynamic properties of the simulation, which are independent of scale, describe both microscopic and macroscopic phenomena, all in accord with physiological findings. Received: 25 June 1996 / Accepted in revised form: 29 November 1996     

The inverse niche model for food webs with parasites

Although parasites represent an important component of ecosystems, few field and theoretical studies have addressed the structure of parasites in food webs. We evaluate the structure of parasitic links in an extensive salt marsh food web, with a new model distinguishing parasitic links from non-parasitic links among free-living species. The proposed model is an extension of the niche model for food web structure, motivated by the potential role of size (and related metabolic rates) in structuring food webs. The proposed extension captures several properties observed in the data, including patterns of clustering and nestedness, better than does a random model. By relaxing specific assumptions, we demonstrate that two essential elements of the proposed model are the similarity of a parasite’s hosts and the increasing degree of parasite specialization, along a one-dimensional niche axis. Thus, inverting one of the basic rules of the original model, the one determining consumers’ generality appears critical. Our results support the role of size as one of the organizing principles underlying niche space and food web topology. They also strengthen the evidence for the non-random structure of parasitic links in food webs and open the door to addressing questions concerning the consequences and origins of this structure.     

Models of metamorphic timing: an experimental evaluation with the pond-dwelling salamander Hemidactylium scutatum (Caudata: Plethodontidae)

Amphibian larvae vary tremendously in size at metamorphosis and length of larval period. We raised pond-dwelling four-toed salamander (Hemidactylium scutatum) larvae to test two models that predict a larva’s age and size at metamorphosis. The Wilbur-Collins model proposes that the developmental rate of a larva responds to changes in growth rate in an adaptive manner throughout the larval period, and that metamorphosis can be initiated after a minimum size has been reached. The Leips-Travis or fixed-rate model states that developmental rate is set early in the larval period, perhaps by early growth rate or food availability and their positive correlation with developmental rate, and that changes in growth rate during the larval period affect size at metamorphosis, but have no effect on the age of an individual at metamorphosis. A modified version of the Wilbur-Collins model suggests that a larva’s developmental rate becomes fixed about two-thirds of the way through the larval period, with changes in growth rate after that point only affecting size at metamorphosis. Larvae were raised on eight different feeding regimes which created two constant and six variable growth histories. Growth history did significantly affect size at metamorphosis. However, an a posteriori statistical test revealed a group of seven and an overlapping group of six treatments with indistinguishable lengths of larval period, indicating a general picture of a fixed developmental rate regardless of growth history. This result is unique among similar studies on invertebrates, fish, and frogs. There was no association between early growth or food level and development rates. Neither the Wilbur-Collins nor the Leips-Travis fixed-rate models were supported. The invariable developmental rate of Hemidactylium and recent osteological evidence from the literature suggest that larvae begin the process of metamorphosis as soon as they hatch, probably a trait selected for by strong predation pressure in the aquatic environment. A variety of different approaches (ecological, developmental, phylogenetic) are necessary to fully evaluate the adaptive nature of the timing of transitions between life cycle stages. Received: 3 June 1999 / Accepted: 18 March 2000     

A Mathematical Model of Cell Cycle Effects in Gastric Cancer Chemotherapy

A mathematical model is presented to investigate the relationship between drug order and treatment response in gastric cancer chemotherapy involving a taxane (either paclitaxel or docetaxel) coupled with flavopiridol. To model treatment effects, we simulate treatment by bolus injection and employ a pulsing condition to indicate cell kill as well as instantaneous changes to the cell’s transition rates. Cell population growth is described using an ordinary differential equation model whereby we examine the treatment effects upon cells in various stages of the cell cycle. Ultimately, the results generated support prior clinical investigations which indicate that for an enhanced synergistic effect, flavopiridol must be administered following taxane therapy.     

Continuous-discrete models of the dynamics of an isolated population and of 2 competing species

The paper presents the analysis of various mathematical models for dynamics of isolated population and for competition between two species. It is assumed that mortality is continuous and birth of individuals of new generations takes place in certain fixed moments. Influence of winter upon the population dynamics and conditions of classic discrete model "deduction" of population dynamics (in particular, Moran-Ricker and Hassel's models) are investigated. Dynamic regimes of models under various assumptions about the birth and death rates upon the population states are also examined. Analysis of models of isolated population dynamics with nonoverlapping generations showed the density changes regularly if the birth rate is constant. Moreover, there exists a unique global stable level and population size stabilizes asymptotically at this equilibrium, i.e. cycle and chaotic regimes in various discrete models depend on correlation between individual productivity and population state in previous time. When the correlation is exponential upon mean population size the discrete Hassel model is realized. Modification of basis model, based on the assumption that during winter survival/death changes are constant, showed that population size at global level is stable. Generally, the dependence of population rate upon "winter parameters" has nonlinear character. Nonparametric models of competition between two species does not vary if the individual productivity is constant. In a phase space there are several stable stationary states and population stabilizes at one or other level asymptotically. So, in discrete models of competition between two species oscillation can be explained by dependence of population growth rate on the population size at previous times.     

Dynamics of the ‘echo’ effect in a phytoplankton system with nitrogen fixation

Consideration of nitrogen fixation adds a positive nonlinear feedback to plankton ecosystem models. We investigate the consequences of this feedback for secondary phytoplankton blooms and the response of phytoplankton dynamics to physical forcing. The dynamics of phytoplankton, Trichodesmium (the nitrogen fixer), and nutrients is modeled with a system of three differential equations. The model includes two types of nonlinear interactions: the competition of phytoplankton and Trichodesmium for light, and the positive feedback resulting from Trichodesmium recycling. A typical simulation of the model in time, with forcing by a varying mixed-layer depth, reveals a clear successional sequence including a secondary or ‘echo’ bloom of the phytoplankton. We explain this sequence of events through the stability analysis of three different steady states of the model. Our analysis shows the existence of a critical biological parameter, the ratio of normalized growth rates, determining the occurrence of ‘echo’ blooms and the specific sequence of events following a physical perturbation. The interplay of positive and negative feedbacks appears essential to the timing and the type of events following such a perturbation.     

Mechanical behavior of human embryonic stem cell pellet under unconfined compression

As a prelude to the understanding of mechanotransduction in human embryonic stem cell (hESC) differentiation, the mechanical behavior of hESCs in the form of cell pellet is studied. The pellets were tested after 3 or 5 weeks of cell culture in order to demonstrate the effect of the duration of cell culture on the mechanical properties of the pellets. A micromechanical tester was used to conduct unconfined compression on hESC pellet, and experimental, numerical, and analytical methods were combined to determine the mechanical properties of hESC pellet. It is assumed that the mechanical behavior of hESC pellets can be described by an isotropic, linear viscoelastic model consisting of a spring and two Maxwell units in parallel, and the Poisson’s ratio of the hESC pellet is constant based on pellet deformation in the direction perpendicular to the compression direction. Finite element method (FEM) simulation was adopted to determine the values of Poisson’s ratio and the five parameters contained in the viscoelastic model. The variations of Poisson’s ratio and the initial elastic modulus are found to be larger compared with those of the four other parameters. Results show that longer duration of cell culture leads to higher modulus of hESC pellet. The effect of pellet size error on the values of mechanical parameters determined is studied using FEM simulation, and it is found that the effect of size error on Poisson’s ratio and initial elastic modulus is much larger than that on the other parameters.     

Annual mitotic activity in the intestinal epithelium of the musselCrenomytilus grayanus

The seasonal dynamics of cell reproduction in the intestinal epithelium of the musselCrenomytilus grayanus are described in detail. Mitotic indices in the intestinal epithelium varied throughout the year from 0.005 to 0.26% (averaged data) and from 0.003 to 0.37% (individual data). Cyclic seasonal changes were found in the mussel’s intestinal epithelium. In general, the average values of mitotic activity in the intestinal epithelium were low (the mitotic index was 0.13%); there was a rise in activity in late April–June and September and a decline in July–August and especially in January–March. The winter-early spring period was characterized by a profound inhibition of cell reproduction and the transition of cells to the resting state. An outburst of proliferation occurred in the spring, due to a manifold increase in the number of cells in the mitotic cycle. The musselC. grayanus may be a good model for the study of the two extreme states of proliferation and their alternation in marine animals in nature. The diel dynamics of mitotic activity in the intestinal epithelium were followed during the most active growth period (May). The mitotic index (MI) varied during the day within a narrow range, deviating from the daily average value by no more than one third; no pronounced diel rhythm was found. Optimum water temperatures for cell reproduction ranged from 5 to 18°C.     

Modeling the pre- and post-synaptic components involved in the synaptic modification between cones and horizontal cells in carp retina

In retinal synapses between cones and luminosity type horizontal cells (LHC), it was previously found in this laboratory that repetitive red flashes progressively strengthened the LHC’s response to red flash, whereas weakened the LHC’s response to green flash; repetitive green flash remarkably depressed the LHC’s red response, but caused little changes in the cell’s green response. However, the detailed mechanisms underlying these phenomena are not entirely clear. In the present study, based on an ion-channel model described mainly in the form of Hodgkin–Huxley equations, possible mechanisms of the short-term synaptic modification are investigated. The simulation results suggest that: (1) the auto-enhancement effect might be induced by the Ca²⁺-dependent process on the post-synaptic AMPA receptors, which could lead to changes of the ionic channel’s properties; (2) the asymmetric response to red- and green-flashes and the mutual-chromatic suppression effects might be attributed to the regulatory effects on the presynaptic glutamate release.     

Structure and elastic properties of tunneling nanotubes

We investigate properties of a reported new mechanism for cell–cell interactions, tunneling nanotubes (TNT’s). TNT’s mediate actin-based transfer of vesicles and organelles and they allow signal transmission between cells. The effects of lateral pulling with polystyrene beads trapped by optical tweezers on TNT’s linking separate U-87 MG human glioblastoma cells in culture are described. This cell line was chosen for handling ease and possible pathology implications of TNT persistence in communication between cancerous cells. Observed nanotubes are shown to have the characteristic features of TNT’s. We find that pulling induces two different types of TNT bifurcations. In one of them, termed V-Y bifurcation, the TNT is first distorted into a V-shaped form, following which a new branch emerges from the apex. In the other one, termed I-D bifurcation, the pulled TNT is bent into a curved arc of increasingly broader span. Curves showing the variation of pulling force with displacement are obtained. Results yield information on TNT structure and elastic properties. Electronic supplementary material The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

A model for the emergence of adaptive subsystems

We investigate the interaction of learning and evolution in a changing environment. A stable learning capability is regarded as an emergent adaptive system evolved by natural selection of genetic variants. We consider the evolution of an asexual population. Each genotype can have ‘fixed’ and ‘flexible’ alleles. The former express themselves as synaptic connections that remain unchanged during ontogeny and the latter as synapses that can be adjusted through a learning algorithm. Evolution is modelled using genetic algorithms and the changing environment is represented by two optimal synaptic patterns that alternate a fixed number of times during the ‘life’ of the individuals. The amplitude of the change is related to the Hamming distance between the two optimal patterns and the rate of change to the frequency with which both exchange roles. This model is an extension of that of Hinton and Nowlan in which the fitness is given by a probabilistic measure of the Hamming distance to the optimum. We find that two types of evolutionary pathways are possible depending upon how difficult (costly) it is to cope with the changes of the environment. In one case the population loses the learning ability, and the individuals inherit fixed synapses that are optimal in only one of the environmental states. In the other case a flexible subsystem emerges that allows the individuals to adapt to the changes of the environment. The model helps us to understand how an adaptive subsystem can emerge as the result of the tradeoff between the exploitation of a congenital structure and the exploration of the adaptive capabilities practised by learning.     

Simulation of EEG: dynamic changes in synaptic efficacy, cerebral rhythms, and dissipative and generative activity in cortex

A simulation of electrocortical activity based upon coupled local aggregates of excitatory and inhibitory cells was modified to include rapid dynamic variations of synaptic efficacy attributable to reversal potentials and related effects. The modified simulation reproduces the rhythmic phenomena observed in real EEG, including the theta, alpha, beta and gamma rhythms, in association with physiologically realistic pulse densities. At high levels of cortical activation, generative activity with a 40-Hz center frequency emerges, suggesting a basis for the occurrence of phase changes and “edge of chaos” dynamics. These local oscillation properties complement the dissipative travelling wave and synchronous oscillation effects attributable to longer range excitatory couplings, as previously demonstrated in related simulations. Results of variation of parameters provide a first approximation to the anticipated effects of slow physiological time variations in gains and lags, and some predictions of the model are described. Received: 25 May 1998 / Accepted in revised form: 1 March 1999     

Estimating Behavioral Transition Rates: Problems and Solutions

Continuous time Markov chain (CTMC) models offer ethologists a powerful tool. The methods are based on well-established procedures for estimating the rates at which one state (e.g. resting) changes to some other set of states (e.g. feeding, fighting, etc.). Unfortunately, ethological data typically differ in a very critical manner from the type of data to which these methods are usually applied: ethological data are usually heavily censored in the sense that each behavioral state shows frequent transitions to several other possible states. This occurs when several competing processes can each end a bout.
We used computer simulation of various behavioral models with known transition rates to investigate the unknown performance of four of the most popular statistical tests for screening data prior to application of CTMC models; this included a modification of one of these tests derived under the assumption of random censoring. Two of the four tests failed completely and would result in rejection of nearly all data even if the model did fit the assumptions of the CTMC methods. Only Barlow's total-time-on test performed with an acceptable α error rate under all conditions. None of the tests were particularly effective at detecting certain types of departures from the CTMC assumptions.
Guidelines are given as to how much confidence should be attached to apparent changes in transition rates.