A cellular automata model for avascular solid tumor growth under the effect of therapy

Tumor growth has long been a target of investigation within the context of mathematical and computer modeling. The objective of this study is to propose and analyze a two-dimensional stochastic cellular automata model to describe avascular solid tumor growth, taking into account both the competition between cancer cells and normal cells for nutrients and/or space and a time-dependent proliferation of cancer cells. Gompertzian growth, characteristic of some tumors, is described and some of the features of the time-spatial pattern of solid tumors, such as compact morphology with irregular borders, are captured. The parameter space is studied in order to analyze the occurrence of necrosis and the response to therapy. Our findings suggest that transitions exist between necrotic and non-necrotic phases (no-therapy cases), and between the states of cure and non-cure (therapy cases). To analyze cure, the control and order parameters are, respectively, the highest probability of cancer cell proliferation and the probability of the therapeutic effect on cancer cells. With respect to patterns, it is possible to observe the inner necrotic core and the effect of the therapy destroying the tumor from its outer borders inwards.     

Nonequilibrium dynamics in lattice ecosystems: Chaotic stability and dissipative structures

A generalized coupled map lattice (CML) model of ecosystem dynamics is presented. We consider the spatiotemporal behavior of a prey-predator map, a model of host-parasitoid interactions, and two-species competition. The latter model can show phase separation of domains (Turing-like structures) even when chaos is present. We also use this CML model to explore the time evolution and structural properties of ecological networks built with a set of N competing species. The May-Wigner criterion is applied as a measure of stability, and some regularities in the stable networks observed are discussed.     

Theoretical study of mesoscopic stochastic mechanism and effects of finite size on cell cycle of fission yeast

Based on a deterministic cell cycle model, the mesoscopic stochastic differential equations are theoretically derived from the biochemical reactions. The effects of the finite cell size on the cell cycle regulation in the wild type and wee1^-cdc25Δ double mutant type are numerically studied by virtue of the chemical Langevin equations. (i) When the system is driven only by the internal noise, our numerical results are in qualitative agreement well with some experimental observations and data. (ii) A parameter, which is sensitive to two resettings of M-phase promoting factor to G₂, is treated as a stochastic variable, and the system driven only by the external noise for double mutant type is investigated. (iii) When the system is driven by both the internal and external noise, a simple discussion about the combined effect for double mutant type is given. Our results imply some experimental results would be explained by introducing the appropriate internal or external noise into the cell cycle system.     

Linked by loops: Network structure and switch integration in complex dynamical systems

Simple nonlinear dynamical systems with multiple stable stationary states are often taken as models for switchlike biological systems. This paper considers the interaction of multiple such simple multistable systems when they are embedded together into a larger dynamical “supersystem.” Attention is focused on the network structure of the resulting set of coupled differential equations, and the consequences of this structure on the propensity of the embedded switches to act independently versus cooperatively. Specifically, it is argued that both larger average and larger variance of the node degree distribution lead to increased switch independence. Given the frequency of empirical observations of high variance degree distributions (e.g., power-law) in biological networks, it is suggested that the results presented here may aid in identifying switch-integrating subnetworks as comparatively homogenous, low-degree, substructures. Potential applications to ecological problems such as the relationship of stability and complexity are also briefly discussed.     

On the equilibria of the MAPK cascade: Cooperativity, modularity and bistability

In this paper we present a discussion of a phenomenological model of the MAPK cascade which was originally proposed by Angeli et al. [D. Angeli, J.E. Ferrell, Jr., E.D. Sontag, PNAS 101 (2004), 1822]. The model and its solution are extended in several respects: (a) an analytical solution is given for the cascade equilibria, exploiting a parameter-based symmetry of the rate equations; (b) we discuss the cooperativity (Hill coefficients) of the cascade and show that a feedforward loop within the cascade increases its cooperativity. The relevance of this result for the notion of modularity is discussed; (c) the feedback model for cascade bistability by Angeli et al. is reconsidered. We argue that care must be taken in modeling the interactions and a biologically realistic phenomenological model cannot be too reductionist. The inclusion of a time-dependent degradation rate is needed to account for a switching of the cascade.     

Control Chaos in Hindmarsh--Rose Neuron by Using Intermittent Feedback with One Variable

The mechanism of the famous phase compression is discussed, and it is used to control the chaos in the Hindmarsh-Rose （H-R） model. It is numerically confirmed that the phase compression scheme can be understood as one kind of intermittent feedback scheme, which requires appropriate thresholds and feedback coeffcient, and the intermittent feedback can be realized with the Heaviside function. In the case of control chaos, the output variable （usually the voltage or the membrane potential of the neuron） is sampled and compared with the external standard signal of the electric electrode. The error between the sampled variable and the external standard signal of the electrode is input into the system only when the sampled variable surpasses the selected thresholds. The numerical simulation results confirm that the chaotic H-R system can be controlled to reach arbitrary n-periodical （n = 1, 2, 3, 4, 5, 6,...） orbit or stable state even when just one variable is feed backed into the system intermittently. The chaotic Chua circuit is also investigated to check its model independence and effectiveness of the schemes and the equivalence of the two schemes are confirmed again.     

Can spontaneous cell movements be modelled as Lévy walks?

Spontaneous cell movement is a random motion that takes place in the absence of external guiding stimuli. The spontaneous movements of HaCaT and NHDF cells (cells of the epidermis) are well represented as continuous Markovian processes driven by multiplicative noise [D. Selmeczi, S. Mosler, P.H. Hagedorn, N.B. Larsen, H. Flyvbjerg, Biophysical Journal 89 (2005) 912]. Model components are, however, ad hoc as they are inspired by fits to experimental data. As a consequence, model agreement with experimental data does not add much to our understanding of spontaneous movements of these cells beyond demonstrating that they can be modelled phenomenologically. Here it is noted that a slight re-parameterization and re-interpretation of the driving noise leads to the model of Lubashevsky et al. (2009) [I. Lubashevsky, R. Friedrich, A. Heuer, Physical Review E 79 (2009) 011110] that realises Lévy walks as Markovian stochastic processes. This brings forth new biological insight as Lévy walks are advantageous when searching in the absence of external stimuli and without knowledge of the target distribution, as may be the case with cells of the epidermis that form new tissue by locating and then attaching on to one another. The Hänggi-Klimontovich interpretation of the driving noise in the model of Lubashevsky et al. (2009) and Cauchy distributions of predicted velocities do, however, appear problematic, even unphysical. Here it is shown that these are perceived rather than actual difficulties. Intermittent stop-start motions of the kind displayed by some cells and protozoan are found to underlie the formulation of the model of Lubashevsky et al. (2009) and the velocities of starved Dictyostelium discoideum (a unicellular organism) are found to be Cauchy distributed to a good approximation. It is therefore suggested that the model of Lubashevsky et al. (2009) can describe the spontaneous movements of some cells, and that some cells have spontaneous movement patterns that can be approximated by Lévy walks, as first proposed by Schuster and Levandowsky (1996) [F.L. Schuster, M. Levandowsky, Journal of Eukaryotic Microbiology 43 (1996) 150].     

Stationary cell size distributions and mean protein chain length distributions of Archaea,Bacteria and Eukaryotes described with an increment model in terms of irreversible thermodynamics

In terms of an increment model irreversible thermodynamics allows to formulate general relations of stationary cell size distributions observed in growing colonies. The treatment is based on the following key postulates: i) The growth dynamics covers a broad spectrum of fast and slow processes. ii) Slow processes are considered to install structural patterns that operate in short periods as temporary stationary states of reference in the sense of irreversible thermodynamics. iii) Distortion during growth is balanced out via the many fast processes until an optimized stationary state is achieved. The relation deduced identifies the numerous different stationary patterns as equivalents, predicting that they should fall on one master curve. Stationary cell size distributions of different cell types, like Hyperphilic archaea, E. coli (Prokaryotes) and S. cerevisiae (Eukaryotes), altogether taken from the literature, are in fact consistently described. As demanded by the model they agree together with the same master curve. Considering the “protein factories” as subsystems of cells the mean protein chain length distributions deduced from completely sequenced genomes should be optimized. In fact, the mean course can be described with analogous relations as used above. Moreover, the master curve fits well to the patterns of different species of Archaea, Bacteria and Eukaryotes. General consequences are discussed.     

Microscopic noise,adaptation and survival in hostile environments

The survival of autocatalytic agents in hostile environments depends on their ability to adapt their spatial configuration to local fluctuations. A model of diffusive reactants that extract the advantage of spatio-temporal fluctuations associated with the stochastic wandering of diffusive catalysts is discussed. Two arguments are presented for the basic processes behind this extraordinary behavior. In the first, the local colonies that evolve around any spatially advantageous region overlap in space-time and an infinite directed percolation cluster emerges. The second argument is based on the return probability of a diffusive agent that is shown to yield finite density of active “oases" with an exponentially large contribution to the reactant population. The different range of applicability of these survival lower bounds to small systems is discussed.     

Mean field model for synchronization of coupled two-state units and the effect of memory

A prototypical model for a mean field second order transition is presented, which is based on an ensemble of coupled two-states units. This system is used as a basic model to study the effect of memory. To wit, we distinguish two types of memories: weak and strong, depending on the feasibility of linearizing the generalized mean field master equation. For weak memory we find static solutions that behave much like those of the memoryless (Markovian) system. The latter exhibits a pitchfork bifurcation as the control parameter is increased, with two stable and one unstable solution. The former exhibits an imperfect pitchfork bifurcation to states with the same behaviors. In both cases, the stability of the static solutions is analyzed via the usual linearization around the equilibrium solution. For strong memories we again find an imperfect pitchfork bifurcation, with two stable and one unstable branch. However, it is no longer possible to analyze these behaviors via the usual linearization, which is local in time, because a strong memory requires knowledge of the system for its entire past. Finally, we are pleased to dedicate this publication to Helmut Brand on the occasion of his 60^th birthday.     

Rhythm Synchronization of Coupled Neurons with Temporal Coding Scheme

Encoding information by firing patterns is one of the basic neural functions, and synchronization is important collective behaviour of a group of coupled neurons. Taking account of two schemes for encoding information （that is, rate coding and temporal coding）, rhythm synchronization of coupled neurons is studied. There are two types of rhythm synchronization of neurons： spike and burst synchronizations. Firstly, it is shown that the spike synchronization is equivalent to the phase synchronization for coupled neurons. Secondly, the similarity function of the slow variables of neurons, which have relevant to the bursting process, is proposed to judge the burst synchronization. It is also found that the burst synchronization can be achieved more easily than the spike synchronization, whatever the firing patterns of the neurons are. Hence the temporal encoding scheme, which is closely related to both the spike and burst synchronizations, is more comprehensive than the rate coding scheme in essence.     

Entropy measures of cellular aggregation

In this paper a measure is proposed of the rate with which the collective motion of cells leads to aggregation and structure formation. It will be shown that the spatial entropy of the cells tends to decrease during aggregation and an index will be derived to quantify the rate with which this process takes place. Finally, applications will be presented to experiments on cellular migration and aggregation.     

Surface modification of titanium by nano-TiO2/HA bioceramic coating

A nano-TiO₂/hydroxyapatite composite bioceramic coating was developed and applied to the surfaces of pure titanium discs by the sol-gel method. A TiO₂ anatase bioceramic coating was utilized in the inner layer, which could adhere tightly to the titanium substrate. A porous hydroxyapatite (HA) bioceramic coating was utilized in the outer layer, which has higher solubility and better short-term bioactivity. Conventional HA coatings and commercially pure titanium were used as controls. X-ray diffraction (XRD) and scanning electron microscopy (SEM) were employed to characterize the crystallization, surface morphology, and thickness of the coatings. The bioactivities of the coatings were evaluated by in vitro osteoblast cultures. Results showed that the nano-TiO₂/HA composite bioceramic coating exhibited good crystallization and homogeneous, nano-scale surface morphology. In addition, the nano-TiO₂/HA coating adhered tightly to the substrate, and the in vitro osteoblast cultures exhibited satisfactory bioactivity.     

Osteoblastic cell response on magnesium-incorporated apatite coatings

Magnesium is one of the most important bivalent ions associated with biological apatite. A series of magnesium-substituted calcium apatite coatings (Ca_10−xMg_x)(PO₄)₆(OH)₂, where x = 0, 0.50, 1.00, 1.50 and 2.00, are synthesized onto Ti6Al4V substrate by sol-gel dip-coating method to determine how magnesium influences the synthesis and the resulting structural and biological properties. X-ray diffraction (XRD) analysis shows that the incorporation of magnesium helps formation of Mg-containing β-TCP (β-TCMP) phase. X-ray photoelectron spectroscopy (XPS) is used to study the chemical composition and the results show that the apatite structure can only host magnesium less than ∼2.4 wt.% beyond which magnesium aggregates on the surfaces. The incorporation of magnesium slows down the dissolution of Ca²⁺ from the coating. The in vitro behavior of the coatings is evaluated with human osteosarcoma MG63 cells for cell morphology and proliferation. Similar cell morphologies are observed on all coatings. The cell proliferation results show that the incorporation of magnesium up to x = 2 has no adverse effect on cell growth.     

Paradox of enrichment: A fractional differential approach with memory

The paradox of enrichment (PoE) proposed by Rosenzweig [M. Rosenzweig, The paradox of enrichment, Science 171 (1971) 385–387] is still a fundamental problem in ecology. Most of the solutions have been proposed at an individual species level of organization and solutions at community level are lacking. Knowledge of how learning and memory modify behavioral responses to species is a key factor in making a crucial link between species and community levels. PoE resolution via these two organizational levels can be interpreted as a microscopic- and macroscopic-level solution. Fractional derivatives provide an excellent tool for describing this memory and the hereditary properties of various materials and processes. The derivatives can be physically interpreted via two time scales that are considered simultaneously: the ideal, equably flowing homogeneous local time, and the cosmic (inhomogeneous) non-local time. Several mechanisms and theories have been proposed to resolve the PoE problem, but a universally accepted theory is still lacking because most studies have focused on local effects and ignored non-local effects, which capture memory. Here we formulate the fractional counterpart of the Rosenzweig model and analyze the stability behavior of a system. We conclude that there is a threshold for the memory effect parameter beyond which the Rosenzweig model is stable and may be used as a potential agent to resolve PoE from a new perspective via fractional differential equations.     

Osteoblast adhesion to clodronate-hydroxyapatite composite

The present study combined clodronate, one of bisphosphonates, to the surface of HA scaffold by chelation. The cell culture test and gene expression test were performed and evaluated the effect of clodronate modifying the HA on co-culturing osteoblast with HA scaffold in vitro. Characteristic peaks in XPS and FT-IR spectra indicated clodronate being immobilized on the surface of HA. The cell culture test indicated that the cells actively proliferated on the scaffolds. The results of RT-PCR showed there was no significant difference between two groups in expression of RANKL gene, while the expression of OPG gene showed higher expression in Clodronate-HA group. Clodronate had no obvious effect on the cytocompatibility of HA, and Clodronate-HA might be used as bone scaffold with potential ability to improve osteogenesis.     

Sensitisation waves in a bidomain fire-diffuse-fire model of intracellular Ca dynamics

We present a bidomain threshold model of intracellular calcium (Ca²⁺) dynamics in which, as suggested by recent experiments, the cytosolic threshold for Ca²⁺ liberation is modulated by the Ca²⁺ concentration in the releasing compartment. We explicitly construct stationary fronts and determine their stability using an Evans function approach. Our results show that a biologically motivated choice of a dynamic threshold, as opposed to a constant threshold, can pin stationary fronts that would otherwise be unstable. This illustrates a novel mechanism to stabilise pinned interfaces in continuous excitable systems. Our framework also allows us to compute travelling pulse solutions in closed form and systematically probe the wave speed as a function of physiologically important parameters. We find that the existence of travelling wave solutions depends on the time scale of the threshold dynamics, and that facilitating release by lowering the cytosolic threshold increases the wave speed. The construction of the Evans function for a travelling pulse shows that of the co-existing fast and slow solutions the slow one is always unstable.     

阻力对透平压缩机系统稳定性的影响

建立了一个考虑管道阻力作用的透平压缩机系统喘振点计算模型，通过算例计算分析了阻力对喘振点的影响．结果表明阻力对系统具有稳定性作用，较大的阻力可以明显地推迟喘振的发生。对于管道很长的压缩系统，在估算喘振点时应当考虑沿程阻力的影响．另外还提出了一种提高压缩系统稳定性的结构，通过进口阀门提供局部阻力可以构做出几乎无喘振的压缩系统，同时计算分析表明此结构对于低压低速压缩机系统具有一定的实用意义，而对于高压高速系统则造成性能的严重下降．     

The influence of band Jahn-Teller effect and magnetic order on the magneto-resistance in manganite systems

A model calculation is presented in order to study the magneto-resistivity through the interplay between magnetic and structural transitions for the manganite systems. The model consists of an orbitally doubly degenerate conduction band and a periodic array of local moments of the t_2g electrons. The band electrons interact with the local t_2g electrons via the s-f hybridization. The phonons interact with the band electrons through static and dynamic band Jahn-Teller (J-T) interaction. The model Hamiltonian including the above terms is solved for the single particle Green's functions and the imaginary part of the self-energy gives the electron relaxation time. Thus the magneto-resistivity (MR) is calculated from the Drude formula. The MR effect is explained near the magnetic and structural transition temperatures.