The steady-state phase distribution of the motor switch complex model of Halobacterium salinarum

Steady-state analysis is performed on the kinetic model for the switch complex of the flagellar motor of Halobacterium salinarum (Nutsch et al. [16]). The existence and uniqueness of a positive steady-state of the system is established and it is demonstrated why the steady-state is centered around the competent phase, a state of the motor in which it is able to respond to light stimuli. It is also demonstrated why the steady-state shifts to the refractory phase when the steady-state value of the response regulator CheYP increases. This work is one aspect of modeling in systems biology wherein the mathematical properties of a model are established.     

Fingerprint formation

Fingerprints (epidermal ridges) have been used as a means of identifications for more than 2000 years. They have also been extensively studied scientifically by anthropologists and biologists. However, despite all the empirical and experimental knowledge, no widely accepted explanation for the development of epidermal ridges on fingers, palms and soles has yet emerged. In this article we argue that fingerprint patterns are created as the result of a buckling instability in the basal cell layer of the fetal epidermis. Analysis of the well-known von Karman equations informs us that the buckling direction is perpendicular to the direction of greatest stress in the basal layer. We propose that this stress is induced by resistance of furrows and creases to the differential growth of the basal layer and regression of the volar pads during the time of ridge formation. These ideas have been tested by computer experiments. The results are in close harmony with observations. Specifically, they are consistent with the well-known observation that the pattern type is related to the geometry of the fingertip surface when fingerprint patterns are formed.     

Spatial distribution of VEGF isoforms and chemotactic signals in the vicinity of a tumor

We propose a mathematical model that describes the formation of gradients of different isoforms of vascular endothelial growth factor (VEGF). VEGF is crucial in the process of tumor-induced angiogenesis, and recent experiments strongly suggest that the molecule is most potent when bound to the extracellular matrix (ECM). Using a system of reaction-diffusion equations, we study diffusion of VEGF, binding of VEGF to the ECM, and cleavage of VEGF from the ECM by matrix metalloproteases (MMPs). We find that spontaneous gradients of matrix-bound VEGF are possible for an isoform that binds weakly to the ECM (i.e. VEGF₁₆₅), but cleavage by MMPs is required to form long-range gradients of isoforms that bind rapidly to the ECM (i.e. VEGF₁₈₉). We also find that gradient strengths and ranges are regulated by MMPs. Finally, we find that VEGF molecules cleaved from the ECM may be distributed in patterns that are not conducive to chemotactic migration toward a tumor, depending on the spatial distribution of MMP molecules. Our model elegantly explains a number of in vivo observations concerning the significance of different VEGF isoforms, points to VEGF₁₆₅ as an especially significant therapeutic target and indicator of a tumor's angiogenic potential, and enables predictions that are subject to testing with in vitro experiments.     

Energy efficiency of information transmission by electrically coupled neurons

The generation of spikes by neurons is energetically a costly process. This paper studies the consumption of energy and the information entropy in the signalling activity of a model neuron both when it is supposed isolated and when it is coupled to another neuron by an electrical synapse. The neuron has been modelled by a four-dimensional Hindmarsh–Rose type kinetic model for which an energy function has been deduced. For the isolated neuron values of energy consumption and information entropy at different signalling regimes have been computed. For two neurons coupled by a gap junction we have analyzed the roles of the membrane and synapse in the contribution of the energy that is required for their organized signalling. Computational results are provided for cases of identical and nonidentical neurons coupled by unidirectional and bidirectional gap junctions. One relevant result is that there are values of the coupling strength at which the organized signalling of two neurons induced by the gap junction takes place at relatively low values of energy consumption and the ratio of mutual information to energy consumption is relatively high. Therefore, communicating at these coupling values could be energetically the most efficient option.     

Non-Gaussian noise-optimized intracellular cytosolic calcium oscillations

We have numerically studied the effect of a particular kind of non-Gaussian colored noise (NGN), characterized by the deviation q from Gaussian noise (q = 1), on intracellular cytosolic calcium (Ca²⁺) oscillations. It is found that, as q is increased, the Ca²⁺ oscillation regularity increases and reaches a best performance at an optimal q, and then decreases with further increasing q, which represents the occurrence of coherence resonance, i.e., the most regular Ca²⁺ oscillations. Similar phenomena occur for different values of noise intensity and correlation time of the NGN. This phenomenon of deviation-optimized Ca²⁺ oscillations show that, external non-Gaussian noises of different types can enhance and even optimize the intracellular Ca²⁺ oscillations. This result provides new insights into the constructive roles and potential applications of non-Gaussian noises in intracellular cytosolic Ca²⁺ oscillations.     

One-hit stochastic decline in a mechanochemical model of cytoskeleton-induced neuron death I: cell-fate arrival times

Much experimental evidence shows that the cytoskeleton is a downstream target and effector during cell death in numerous neurodegenerative diseases, including Parkinson's, Huntington's, and Alzheimer's diseases. However, recent evidence indicates that cytoskeletal dysfunction can also trigger neuronal death, by mechanisms as yet poorly understood. This is the first of two papers in which we study a mathematical model of cytoskeleton-induced neuron death. In our model, assembly control of the neuronal cytoskeleton interacts with both cellular stress levels and cytosolic free radical concentrations to trigger neurodegeneration. This trigger mechanism is further modulated by the presence of cell interactions in the form of a diffusible toxic factor released by dying neurons. We find that, consistent with empirical observations, our model produces one-hit exponential and sigmoid patterns of cell dropout. In all cases, cell dropout is exponential-tailed and described accurately by a gamma distribution. The transition between exponential and sigmoidal is gradual, and determined by a synergetic interaction between the magnitude of fluctuations in cytoskeleton assembly control and by the degree of cell coupling. We conclude that a single mechanism involving neuron interactions and fluctuations in cytoskeleton assembly control is compatible with the experimentally observed range of neuronal attrition kinetics.     

One-hit stochastic decline in a mechanochemical model of cytoskeleton-induced neuron death III: diffusion pulse death zones

This is the third of three papers in which we study a mathematical model of cytoskeleton-induced neuron death. In the first two papers of this suite [Lomasko, T., Clarke, G., Lumsden, C., 2007a. One-hit stochastic decline in a mechanochemical model of cytoskeleton-induced neuron death I: cell fate arrival times. J. Theor. Biol. 249, 1-17, doi:10.1016/j.jtbi.2007.05.031; Lomasko, T., Clarke, G., Lumsden, C., 2007b. One-hit stochastic decline in a mechanochemical model of cytoskeleton-induced neuron death II: transition state metastability. J. Theor. Biol. 249, 18-28, doi:10.1016/j.jtbi.2007.05.032], we established that the mean-field limit of our model relates the known patterns of neuron decline to specific scales of cytoskeleton reorganization and cell-cell interaction by diffusible death factors. In the mean-field limit, the spatially variable concentration of diffusing death factor is replaced by a constant average value. Recent empirical advances now permit the actual diffusion of such factors to be followed in intact neuropil. In this paper we therefore extend the model beyond the mean-field limit, to include the diffusion dynamics of death factor bursts released from dying neurons. A range of novel tissue degeneration patterns is observed, for which we confirm and extend the mean-field prediction that sigmoidal patterns of neuron population decay are a principal hallmark of cell death in the presence of death factor release.     

Morphogenetic apoptosis: a mechanism for correcting discontinuities in morphogen gradients

Smooth gradients of the morphogens Hh, Dpp, and Wg are required for proper development of Drosophila imaginal discs. Here, it is reported that, when a discontinuity is generated between two adjacent cells in the reception of either the Dpp or Wg signal, then cells on either side of the discontinuity boundary undergo apoptosis by activating the c-Jun N-terminal Kinase (JNK) pathway. Furthermore, in the medial region of the wing imaginal disc, the JNK pathway is also activated if cells do not receive the proper levels of Dpp and Hh signals. These observations suggest that cells within a developing field have the ability to access their spatial positions by comparing the level of morphogen signal they receive with that of their neighbors. This phenomenon is likely related to the process of cell competition, and we suggest that it is an evolutionarily important mechanism that helps prevent abnormal tissue specification and growth during development.     

Temporal control of self‐organized pattern formation without morphogen gradients in bacteria

Diverse mechanisms have been proposed to explain biological pattern formation. Regardless of their specific molecular interactions, the majority of these mechanisms require morphogen gradients as the spatial cue, which are either predefined or generated as a part of the patterning process. However, using Escherichia coli programmed by a synthetic gene circuit, we demonstrate here the generation of robust, self‐organized ring patterns of gene expression in the absence of an apparent morphogen gradient. Instead of being a spatial cue, the morphogen serves as a timing cue to trigger the formation and maintenance of the ring patterns. The timing mechanism enables the system to sense the domain size of the environment and generate patterns that scale accordingly. Our work defines a novel mechanism of pattern formation that has implications for understanding natural developmental processes.     

Aging and immortality in a cell proliferation model

We investigate a model of cell division in which the length of telomeres within a cell regulates its proliferative potential. At each division, telomeres undergo a systematic length decrease as well as a superimposed fluctuation due to exchange of telomere DNA between the two daughter cells. A cell becomes senescent when one or more of its telomeres become shorter than a critical length. We map this telomere dynamics onto a biased branching-diffusion process with an absorbing boundary condition whenever any telomere reaches the critical length. Using first-passage ideas, we find a phase transition between finite lifetime and immortality (infinite proliferation) of the cell population as a function of the influence of telomere shortening, fluctuations, and cell division.     

Binding equations for interacting systems comprising multivalent acceptor and bivalent ligand: application to antigen-antibody systems

Consideration is given to the reversible interaction of a bivalent ligand, B, with a multivalent acceptor, A (possessing f reactive sites) which leads to the formation of a series of complexes, A_iB_j, comprising networks of alternating acceptor and ligand molecules. A binding equation is derived on the basis of a site association constant, k, defined in terms of reacted site probability functions. This equation, which relates the binding function, r (the moles of ligand bound per mole of acceptor) to the concentration of unbound ligand, m_b, is used to show that plots of r vs. 2km_B constructed with fixed but different values of $\text{k}\text{m}{}_{\mathrm{A}}$ intersect at the point ($\text{m}{}_{\mathrm{B}}\text{=}\text{1}\text{2k}\text{, r =}\text{f}\text{2}$) where the extent of reaction and the concentrations of those complexes for which $\text{j}\text{i}\text{=}\text{f}\text{2}$ attain maximal values. Corresponding Scatchard plots are shown by numerical example to be non-linear, their second derivative being positive for all r. It follows that such deviations from linearity cannot be taken alone as evidence for site heterogeneity in cross-linking systems. The binding equation obtained directly is shown to be identical with that obtained with f = 2 by summation procedures involving the general expression for concentrations of complexes, m_AiBj, formulated in terms of appropriate statistical factors. In this way, previous findings on precipitation and gel formation in cross-linking systems are correlated with the present development of binding theory.     

Information flow in heterogeneously interacting systems

Motivated by studies on the dynamics of heterogeneously interacting systems in neocortical neural networks, we studied heterogeneously-coupled chaotic systems. We used information-theoretic measures to investigate directions of information flow in heterogeneously coupled Rössler systems, which we selected as a typical chaotic system. In bi-directionally coupled systems, spontaneous and irregular switchings of the phase difference between two chaotic oscillators were observed. The direction of information transmission spontaneously switched in an intermittent manner, depending on the phase difference between the two systems. When two further oscillatory inputs are added to the coupled systems, this system dynamically selects one of the two inputs by synchronizing, selection depending on the internal phase differences between the two systems. These results indicate that the effective direction of information transmission dynamically changes, induced by a switching of phase differences between the two systems.     

Equilibria in systems of interacting structured populations

The existence of a stable positive equilibrium density for a community of k interacting structured species is studied as a bifurcation problem. Under the assumption that a subcommunity of k–1 species has a positive equilibrium and under only very mild restrictions on the density dependent vital growth rates, it is shown that a global continuum of equilibria for the full community bifurcates from the subcommunity equilibrium at a unique critical value of a certain inherent birth modulus for the kth species. Local stability is shown to depend upon the direction of bifurcation. The direction of bifurcation is studied in more detail for the case when vital per unity birth and death rates depend on population density through positive linear functionals of density and for the important case of two interacting species. Some examples involving competition, predation and epidemics are given.     

Extended Holstein small polaron model for charge transfer in dry DNA

In this paper, the charge transfer problem in dry DNA was investigated by employing an extended Holstein small polaron model with external potential traps being involved in consideration. The ground state energy and the probability amplitude of polaron in various DNA chains with different external trap potentials were obtained by variational method with the trial function being taken in coherent state form. The stability of transfered charges in various circumstances was discussed accordingly.     

Thermodynamic compensation process in interacting protein systems: definition of thermodynamic compensatory temperature, (Tc).

In our thermodynamic analysis of the non-linear Van't Hoff expression as applied to several self-associating systems -- specifically in the cases of bovine liver L-glutamate dehydrogenase (GDH), glucagon and S-carboxymethylated apo A-II protein from human high density lipoprotein -- we have examined the interrelationships of a number of thermodynamic temperatures as they affect the association process. We found the principal determinants of the linear thermodynamic compensation process to be delta S0(T)/delta C0p(T) = (delta T'C)/(Texp), where (delta T'C) = Texp). We have defined the unique compensatory temperature, (TC), for any interacting system, at which the contributions of enthalpy and entropy to the association process are balanced.     

Transmembrane pH gradients and functional heterogeneity in reconstituted vesicle systems

The pH-sensitive, membrane impermeant fluorescence probes 8-hydroxy-1,3,6-pyrenetrisulfonate (pyranine; pK_a = 7.2) and 1-naphthol-3,6-disulfonate (Naps pK_a = 8.2) can be simultaneously entrapped within the intravesicular aqueous compartment of unilamellar vesicles and reconstituted proteoliposomes, where they function as reliable reporters of the intravesicular pH. Because the two probes are sensitive to pH over different but overlapping ranges, the useful monitoring range for the co-trapped probe pair extends from pH 6.5 to 9. In vesicles pre-equilibrated at a given pH and then subjected to a sudden change in external pH, the rate and extent of the subsequent change in internal pH are identical at all times during the re-equilibration, regardless of which probe is used to monitor the change. However, in reconstituted bacteriorhodopsin proteoliposomes, the size of the transmembrane pH gradient generated in the light always appears greater when pyranine is used to monitor internal pH. This discrepancy can most readily be understood in terms of heterogeneity in the vesicle suspension, with at least two populations of vesicles, one active in proton and one inactive. A simple algorithm was developed which generates, from the observed internal pH changes for two probes of different pK_a, the percentage of vesicles which are inactive, as well as the actual internal pH of the active fraction. The applicability of this algorithm was subsequently confirmed using a suspension of vesicles in which the level of heterogeneity was deliberately altered by the addition of various amounts of gramicidin. The apparent transmembrane pH gradient for the vesicle population as a whole decreased with increasing gramicidin, as did the calculated percentage of vesicles able to maintain a pH gradient, while the transmembrane gradient calculated for the active vesicle fraction only was essentially unaffected by gramicidin.     

Scanning molecular sieve chromatography of interacting protein systems. II. Determination of large zone transport parameters by the difference profile method at low solute concentration

The experimental determination of difference profiles for the study of large zone transport processes by scanning molecular sieve chromatography is described. Using the difference profile method, the progesterone-induced purple glycoprotein of the porcine uterus was found to exist as monomeric units in high ionic environment, with a partition coefficient of 0.269, partition cross-section of 0.488, partition radius of 25 A and a molecular weight of 33500 g mole . The technique was further applied in examining the association-dissociation properties of oxyhemoglobin. In a high tonic environment, the partition coefficient was found to be 0.365 for dimer and the partition cross-section, 0.419; for the tetramer in low ionic strength solution, the partition coefficient was 0.275 and the partition cross-section 0.377, with a dissociation constant of 1.03 x 10(-6) mole/1. This new technique should prove applicable in (1) readily locating the centroid positions of transport boundary profiles at the lowest practicable protein concentration limits, (2) demonstrating the characteristic boundary shape and concentration-dependent centroid position for an interacting solute, (3) determining the axial dispersion coefficient characteristic of solute turbulence within the gel matrix, and (4) distinguishing the boundary between low and high ionic strength solvent phases in the gel column.     

A collocation approach to the numerical calculation of simple gradients in reaction-diffusion systems

Reaction diffusion equations are frequently used to model pattern formation problems in biology, but numerical experiments in two or three space dimensions can be expensive in computing time. We show that the spectral method with collocation is a particularly efficient method for the numerical study of the evolution of simple patterns in such models. In many cases of interest, the scheme is sufficiently simple and efficient for calculations to be carried out on a micro-computer.     

Specialization in phagocytosis

When phagocytes have been incubated with a mixture of two types of particles (say A and B), a microscopic count can be made of cells having ingested iA-particles and jB-particles, for various i and j. Simple assumptions (the validity of which has been checked in a previous work) allow derivation of formulae that give a fair account of experimental results.Only such a quantitative study can determine whether there is a specialization among phagocytic cells, which question is of great immunological interest.