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  • 1
    Keywords: cell signaling ; Mixed differential equations ; Finite elements ; Mesh generation ; 34A34 ; 34D05 ; 34D20 ; 35K57 ; 65N30 ; 65N50
    Abstract: Among other approaches, differential equations are used for a deterministic quantitative description of time-dependent biological processes. For intracellular systems, such as signaling pathways, most existing models are based on ordinary differential equations. These models describe temporal processes, while they neglect spatial aspects. We present a model for the SMAD signaling pathway, which gives a temporal and spatial description on the basis of reaction diffusion equations to answer the question whether cell geometry plays a role in signaling. In this article we simulate the ordinary differential equations as well as partial differential equations of parabolic type with suile numerical methods, the latter on different cell geometries. In addition to manual construction of idealized cells, we also construct meshes from microscopy images of real cells. The main focus of the paper is to compare the results of the model without and with spatial aspects to answer the addressed question. The results show that diffusion in the model can lead to significant intracellular gradients of signaling molecules and changes the level of response to the signal transduced by the signaling pathway. In particular, the extent of these observations depends on the geometry of the cell.
    Type of Publication: Journal article published
    PubMed ID: 22986890
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  • 2
    ISSN: 1432-1416
    Keywords: NK cells ; Immune surveillance
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Summary The theory of immune surveillance of Thomas and Burnet stated in part that antigenic differences between neoplastic and normal cells provide the stimulus for their destruction by cells of the immune system. Burnet pointed to the T lymphocyte as the cell which mediated this surveillance. The existence of some form of surveillance in cases of no T lymphocyte functioning presents the possibility that surveillance, if present at all, is mediated by non T cells. Cells identified as naturally cytotoxic killer (NK) cells appear to have properties required of a surveillance effector population. This paper utilizes properties of NK cells and the effects of interferon on this population to construct a mathematical model of the characteristics that an NK cell surveillance would have. A two level theory of immune surveillance is proposed.
    Type of Medium: Electronic Resource
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  • 3
    ISSN: 1432-1416
    Keywords: Water recycling ; Lactation ; Mus ; Modelling
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract 1. The exchanges of water between lactating female and young Mus musculus were modelled on the computer. 2. The model was used to estimate rates of milk production and water recycling in various litter sizes under various water regimes by following the time course of injected tritiated water. 3. The high correlation between estimated rates of milk production and actual growth rates of young was taken to indicate that the method gave if not the actual rates of milk production a very constant proportion of it. 4. Approximately 50% of the water secreted in milk is returned to the mother by recycling.
    Type of Medium: Electronic Resource
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  • 4
    ISSN: 1432-1416
    Keywords: Immune response ; Delay ; Model
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract A model of the humoral immune response, proposed by Dibrov, Livshits and Volkenstein (1977b), in which the antibody production by a constant target cell population depends on the antigenic stimulation at earlier times, is considered from an analytic standpoint. A method of approximation based on a consideration of the asymptotic limit of “large” delay in the antibody response is shown to be applicable, and to give results similar to those obtained numerically by the above authors. The relevance of this type of approximation to other systems exhibiting “outbreak” phenomena is discussed.
    Type of Medium: Electronic Resource
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  • 5
    ISSN: 1432-1416
    Keywords: Stochastic equilibrium ; Population dynamics
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Existence, uniqueness and asymptotic stability of stochastic equilibrium are established in multi-dimensional population models with monotone dynamics.
    Type of Medium: Electronic Resource
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  • 6
    Electronic Resource
    Electronic Resource
    Springer
    Journal of mathematical biology 13 (1981), S. 47-65 
    ISSN: 1432-1416
    Keywords: Pattern recognition systems ; Structure of biological models ; Classifier and memory ; Detection experiments
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract This paper addresses the problem of analyzing biological pattern recognition systems. As no complete analysis is possible due to limited observability, the theoretical part of the paper examines some principles of construction for recognition systems. The relations between measurable and characteristic variables of these systems are described. The results of the study are: 1. Human recognition systems can always be described by a model consisting of an analyzer (F A) and a linear classifier. 2. The linearity of the classifier places no limits on the universal validity of the model. The principle of organization of such a system may be put into effect in many different ways. 3. The analyzer function F A determines the transformation of external patterns into their internal representations. For the experiments described in this paper, F A can be approximated by a filtering operation and a transformation of features (contour line filter). 4. Narrow band filtering (comb filter) in the space frequency domain is inadequate for pattern recognition because noise of different bandwidths and mean frequencies affects sinusoidal gratings differently. This excludes the use of a Fourier analyzer. 5. The relations between the measurable variables, which are the probabilities of detection (P D curves), and the characteristic variables of the recognition system are established analytically. 6. The probability of detection not only depends on signal energy but also on signal structure. This would not be the case in a simple matched filter system. 7. The differing probabilities of error in multiple detection experiments show that the interference is pattern specific and the bandwidth (steepness of the P D curves) is different for the different sets of patterns. 8. The distance between the reference vectors in feature space can be determined from the internal representation of the patterns defined by the model. Through multiple detection experiments it is possible to determine not only the relative distances between the patterns but also their absolute position in feature space.
    Type of Medium: Electronic Resource
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  • 7
    ISSN: 1432-1416
    Keywords: Antibody response dynamics ; B lymphocyte differentiation ; Control by antigen
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract A mathematical model of B lymphocyte differentiation, based on experimental results, has been developed. The model focuses on the role of antigen in initiating and regulating B cell differentiation while other mechanisms, acting in concert with antigen but the functioning of which can be circumvented under appropriate conditions, are not considered. The importance of presence of antigen at individual stages of B cell differentiation was studied in experiments with an easily metabolizable antigen. Immunocompetent cells (ICC), arising by antigen-independent differentiation of stem cells, are activated by antigen (they become immunologically activated cells — IAC). Excess of antigen drives IAC into the terminal stage (antibody-forming cells — AFC) thereby restricting proliferation. Exhaustive terminal differentiation results in tolerance. A low primary dose permits IAC to escape antigen; IAC proliferate and later give rise to resting memory cells (MC) which are amenable to reactivation. MC have higher avidity for antigen (due to higher affinity, number and density of receptors) and the effect of different doses of antigen on MC is diverse. A very low secondary dose induces tolerance, a medium dose secondary response, and the administration of a high dose of antigen also brings about tolerance. The model suggests that the fate of memory cells is controlled by the ratio R∶Ag, of the number of immunoglobulin receptors on B cells (R) to the number of available antigenic molecules (Ag), low values of R∶Ag favouring stimulation to differentiation while high values of R∶Ag favouring inactivation. A nonlinear system of ordinary differential equations, describing the development of the populations involved in antigen driven B cell differentiation, was used to simulate experiments and good qualitative agreement was achieved.
    Type of Medium: Electronic Resource
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  • 8
    ISSN: 1432-1416
    Keywords: Drug receptor interaction ; Stochastic model ; Birth and death processes ; Postsynaptic current noise ; Muscle relaxants ; Competition ; Ionic channel blocking
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Stochastic modelling of drug receptor interaction at the neuromuscular junction with birth and death processes is presented. Systems with only one drug and two drugs interacting are investigated separately. The total number of receptors is assumed finite and only single receptor binding is considered. It has been possible to obtain exact analytical solutions to forward Kolmogorov equations yielding the transition probability densities and the associated probability generating functions.
    Type of Medium: Electronic Resource
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  • 9
    Electronic Resource
    Electronic Resource
    Springer
    Journal of mathematical biology 12 (1981), S. 199-214 
    ISSN: 1432-1416
    Keywords: Reaction-diffusion equations ; Bifurcation analysis
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract We investigate the stability of rotating waves of reaction-diffusion equations by deriving the bifurcation equations for the simplest time-periodic patterns defined in the r, θ plane of polar coordinates. We prove that stable rotating waves can effectively be expected either after a primary or a secondary bifurcation point.
    Type of Medium: Electronic Resource
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  • 10
    ISSN: 1432-1416
    Keywords: Harmonic balance ; Stiff differential equations ; Non-linear algebraic equations ; Enzyme catalysis
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The method of unrestricted harmonic balance (UHB) is applied to a model system of oscillations in enzyme catalysis, which in view of the stiffness of the differential equations cannot be solved by direct simulation and requires special methods. The solution by UHB is achieved in two steps: first the reduced 3-dimensional system gained by assumption of a partial steady state of the fast variables with respect to the slow ones is treated, then this solution is taken as a starting vector for the complete 5-dimensional system.
    Type of Medium: Electronic Resource
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