Subtype-specific network organization of molecular complexes in breast cancer.

Breast cancer, a leading cause of death in women, is a complex heterogeneous disease comprising multiple molecular subtypes with different treatment responses and hence clinical outcomes. The present study aims to gain a deeper insight into the disease complexities at the level of molecular subtypes. For this, first, three subtype networks of breast cancer, viz.

Modelling of indirect cell-cell interaction networks mediated by IFNγ/IL-4 cytokine involved in atopic dermatitis.

Atopic dermatitis (AD) is an immune-driven inflammatory skin disease that is known to have a significantly high life-time prevalence in the human population. T-helper (Th) immune cells play a key role in the pathogenesis of AD which is marked by defects in the skin barrier function along with a significant increase in the population of either Th1 or Th2 sub-types of Th cells. The progression of AD from the acute to chronic phase is still poorly understood, and here we explore the mechanism of this transition through the study of a mathematical model for indirect cell-cell interactions among Th and skin cells via the secreted cytokines IFNγ and IL-4, both known to have therapeutic potential.

General mechanism for the 1/f noise.

We consider the response of a memoryless nonlinear device that acts instantaneously, converting an input signal ξ(t) into an output η(t) at the same time t. For input Gaussian noise with power-spectrum 1/f^{α}, the nonlinearity can modify the spectral index of the output to give a spectrum that varies as 1/f^{α^{'}} with α^{'}≠α. We show that the value of α^{'} depends on the nonlinear transformation and can be tuned continuously.

Emergence of chimeras through induced multistability.

Chimeras, namely coexisting desynchronous and synchronized dynamics, are formed in an ensemble of identically coupled identical chaotic oscillators when the coupling induces multiple stable attractors, and further when the basins of the different attractors are intertwined in a complex manner. When there is coupling-induced multistability, an ensemble of identical chaotic oscillators-with global coupling, or also under the influence of common noise or an external drive (chaotic, periodic, or quasiperiodic)-inevitably exhibits chimeric behavior. Induced multistability in the system leads to the formation of distinct subpopulations, one or more of which support synchronized dynamics, while in others the motion is asynchronous or incoherent.

Generalized synchrony of coupled stochastic processes with multiplicative noise.

We study the effect of multiplicative noise in dynamical flows arising from the coupling of stochastic processes with intrinsic noise. Situations wherein such systems arise naturally are in chemical or biological oscillators that are coupled to each other in a drive-response configuration. Above a coupling threshold we find that there is a strong correlation between the drive and the response: This is a stochastic analog of the phenomenon of generalised synchronization.

Phase oscillators in modular networks: The effect of nonlocal coupling.

We study the dynamics of nonlocally coupled phase oscillators in a modular network. The interactions include a phase lag, α. Depending on the various parameters the system exhibits a number of different dynamical states.

Bipartite networks of oscillators with distributed delays: Synchronization branches and multistability.

We study synchronization in bipartite networks of phase oscillators with general nonlinear coupling and distributed time delays. Phase-locked solutions are shown to arise, where the oscillators in each partition are perfectly synchronized among themselves but can have a phase difference with the other partition, with the phase difference necessarily being either zero or π radians. Analytical conditions for the stability of both types of solutions are obtained and solution branches are explicitly calculated, revealing that the network can have several coexisting stable solutions.

Delay-induced remote synchronization in bipartite networks of phase oscillators.

We study a system of mismatched oscillators on a bipartite topology with time-delay coupling, and analyze the synchronized states. For a range of parameters, when all oscillators lock to a common frequency, we find solutions such that systems within a partition are in complete synchrony, while there is lag synchronization between the partitions. Outside this range, such a solution does not exist and instead one observes scenarios of remote synchronization-namely, chimeras and individual synchronization, where either one or both of the partitions are synchronized independently.

Phase-locked regimes in delay-coupled oscillator networks.

For an ensemble of globally coupled oscillators with time-delayed interactions, an explicit relation for the frequency of synchronized dynamics corresponding to different phase behaviors is obtained. One class of solutions corresponds to globally synchronized in-phase oscillations. The other class of solutions have mixed phases, and these can be either randomly distributed or can be a splay state, namely with phases distributed uniformly on a circle.

Synchronization and amplitude death in hypernetworks.

We study dynamical systems on a hypernetwork, namely by coupling them through several variables. For the case when the coupling(s) are all linear, a comprehensive analysis of the master stability function (MSF) for synchronized dynamics is presented and, through application to a number of paradigmatic examples, the typical forms of the MSF are discussed. The MSF formalism for hypernetworks also provides a framework to study synchronization in systems that are diffusively coupled through dissimilar variables-the so-called conjugate coupling that can lead to amplitude or oscillation death.

Two-layer modular analysis of gene and protein networks in breast cancer.

Background: Genomic, proteomic and high-throughput gene expression data, when integrated, can be used to map the interaction networks between genes and proteins. Different approaches have been used to analyze these networks, especially in cancer, where mutations in biologically related genes that encode mutually interacting proteins are believed to be involved. This system of integrated networks as a whole exhibits emergent biological properties that are not obvious at the individual network level.

Chimeras with multiple coherent regions.

We study chimeric states in a coupled phase oscillator system with piecewise linear nonlocal coupling. By modifying the details of the coupling, it is possible to obtain multiple chimeric states with a specified number of coherent regions and with specified phase relationships. The case of a two-component chimera is illustrated and the generalization to arbitrary chimeric configurations is discussed.

Scaling behavior in probabilistic neuronal cellular automata.

We study a neural network model of interacting stochastic discrete two-state cellular automata on a regular lattice. The system is externally tuned to a critical point which varies with the degree of stochasticity (or the effective temperature). There are avalanches of neuronal activity, namely, spatially and temporally contiguous sites of activity; a detailed numerical study of these activity avalanches is presented, and single, joint, and marginal probability distributions are computed.

Phantom instabilities in adiabatically driven systems: dynamical sensitivity to computational precision.

We study the robustness of dynamical phenomena in adiabatically driven nonlinear mappings with skew-product structure. Deviations from true orbits are observed when computations are performed with inadequate numerical precision for monotone, periodic, or quasiperiodic driving. The effect of slow modulation is to "freeze" orbits in long intervals of purely contracting or purely expanding dynamics in the phase space.

Power spectrum of mass and activity fluctuations in a sandpile.

We consider a directed Abelian sandpile on a strip of size 2×n, driven by adding a grain randomly at the left boundary after every T timesteps. We establish the exact equivalence of the problem of mass fluctuations in the steady state and the number of zeros in the ternary-base representation of the position of a random walker on a ring of size 3^{n}. We find that while the fluctuations of mass have a power spectrum that varies as 1/f for frequencies in the range 3^{-2n}≪f≪1/T, the activity fluctuations in the same frequency range have a power spectrum that is linear in f.

MicroRNAs modulate the dynamics of the NF-κB signaling pathway.

Background: NF-κB, a major transcription factor involved in mammalian inflammatory signaling, is primarily involved in regulation of response to inflammatory cytokines and pathogens. Its levels are tightly regulated since uncontrolled inflammatory response can cause serious diseases. Mathematical models have been useful in revealing the underlying mechanisms, the dynamics, and other aspects of regulation in NF-κB signaling.

Order parameter for the transition from strong to weak generalized synchrony from empirical mode decomposition analysis.

We examine driven nonlinear dynamical systems that are known to be in a state of generalized synchronization with an external drive. The chaotic time series of the response system are subject to empirical mode decomposition analysis. The instantaneous intrinsic mode frequencies (and their variance) present in these signals provide suitable order parameters for detecting the transition between the regimes of strong and weak generalized synchrony.

Dynamical effects of integrative time-delay coupling.

We study coupled dynamical systems wherein the influence of one system on the other is cumulative: coupling signals are integrated over a time interval τ. A major consequence of integrative coupling is that amplitude death occurs over a wider range and in a single region in parameter space. For coupled limit cycle oscillators (the Landau-Stuart model) we obtain an analytic estimate for the boundary of this region while for coupled chaotic Lorenz oscillators numerical results are presented.

Amplitude death in nonlinear oscillators with nonlinear coupling.

Amplitude death is the cessation of oscillations that occurs in coupled nonlinear systems when fixed points are stabilized as a consequence of the interaction. We show here that this phenomenon is very general: it occurs in nonlinearly coupled systems in the absence of parameter mismatch or time delay although time-delayed interactions can enhance the effect. Application is made to synaptically coupled model neurons, nonlinearly coupled Rössler oscillators, as well as to networks of nonlinear oscillators with nonlinear coupling.

Transition to weak generalized synchrony in chaotically driven flows.

We study regimes of strong and weak generalized synchronization in chaotically forced nonlinear flows. The transition between these dynamical states can occur via a number of different routes, and here we examine the onset of weak generalized synchrony through intermittency and blowout bifurcations. The quantitative characterization of this dynamical transition is facilitated by measures that have been developed for the study of strange nonchaotic motion.

Characterisation of inactivation domains and evolutionary strata in human X chromosome through Markov segmentation.

Markov segmentation is a method of identifying compositionally different subsequences in a given symbolic sequence. We have applied this technique to the DNA sequence of the human X chromosome to analyze its compositional structure. The human X chromosome is known to have acquired DNA through distinct evolutionary events and is believed to be composed of five evolutionary strata.

miRNA-regulated dynamics in circadian oscillator models.

Background: We have studied the dynamics of miRNA regulation in two models of circadian oscillators. miRNAs are a class of small RNA molecules (18-24 nucleotides) that are known to regulate gene expression at the post-transcriptional level by reducing the amount of proteins produced by translation. This is done either by blocking translation or by degradation of mRNAs, the latter being mainly due to the initiation of a set of processes induced by formation of the miRNA:mRNA complex.

Scenarios for generalized synchronization with chaotic driving.

In chaotically driven nonlinear dynamical systems, weak generalized synchrony can arise through distinct scenarios or routes in a manner similar to the onset of low-dimensional chaos or the creation of strange nonchaotic attractors in quasiperiodically driven systems. The limit sets of the dynamics for weak generalized synchronization are nonchaotic-the Lyapunov exponent is nonpositive-and are geometrically strange. Quantitative measures related to the parameter sensitivity exponent and finite-time Lyapunov exponent distributions can be defined in order to characterize generalized synchronization.

Universal occurrence of the phase-flip bifurcation in time-delay coupled systems.

Recently, the phase-flip bifurcation has been described as a fundamental transition in time-delay coupled, phase-synchronized nonlinear dynamical systems. The bifurcation is characterized by a change of the synchronized dynamics from being in-phase to antiphase, or vice versa; the phase-difference between the oscillators undergoes a jump of pi as a function of the coupling strength or the time delay. This phase-flip is accompanied by discontinuous changes in the frequency of the synchronized oscillators, and in the largest negative Lyapunov exponent or its derivative.

Analytical signal analysis of strange nonchaotic dynamics.

We apply an analytical signal analysis to strange nonchaotic dynamics. Through this technique it is possible to obtain the spectrum of instantaneous intrinsic mode frequencies that are present in a given signal. We find that the second-mode frequency and its variance are good order parameters for dynamical transitions from quasiperiodic tori to strange nonchaotic attractors (SNAs) and from SNAs to chaotic attractors.