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., ER-/HER2-, ER+/HER2-, and HER2+, were constructed utilizing mRNA expression profiles of tumor tissues. Subsequently, these networks were used to construct three exclusively subtype-specific networks. Further, the mRNA expression profiles of all three subtypes were analyzed using differential correlations based on z-statistics of the F-test. Finally, functional enrichment analysis was carried out to elucidate functions and processes of important genes involved in subtype networks. From this analysis, it was observed that these subtype networks share a commonality among them in terms of preserved patterns. However, these networks possess specific patterns that result in exclusively subtypespecific networks having unique sets of wiring among the genes. Additionally, the significantly differentially correlated gene pairs between two subtypes demonstrate subtype-specific expressional patterns which make them different at the molecular level. Furthermore, the network analysis also revealed ER-/HER2--specific genes, viz., LUM, RARB, and ERCC6. Thus, the present analysis provides new insights for further research on breast cancer subtypes and hence the development of the most effective diagnosis and treatment.

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. An increase in the level of cytokine IFN γ can catalyse the transition of AD from an acute to a chronic stage, while an increase in the level of cytokine IL-4 has the reverse effect. In our model, the transition of AD from the acute to chronic stage and vice versa can be abrupt (switch-like) with hysteresis: this bistable behaviour can potentially be used to keep AD in the acute phase since therapy based on suppression of IFNγ can retard the transition to the chronic phase.

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. This provides a general mechanism for the ubiquitous 1/f noise found in nature.

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. We study the mechanism for the emergence of such chimeric states, and we discuss the generality of our results.

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. Since the dynamical fluctuations are large when there is intrinsic noise, it is necessary to employ measures that are sensitive to correlations between the variables of drive and the response, the permutation entropy, or the mutual information in order to detect the transition to generalized synchrony in such systems.

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. In addition to global synchrony there can also be modular synchrony when each module can synchronize separately to a different frequency. There can also be multicluster frequency chimeras, namely coherent domains consisting of modules that are separately synchronized to different frequencies, coexisting with modules within which the dynamics is desynchronized. We apply the Ott-Antonsen ansatz in order to reduce the effective dimensionality and thereby carry out a detailed analysis of the 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. With increasing value of the mean delay, the system exhibits hysteresis, phase flips, final state sensitivity, and an extreme form of multistability where the numbers of stable in-phase and antiphase synchronous solutions with distinct frequencies grow without bound. The theory is applied to networks of Landau-Stuart and Rössler oscillators and shown to accurately predict both in-phase and antiphase synchronous behavior in appropriate parameter ranges.

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. In the absence of time delay such states are not observed in phase oscillators.

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. We analyze the system in terms of modules, namely a set of densely interconnected nodes that can be further divided into submodules that are expected to participate in multiple biological activities in coordinated manner. RESULTS: In the present work we construct two layers of the breast cancer network: the gene layer, where the correlation network of breast cancer genes is analyzed to identify gene modules, and the protein layer, where each gene module is extended to map out the network of expressed proteins and their interactions in order to identify submodules. Each module and its associated submodules are analyzed to test the robustness of their topological distribution. The constituent biological phenomena are explored through the use of the Gene Ontology. We thus construct a "network of networks", and demonstrate that both the gene and protein interaction networks are modular in nature. By focusing on the ontological classification, we are able to determine the entire GO profiles that are distributed at different levels of hierarchy. Within each submodule most of the proteins are biologically correlated, and participate in groups of distinct biological activities. CONCLUSIONS: The present approach is an effective method for discovering coherent gene modules and protein submodules. We show that this also provides a means of determining biological pathways (both novel and as well those that have been reported previously) that are related, in the present instance, to breast cancer. Similar strategies are likely to be useful in the analysis of other diseases as well.

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. In the strong coupling limit and for larger networks, the in-phase synchronized configuration alone remains. Upon variation of the coupling strength or the size of the system, the frequency can change discontinuously, when there is a transition from one class of solutions to another. This can be from the in-phase state to a mixed-phase state, but can also occur between two in-phase configurations of different frequency. Analytical and numerical results are presented for coupled Landau-Stuart oscillators, while numerical results are shown for Rössler and FitzHugh-Nagumo systems.

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. The phase relations between the two clusters of phase oscillators is described in some detail.

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. At the critical point, we find that the scaling exponents for the variables are in good agreement with a mean-field theory.

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. When computations are carried out with low precision, numerical errors build up phantom instabilities which ultimately force trajectories to depart from the true motion. Thus, the dynamics observed with finite precision computation shows sensitivity to numerical precision: the minimum accuracy required to obtain "true" trajectories is proportional to an internal timescale that can be defined for the adiabatic system.

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. The recognition that miRNAs are important regulators of gene expression, and that a number of miRNAs target different components of the NF-κB network, motivate the incorporation of miRNA regulated steps in existing mathematical models to help understand the quantitative aspects of miRNA mediated regulation. METHODOLOGY/PRINCIPAL FINDINGS: In this study, two separate scenarios of miRNA regulation within an existing model are considered. In the first, miRNAs target adaptor proteins involved in the synthesis of IKK that serves as the NF-κB activator. In the second, miRNAs target different isoforms of IκB that act as NF-κB inhibitors. Simulations are carried out under two different conditions: when all three isoforms of IκB are present (wild type), and when only one isoform (IκBα) is present (knockout type). In both scenarios, oscillations in the NF-κB levels are observed and are found to be dependent on the levels of miRNAs. CONCLUSIONS/SIGNIFICANCE: Computational modeling can provide fresh insights into intricate regulatory processes. The introduction of miRNAs affects the dynamics of the NF-κB signaling pathway in a manner that depends on the role of the target. This "fine-tuning" property of miRNAs helps to keep the system in check and prevents it from becoming uncontrolled. The results are consistent with earlier experimental findings.

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. Application is made to a variety of chaotically driven flows as well as maps.

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. For given τ we find that there is a critical coupling strength at which the frequency of oscillations changes discontinuously.

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. Weak and strong generalized synchronous motion show contrasting sensitivity to parametric variation and have distinct distributions of finite-time Lyapunov exponents.

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. By suitably designing the nonlinear coupling, arbitrary steady states can be stabilized.