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Prions are proteinaceous infectious particles that cause neurodegenerative diseases in humans and animals. The complex nature of prions, with respect to their conformations and aggregations, has been an important area of research for quite some time. Here, we develop a model of prion dynamics prior to the formation of oligomers and subsequent development of prion diseases within a stochastic framework, based on the analytical Master Equation and Stochastic Simulation Algorithm by Gillespie. The results that we obtain shows that solvent water molecules act as driving agents in the dynamics of prion aggregation. Further, it is found that aggregated and non-aggregated proteins tend to co-exist in an equilibrium state, depending upon the reaction rate constants. These results may provide a theoretical and qualitative contexts of possible therapeutic strategies in the treatment of prion diseases., Comment: 10 pages, 6 figures

Fluctuations-driven complex oscillations are experimentally observed in cellular systems such as hepatocytes, cardiac cells, neuronal cells, etc. These systems are generally operating in regimes far from thermodynamic equilibrium. To study nonequilibrium thermodynamic properties such as energy dissipation in stochastic complex oscillations, we consider stochastic modeling of two nonlinear biological oscillators, namely, the intracellular calcium (Ca$^{2+}$) oscillation model and the Hindmarsh-Rose model for neuronal dynamics. These models exhibit various types of complex oscillations like bursting and quasi-periodic oscillations for various system parameter values. In this work, we formulate open chemical reaction schemes for the two model systems driving the systems far from thermodynamic equilibrium. We then analyze the steady-state total entropy production rate (EPR) in the various types of stochastic complex oscillations. Our results show higher values of steady-state total EPR in stochastic complex oscillations than simple periodic oscillations. Moreover, in the Hindmarsh-Rose neuronal model, we observe an order-to-disorder transition from periodic (organized) bursts of spikes to chaotic (unorganized) oscillations with distinct behaviors of steady-state total EPR. Our results reveal that stochastic complex oscillations are produced at the cost of higher energy consumption and that it requires a higher thermodynamic cost to maintain the periodic bursts than chaotic oscillations. Our findings indicate that complex cellular regulatory or signaling processes by Ca$^{2+}$ that help perform complex tasks of the nervous system or rich information coding by neurons involve a higher thermodynamic cost. The results deepen our understanding of energy dissipation in nonlinear, nonequilibrium biological systems with stochastic complex oscillatory dynamics.

The concentration of intracellular calcium ion (Ca$^{2+}$) exhibits complex oscillations, including bursting and chaos, as observed experimentally. These dynamics are influenced by inherent fluctuations within cells, which serve as crucial determinants in cellular decision-making processes and fate determination. In this study, we systematically explore the interplay between intrinsic fluctuation and the complexity of intracellular cytosolic Ca$^{2+}$ dynamics using complexity measures such as permutation entropy (PE) and statistical complexity (SC). Using the chemical Langevin equation, we simulate the stochastic dynamics of cytosolic Ca$^{2+}$. Our findings reveal that PE and SC effectively characterize the diverse, dynamic states of cytosolic Ca$^{2+}$ and illustrate their interactions with intrinsic fluctuation. PE analysis elucidates that the chaotic state is more sensitive to intrinsic fluctuation than the other periodic states. Furthermore, we identify distinct states of cytosolic Ca$^{2+}$ occupying specific locations within the theoretical bounds of the complexity-entropy causality plane. These locations indicate varying complexity and information content as intrinsic fluctuation varies. When adjusting the permutation order, the SC for the different states exhibits peaks in an intermediate range of intrinsic fluctuation values. Additionally, we identify scale-free or self-similar patterns in this intermediate range, which are further corroborated by multifractal detrended fluctuation analysis. These high-complexity states likely correspond to optimal Ca$^{2+}$ dynamics with biological significance, revealing rich and complex dynamics shaped by the interplay of intrinsic fluctuation and complexity. Our investigation enhances our understanding of how intrinsic fluctuation modulates the complexity of intracellular Ca$^{2+}$ dynamics that play crucial roles in biological cells.

Control studies in ecological models of networks involving invasive plants can offer tangible management strategies. Framing control policies to tackle problems posed by invasive plants on indigenous plant diversity under the sustainable developmental goals framework is the main motivation of this paper. Based on the reported experimental and theoretical observations, we propose minimal ecological models of direct interactions of \textit{Lantana camara} with other entitites e.g. other indigenous plants (denoted by Control-plant) and soil microbes. The species' abundance variables represented by the nodes in the model networks follow generalized Lotka-Volterra dynamics and input signals are incorporated at appropriate driver nodes to control the \textit{Lantana camara} outgrowth. Analysis with Lie algebra and controllability criteria showed that our model systems are controllable with minimum of two control input signals. Analytical results of our model equations indicated that successive interventions are required to increase the abundance of control-plant, the microbes and suppress \textit{Lantana camara}. We observed from the numerical analysis with LQR algorithm that the number of interventions to be taken to control the \textit{Lantana camara} outgrowth depends on the choice of various important parameters such as intervention time, effort and penalty etc. Further, the increase in nonlinear terms contributed from self-loops in the network model shows the persistence of \textit{L. camara} and how it drives the coexistence of the species. We propose that parameters we have identified in our study needed to be considered and analyzed extensively for designing sustainable policies to control invasive species in an ecosystem., Comment: 11 Figures, 30 pages

Patterns in complex systems store hidden information of the system which is needed to be explored. We present a simple model of cytokine and T-cells interaction and studied the model within stochastic framework by constructing Master equation of the system and solving it. The solved probability distribution function of the model show classical Poisson pattern in the large population limit $M,Z\rightarrow large$ indicating the system has the tendency to attract a large number small-scale random processes of the cytokine population towards the basin of attraction of the system by segregating from nonrandom processes. Further, in the large $\langle Z\rangle$ limit, the pattern transform to classical Normal pattern, where, uncorrelated small-scale fluctuations are wiped out to form a regular but memoryless spatiotemporal aggregated pattern. The estimated noise using Fano factor shows clearly that the cytokine dynamics is noise induced process driving the system far away from equilibrium., Comment: 14 pages, 1 figure

Modeling and analysis of the large scale Covid-19 pandemic data can yield inferences about it's dynamics and characteristics of disease propagation. These inferences can then be correlated with contextual factors like population density, effects of strategic interventions, heterogeneous disease propagation etc, and such set of validated inferences can serve as precedents for designing of subsequent mitigation strategies. In this work, we present the analysis of Covid-19 pandemic data in Indian context using growth functions fitting procedure and harmonic analysis method. Our results of growth function fitting to the data indicate that the growth function parameters are quite sensitive to the growth of the infected population indicating positive impact of lockdown strategy, identification of inflection point and nearly synchronous statistical features of disease spreading. The harmonic analysis of the data shows the countrywide synchronous incident features due to simultaneous implementation of control strategies. However, if one analyzes the data from each state of the India, one can see various forms of travelling waves in the countrywide wave pattern. Hence, one needs to do these analysis from time to time to understand the effectiveness of any control strategy and to closely look at the disease propagation to devise the required type of mitigation strategies., Comment: 14 pages, 7 figures

The Crab Nebula is an astrophysical system that exhibits complex morphological patterns at different observing frequencies. We carry out a systematic investigation of the structural complexity of the nebula using publicly available imaging data at radio frequency. For the analysis, we use the well-known multifractal detrended fluctuation analysis in two dimensions. We find that radio data exhibit long-range correlations, as expected from the underlying physics of the supernova explosion and evolution. The correlations follow a power-law scaling with length scales. The structural complexity is found to be multifractal in nature, as evidenced by the dependence of the generalized Hurst exponent on the order of the moments of the detrended fluctuation function. By repeating the analysis on shuffled data, we further probe the origin of the multifractality in the radio imaging data. For the radio data, we find that the probability density function is close to a Gaussian form. Hence, the multifractal behavior is due to the differing nature of long-range correlations of the large and small detrended fluctuation field values. We investigate the multifractal parameters across different partitions of the radio image and find that the structures across the image are highly heterogeneous, making the Crab Nebula a structurally complex astrophysical system. Our analysis thus provides a fresh perspective on the morphology of the Crab Nebula from a complexity science viewpoint.

The survival of endangered languages in complex language competition depends on socio-cultural status and honour endowed (by itself and by the other) among them. The restriction in the endorsement of this honour leads to language extinction of one language, and rise of the other. Endorsing proper mutual honour each other trigger the co-existence of language speakers and can save both languages from extinction. The lost of respect to each other drives the death of both languages. We found a minimal or critical mutual honour (a=0.9635) which protects the two languages from extinction. The increase in mutual honour from this minimal value allows increase in the populations of the two languages speakers. The state of co-existence of competing languages abolishes the concept of minority and majority in language competition which can be obtained by mutual honour. Further, excess biased honour to a particular language (minority or majority) force the language to extinct. In mean-field approximation of language competition, magnetization parameter can be taken as an indicator of survival of a language., Comment: 5 pages, 4 figures

We model the well-known interlinked cascade of Rab GTPases found in eukaryotic cells by using a network of Markov states to investigate the universal Thermodynamic Uncertainty Relation for the non-equilibrium system. First, we prove numerically the TUR in both single Rab species model and the interlinked two Rab species model. Our results show that when two Rab GTPase proteins are interlinked at far from equilibrium, the thermodynamic cost and hence precision is greatly enhanced as compared to single species switching. This implies that at far from equilibrium, the proteins tries to optimise the precision of the performance of their biomolecular processes by forming interlinks in the cascade. Again,our results imply that the interlinked cascade (or oscillator) can achieve a range of tunable rate constants (or frequencies) which suggests a means of maintaining its robustness. Lastly, we highlight a close relation between thermodynamic cost-precision, triangular motifs and disease dynamics., Comment: 25 pages, 7 figures

Unfolded protein aggregation in cellular system is a problem causing various types of diseases depending on which type unfolded proteins aggregate. This phenomenon of aggregation may take place during production, storage, shipment or delivery in the cellular medium. In the present work, we studied a simplified and extended version of unfolded protein aggregation model by Lumry and Eyring using stochastic approach. We solved analytically the Master equation of the model for the probability distribution $P(x,t)$ of the unfolded protein population and the solution was found to be time dependent complex binomial distribution. In the large population limit $P(x,t)\sim \Lambda(x,t)\times Pois(x,t)$. Further, the distribution became Normal distribution at large population and mean of the distribution limit: $P(x,t)\sim\Lambda(x,t)\times N(\langle qx\rangle,\langle qx\rangle)$. The fluctuations inherent in the dynamics measured by Fano factor can have sub-Poisson, Poisson and super-Poisson at different situations., Comment: 9 pages, 1 figure

Covid-19 is one of the most dreaded pandemics/epidemics in the world threatening the human population. The dynamics of this pandemic is quite complicated and prediction of pandemic states often fails. In this work, we study and correlate the SIR epidemiological model with the ongoing pandemic and found that pandemic dynamics and states are quite sensitively dependent on model parameters. The analysis of the exact parametric solution of the deterministic SIR model shows that the fixed points ($g^*$) depend on the SIR parameters, where, if $g^*\langle\frac{\gamma}{\beta x_0}$ then $g^*$ is stable and the pandemic can be controlled, whereas, if $g^*\rangle\frac{\gamma}{\beta x_0}$ then $g^*$ is unstable indicating active pandemic state, and $g^*=0$ corresponds to an endemic state. The dynamics show asymptotic stability bifurcation. The analytical solution of the stochastic SIR model allows to predict endemic time $t_E\propto\frac{1}{\gamma}$ and mean infected population becomes constant for small values of $\gamma$. The noise in the system even can modulate pandemic dynamics. Hence, we estimated the values of these parameters during lockdown periods using the ABC SSA algorithm and found strong dependence on control strategy, namely, lockdown and follows power-law behaviors. The numerical results show strong agreement with the real data and have the positive effect of social distancing or lockdown in the pandemic control., Comment: 10 pages, 4 figures

The cognitive states have broadly been divided into waking, rapid eye movement sleep and non-REMS. Although the mechanism of state transition is unknown, it has been proposed that functional activation/deactivation among different brain regions leads to such transition. As analysis of electroencephalogram allows us exploring properties and association among brain regions, we exploited it to address our query. We have recorded the frontal and occipital cortical EEG from surgically prepared chronic freely moving, normally behaving rats and classified their vigilance states and vigilance state-transitions. The complexity analysis carried out by computing multifractal spectrum width categorized VST as highly non-linear and complex than their participating vigilance states. The EEG signals were decomposed into frequency ranges as that of classical human Delta, Theta, Alpha, Beta and Gamma oscillations. The dynamic network attributes of these oscillations has shown compelling topological correlation between the frontal and occipital regions of the brain during conscious states. The topological trends of the underlying hierarchical network organization have been characterized by the percentage of the hub and non-hub nodes, that further determines the global and local connectivity trends. This has revealed interesting insights of functional disconnection during NREMS due to decrease in their number of hub-nodes as compared to waking state. There is also switching behaviour in the ratio of hub/non-hub nodes between frontal and occipital region during NREMS-Wake and Wake-NREMS transitions. Our findings provide support as proof-of-principle of functional regional inactivation or activation of hub-nodes as the gradual switching mechanism towards transitioning of cognitive states in a graded manner, wake to sleep or vice versa; the detailed neuro-physio-chemical mechanism needs further study., Comment: 18 pages, 11 figures

The investigation of topological properties of ovarian cancer network (OCN) and the roles of hubs involved in it by digging the network at various levels of organization are important to understand how OCN is organized to understand disease states. The OCN constructed from the experimentally verified genes exhibits fractal nature in the topological properties of the network and deeply rooted communities. Also, the network properties at all levels of organization obey one parameter scaling law which lacks centrality lethality rule. We then showed that $\langle k\rangle$ can be taken as a scaling parameter, where, power law exponent can be estimated from the ratio of network diameters, $\lambda=1+\rho\left[\frac{d_c}{d_k}\right]$. The betweenness centrality shows two distinct behaviors one shown by high degree hubs and the other by segregated low degree nodes. The $C_B$ distribution follows power law behavior with the exponent connected to exponents of distributions of high and low degree nodes by, $\theta\sim\left[1+\frac{1}{2}\left(\frac{1}{\epsilon_s}+\frac{1}{\epsilon_h}\right)(\lambda-1)\right]$. Absence of rich-club formation leads to the missing of a number of attractors in the network causing formation of weakly tied diverse functional modules to keep optimal network efficiency. The hubs knockout experiment shows that provincial hubs take major responsibility to keep network integrity and organization. The identified key regulators are found to be provincial and connector hubs. Further, two key regulators, EPCAM and CD44 are found to be maximally over expressed at various cancer stages (II-IV). They are also positively correlated with immune infiltrates (CD4+ T cells). Finally, few potential drugs are identified related to the key regulators.

Synaptic plasticity and neuron cross-talk are some of the important key mechanisms underlying formation of dynamic clusters of active neurons. The essence of this study is to model and decipher the mechanism of emergence of a task-specific functional connectivity among a population of neurons. We have used our proposed neuron activity pattern (NAP) model to define an assorted range of interactions and simulated across a 2D lattice with two-state (firing/non-firing) neurons. We observed the emergence of scale-free, hierarchically organized ordered patterns of active neurons, that we call as functional cortical patterns (FCPs), only at the near-critical phase transition. We have done extensive topological characterization of FCPs and tested its congruency with the functional brain networks obtained from the empirical electrophysiological data of a visual stimulus task. Our results of network theory attributes, fractal analysis and functional cartography have confirmed the transitioning effect while generation of FCPs. This gives us strong implication that the functional neuronal system supports far-from-equilibrium dynamics upon receiving a stimulus. Our investigation for the long, critical and short-ranged interactions has lead to an interpretation that near-critical transition phase is a spectrum of critical temperatures (coupling strengths) being affected by the distance between the neurons. The range of interaction among neurons plays an important role in defining the functional severity of the neuronal circuitry. We propose that interaction range as a function of coupling strength drives connectivity in an augmenting fashion as we observed an increase in coupling strength from short to long-range interactions, among the population of neurons. This gives us insight towards the intensity of cognitive behaviour, which is attained by multiple stimulus attempts.

We present a treatment of non-Markovian character of memory by incorporating different forms of Mittag-Leffler (ML) function, which generally arises in the solution of fractional master equation, as different memory functions in the Generalized Kolmogorov-Feller Equation (GKFE). The cross-over from the short time (stretched exponential) to long time (inverse power law) approximations of the ML function incorporated in the GKFE is proven. We have found that the GKFE solutions are the same for negative exponential and for upto frst order expansion of stretched exponential function for very small $\tau \rightarrow 0$. A generalized integro-differential equation form of the GKFE along with an asymptotic case is provided.

India is one of the worst affected countries by the Covid-19 pandemic at present. We studied publicly available data of the Covid-19 patients in India and analyzed possible impacts of quarantine and social distancing within the stochastic framework of the SEQIR model to illustrate the controlling strategy of the pandemic. Our simulation results clearly show that proper quarantine and social distancing should be maintained from an early time just at the start of the pandemic and should be continued till its end to effectively control the pandemic. This calls for a more socially disciplined lifestyle in this perspective in future. The demographic stochasticity, which is quite visible in the system dynamics, has a critical role in regulating and controlling the pandemic.

Monte carlo simulation of paths of a large number of impinging electrons in a multi-layered solid allows to define area of spreading electrons (A) to capture overall behavior of the solid. This parameter 'A' follows power law with electron energy. Further, change in critical energies, which are minimum energies lost corresponding to various electrons, as a function of variation in lateral distance also follows power law nature. This power law behavior could be an indicator of how strong self-organization a solid has which may be used in monitoring efficiency of device fabrication.

SMAR1 is a sensitive signaling molecule in p53 regulatory network which can drive p53 network dynamics to three distinct states, namely, stabilized (two), damped and sustain oscillation states. In the interaction of p53 network with SMAR1, p53 network sees SMAR1 as a sub-network with its new complexes formed by SMAR1, where SMAR1 is the central node, and fluctuations in SMAR1 concentration is propagated as a stress signal throughout the network. Excess stress induced by SMAR1 can drive p53 network dynamics to amplitude death scenario which corresponds to apoptotic state. The permutation entropy calculated for normal state is minimum indicating self-organized behavior, whereas for apoptotic state, the value is maximum showing breakdown of self-organization. We also show that the regulation of SMAR1 togather with other signaling molecules p300 and HDAC1 in the p53 regulatory network can be engineered to extend the range of stress such that the system can be save from apoptosis.

The organization in brain networks shows highly modular features with weak inter-modular interaction. The topology of the networks involves emergence of modules and sub-modules at different levels of constitution governed by fractal laws. The modular organization, in terms of modular mass, inter-modular, and intra-modular interaction, also obeys fractal nature. The parameters which characterize topological properties of brain networks follow one parameter scaling theory in all levels of network structure which reveals the self-similar rules governing the network structure. The calculated fractal dimensions of brain networks of different species are found to decrease when one goes from lower to higher level species which implicates the more ordered and self-organized topography at higher level species. The sparsely distributed hubs in brain networks may be most influencing nodes but their absence may not cause network breakdown, and centrality parameters characterizing them also follow one parameter scaling law indicating self-similar roles of these hubs at different levels of organization in brain networks.

We present generalized delayed neural network (DNN) model with positive delay feedback and neuron history. The local stability analysis around trivial local equilibria of delayed neural networks has applied and determine the conditions for the existence of zero root. We develop few innovative delayed neural network models in different dimensions through transformation and extension of some existing models. We found that zero root can have multiplicity two under certain conditions. We further show how the characteristic equation can have zero root and its multiplicity is dependent on the conditions undertaken. Finally, we generalize the neural network of $N$ neurons through which we determine the general form of Jacobian of the linear form and corresponding characteristic equation of the system., Comment: 12 pages, 1 figure

Recent studies suggest that PML (Promyelocytic Leukemia) suppresses p53 protein degradation by inhibiting MDM2 protein in the nucleus [1], and regulates a number of biological functions [2]. We modeled a PML-MDM2-p53 regulatory network by integrating p53-MDM2 autoregulatory model [3] and related PML pathways. We found that p53 dynamics switched at various dynamical states induced by PML which corresponds to different cellular states. Our results show clear transitions among the stabilized, damped and sustain states under different stress conditions induced by PML in the system. These states in p53 dynamics is the signature of existence of various cellular states and the phase transition like behaviour of these states involve various biological significance.

We study the efficiency of synchronization in ensembles of identical coupled stochastic oscillator systems. By deriving a chemical Langevin equation, we measure the rate at which the systems synchronize. The rate at which the difference in the Hilbert phases of the systems evolve provides a suitable order parameter, and a 2--dimensional recurrence plot further facilitates the analysis of stochastic synchrony. We find that a global mean--field coupling effects the most rapid approach to global synchrony, and that when the number of "information carrying" molecular species increases, the rate of synchrony increases. The Langevin analysis is complemented by numerical simulations.

We present p53-MDM2-Glucose model to study spatio-temporal properties of the system induced by glucose. The variation in glucose concentration level triggers the system at different states, namely, oscillation death (stabilized), sustain and damped oscillations which correspond to various cellular states. The transition of these states induced by glucose is phase transition like behaviour. We also found that the intrinsic noise in stochastic system helps the system to stabilize more effectively. Further, the amplitude of $p53$ dynamics with the variation of glucose concentration level follows power law behaviour, $A_s(k)\sim k^\gamma$, where, $\gamma$ is a constant.

We present the mechanism of interaction of Wnt network module, which is responsible for periodic sometogenesis, with p53 regulatory network, which is one of the main regulators of various cellular functions, and switching of various oscillating states by investigating p53-Wnt model. The variation in Nutlin concentration in p53 regulating network drives the Wnt network module to different states, stabilized, damped and sustain oscillation states, and even to cycle arrest. Similarly, the change in Axin concentration in Wnt could able to modulate the p53 dynamics at these states. We then solve the set of coupled ordinary differential equations of the model using quasi steady state approximation. We, further, demonstrate the change of p53 and Gsk3 interaction rate, due to hypothetical catalytic reaction or external stimuli, can able to regulate the dynamics of the two network modules, and even can control their dynamics to protect the system from cycle arrest (apoptosis).

We study the ordering kinetics in $d=2$ ferromagnets which corresponds to populated neuron activities with long-ranged interactions, $V(r)\sim r^{-n}$ associated with short-ranged interaction. We present the results from comprehensive Monte Carlo (MC) simulations for the nonconserved Ising model with $n\ge 2$. Our results of long-ranged neuron kinetics are consistent with the same dynamical behavior of short-ranged case ($n > 4$). The calculated characteristic length scale in long-ranged interaction is found to be $n$ dependent ($L(t)\sim t^{1/(n-2)}$), whereas short-ranged interaction follows $L(t)\sim t^{1/2}$ law and approximately preserve universality in domain kinetics. Further, we did the comparative study of phase ordering near the critical temperature which follows different behaviours of domain ordering near and far critical temperature but follows universal scaling law.

We present the results of electron delocalization in thin films of finite thickness at finite tem- parature induced by thickness of the film and temparature. The two dimensional results show temparature induced metal-insulator transition (MIT) obtaining both insulating and metallic so- lutions. The localization length in insulating regime and zero frequency conductivity in metallic regime are found to be as a function of temparature and disorder parameter. Similarly, in thin films of finite thickness, delocalization of electronic states takes place induced by temparature as well as film thickness. Further, we could able to get critical temparature Tc for fixed thickness and critical thickness, b_c for fixed temparature. In metallic regime, conductivity is found to be as a function of film thickness, temparature and disorder.

We study the regulating mechanism of p53 on the properties of cell cycle dynamics in the light of the proposed model of interacting p53 and cell cycle networks via p53. Irradiation (IR) introduce to p53 compel p53 dynamics to suffer different phases, namely oscillating and oscillation death (stabilized) phases. The IR induced p53 dynamics undergo collapse of oscillation with collapse time \Delta t which depends on IR strength. The stress p53 via IR drive cell cycle molecular species MPF and cyclin dynamics to different states, namely, oscillation death, oscillations of periods, chaotic and sustain oscillation in their bifurcation diagram. We predict that there could be a critical \Delta t_c induced by p53 via IR_c, where, if \Delta t < \Delta t_c the cell cycle may come back to normal state, otherwise it will go to cell cycle arrest (apoptosis).

We study two important roles of miRNA as stress inducer and synchronizing agent in regulating diffusively coupled genetic oscillators within stochastic formalism and competition between them. We identify optimal value of coupling constant at which the rate of synchronization is strongest, below this value sychronizing activity dominates stress activity, and above it stress activity destroys synchronizing pattern. The concentration of miRNA maintained in individual oscillator modulates the synchronization activity achieved by diffusing miRNA.

We study the temporal and the synchronous behaviours in p53-Mdm2 regulatory network due to the interaction of its complex network components with the nitric oxide molecule. In single cell process, increase in nitric oxide concentration gives rise the transition to various p53 temporal behaviours, namely fixed point oscillation, damped oscillation and sustain oscillation indicating stability, weakly activated and strongly activated states. The noise in stochastic system is found to help to reach these states much faster as compared to deterministic case which is evident from permutation entropy dynamics. In coupled system with nitric oxide as diffusively coupling molecule, we found nitric oxide as strong coupling molecule within a certain range of coupling strength \epsilon beyond which it become weak synchronizing agent. We study these effects by using correlation like synchronization indicator \gamma obtained from permutation entropies of the coupled system, and found five important regimes in (\epsilon-\gamma) phase diagram, indicating desynchronized, transition, strongly synchronized, moderately synchronized and weakly synchronized regimes respectively. We claim that there is the competition between the toxicity and the synchronizing role of nitric oxide that lead the cell in different stressed conditions.

The regulating mechanism of miRNA on p53 dynamics in p53-MDM2-miRNA model network incorporating reactive oxygen species (ROS) is studied. The study shows that miRNA drives p53 dynamics at various states, namely, stabilized states and oscillating states (damped and sustain oscillation). We found the co-existence of these states within certain range of the concentartion level of miRNA in the system. This co-existence in p53 dynamics is the signature of the system's survival at various states, normal, activated and apoptosis driven by a constant concentration of miRNA.

We study how the temporal behaviours of p53 and MDM2 are affected by stress inducing bioactive molecules NO (Nitric Oxide) in the p53-MDM2-NO regulatory network. We also study synchronization among a group of identical stress systems arranged in a three dimensional array with nearest neighbour diffusive coupling. The role of NO and effect of noise are investigated. In the single system study, we have found three distinct types of temporal behaviour of p53, namely, oscillation death, damped oscillation and sustain oscillation, depending on the amount of stress induced by the NO concentration, indicating how p53 responds to the incoming stress. The correlation among the coupled systems increases as the value of coupling constant (\epsilon) is increased (\gamma increases) and becomes constant after certain value of \epsilon. The permutation entropy spectra H(\epsilon) for p53 and MDM2 as a function of \epsilon are found to be different due to direct and indirect interaction of NO with the respective proteins. \gamma versus \epsilon for p53 and MDM2 are found to be similar in deterministic approach, but different in stochastic approach and the separation between \gamma of the respective proteins as a function of \epsilon decreases as system size increases. The role of NO is found to be twofold: stress induced by it is prominent at small and large values of \epsilon but synchrony inducing by it dominates in moderate range of \epsilon. Excess stress induce apoptosis to the system.

We present the numerical estimation of noise parameter induced in the dynamics of the variables by random particle interactions involved in the stochastic chemical oscillator and use it as order parameter to detect the transition from stochastic to deterministic regime. In stochastic regime, this noise parameter is found to be increased as system size decreases, whereas, in deterministic regime it remains constant to minimum value as system size increases. This let the transition from fluctuating to fixed limit cycle oscillation as the system goes from stochastic to deterministic transition. We also numerically estimated the strength of the noise parameter involved both in Chemical Langevin equation and Master equation formalisms and found that strength of this parameter is much smaller in the former than the later., Comment: 6 pages

We study the influence of elements diffusing in and out of a network to the topological changes of the network and characterize it by investigating the behavior of probability of degree distribution ($\Gamma(k)$) with degree $k$. The local memory of the incoming element and its interaction with the elements already present in the network during the growing process significantly affect the network stability which in turn reorganize the network properties. We found that the properties of $\Gamma(k)$ of this network are deviated from scale free type, where the power law behavior contains a exponentially decay factor supporting earlier reported results of Amaral et.al. \cite{ama} and Newman \cite{new1} and recent statistical analysis results on degree distribution data of some scale free network [11]. Our numerical results also support the behavior of this $\Gamma(k)$. However, we found numerically the contribution from exponential factor to the $\Gamma(k)$ to be very weak as compared to the scale free factor showing that the network as a whole carries the scale free properties approximately., Comment: 5 pages

We examine the synchrony of the dynamics of localized [Ca^{2+}]_i oscillations in internal pool of astrocytes via diffusing coupling of a network of such cells in a certain topology where cytosolic Ca^{2+} and inositol 1,4,5-triphosphate (IP3) are coupling molecules; and possible long range interaction among the cells. Our numerical results claim that the cells exhibit fairly well coordinated behaviour through this coupling mechanism. It is also seen in the results that as the number of coupling molecular species is increased, the rate of synchrony is also increased correspondingly. Apart from the topology of the cells taken, as the number of coupled cells around any one of the cells in the system is increased, the cell process information faster., Comment: 7 pages, 7 figures

The temporal behaviour of segmentation clock oscillations show phase synchrony via mean field like coupling of delta protein restricting to nearest neighbours only, in a configuration of cells arranged in a regular three dimensional array. We found the increase of amplitudes of oscillating dynamical variables of the cells as the activation rate of delta-notch signaling is increased, however, the frequencies of oscillations are decreased correspondingly. Our results show the phase transition from desynchronized to synchronized behaviour by identifying three regimes, namely, desynchronized, transition and synchronized regimes supported by various qualitative and quantitative measurements., Comment: 6 pages, 6 figures

We examine microscopic mechanisms for coupling stochastic oscillators so that they display similar and correlated temporal variations. Unlike oscillatory motion in deterministic dynamical systems, complete synchronization of stochastic oscillators does not occur, but appropriately defined oscillator phase variables coincide. This is illustrated in model chemical systems and genetic networks that produce oscillations in the dynamical variables, and we show that suitable coupling of different networks can result in their phase synchronization., Comment: 4 figures, 4 pages

Localization of electronic states in disordered thin layered systems with b layers is studied within the Anderson model of localization using the transfer-matrix method and finite-size scaling of the inverse of the smallest Lyapunov exponent. The results support the one-parameter scaling hypothesis for disorder strengths W studied and b=1,...,6. The obtained results for the localization length are in good agreement with both the analytical results of the self-consistent theory of localization and the numerical scaling studies of the two-dimensional Anderson model. The localization length near the band center grows exponentially with b for fixed W but no localization-delocalization transition takes place., Comment: 6 pages, 5 figures