Absence of Mobility Edge in Short-Range Uncorrelated Disordered Model: Coexistence of Localized and Extended States.

Unlike the well-known Mott's argument that extended and localized states should not coexist at the same energy in a generic random potential, we formulate the main principles and provide an example of a nearest-neighbor tight-binding disordered model which carries both localized and extended states without forming the mobility edge. Unexpectedly, this example appears to be given by a well-studied ß ensemble with independently distributed random diagonal potential and inhomogeneous kinetic hopping terms. In order to analytically tackle the problem, we locally map the above model to the 1D Anderson model with matrix-size- and position-dependent hopping and confirm the coexistence of localized and extended states, which is shown to be robust to the perturbations of both potential and kinetic terms due to the separation of the above states in space. In addition, the mapping shows that the extended states are nonergodic and allows one to analytically estimate their fractal dimensions.

Mean residence times of TF-TF and TF-miRNA toggle switches.

Toggle switch networks are the simplest possible circuits with the ability of making a decision related to cell differentiation during embryonic development and disease progression. A common occurrence of toggle switch circuits is in the epithelial-mesenchymal transition (EMT) and its reverse, the mesenchymal-epithelial transition (MET), pathways which play key roles in phenotypic plasticity during cancer metastasis and therapy resistance. Recent studies have shown that the cells attaining one or more hybrid epithelial/mesenchymal (E/M) phenotypes during EMT/MET are more aggressive than those in either the epithelial or mesenchymal phenotype. In this work we studied the stability of each phenotype for different toggle switch circuits. We considered two-component toggle switch networks comprising either two mutually inhibiting transcription factors (TF-TF) or a TF-microRNA (TF-miR) chimera pair, and from Langevin dynamics, we determined the mean residence time (MRT) of cell phenotypes. MRT can be considered to be an indicator of stability in each cell phenotype and we showed that by replacing one of the TFs of the TF-TF toggle switch with miRNA generically stabilizes the hybrid phenotype. However, in the absence [presence] of a monostable hybrid state, the miRNA with faster [slower] degradation will make the hybrid state more probable. These results help to understand the implications of TF-TF and TF-miR circuits in the dynamics of cell fate decisions.

Nonergodic extended states in the ß ensemble.

Matrix models showing a chaotic-integrable transition in the spectral statistics are important for understanding many-body localization (MBL) in physical systems. One such example is the ß ensemble, known for its structural simplicity. However, eigenvector properties of the ß ensemble remain largely unexplored, despite energy level correlations being thoroughly studied. In this work we numerically study the eigenvector properties of the ß ensemble and find that the Anderson transition occurs at γ=1 and ergodicity breaks down at γ=0 if we express the repulsion parameter as ß=N^{-γ}. Thus other than the Rosenzweig-Porter ensemble (RPE), the ß ensemble is another example where nonergodic extended (NEE) states are observed over a finite interval of parameter values (0<γ<1). We find that the chaotic-integrable transition coincides with the breaking of ergodicity in the ß ensemble but with the localization transition in the RPE or the 1D disordered spin-1/2 Heisenberg model. As a result, the dynamical timescales in the NEE regime of the ß ensemble behave differently than the latter models.

Transport in deformed centrosymmetric networks.

Centrosymmetry often mediates perfect state transfer (PST) in various complex systems ranging from quantum wires to photosynthetic networks. We introduce the deformed centrosymmetric ensemble (DCE) of random matrices H(λ)≡H_{+}+λH_{-}, where H_{+} is centrosymmetric while H_{-} is skew-centrosymmetric. The relative strength of the H_{±} prompts the system size scaling of the control parameter as λ=N^{-γ/2}. We propose two quantities, P and C, quantifying centro and skewcentrosymmetry, respectively, exhibiting second-order phase transitions at γ_{P}≡1 and γ_{C}≡-1. In addition, DCE posses an ergodic transition at γ_{E}≡0. Thus equipped with a precise control of the extent of centrosymmetry in DCE, we study the manifestation of γ on the transport properties of complex networks. We propose that such random networks can be constructed using the eigenvectors of H(λ) and establish that the maximum transfer fidelity F_{T} is equivalent to the degree of centrosymmetry P.

First passage time properties of miRNA-mediated protein translation.

An important function of microRNAs in gene regulation is to repress the protein synthesis in a multi-step process with implications in timing efficiency of the regulatory network. We propose a stochastic description of translation-initiation mechanism and solve for the steady state distribution of protein number in the linear regime. The time-dependent moments have been approximately calculated and the role of miRNAs in determining the First Passage Time (FPT) properties of protein dynamics has been established. We analytically show that the modulation of slow rates of the translation process will result in efficient and robust timing mechanism. For a general nonlinear model our numerical simulation results are in qualitative agreement with the linear model.

First passage time in post-transcriptional regulation by multiple small RNAs.

The post-transcriptional regulation of a protein by multiple small RNA molecules has been formulated as a stochastic process. An approximate solution of the master equation shows that the protein statistics can exhibit a generic form applicable for many regulatory scenarios. The first passage time (FPT) statistics has been obtained for regulation by single sRNA, with negative and positive regulations as limiting cases, as well as regulation by multiple sRNAs. The multiple sRNAs are able to independently control protein mean and variance, and we show that this is an advantageous mechanism to control FPT fluctuations in order to improve timing efficiency in post-transcriptional regulation.

NFATc Acts as a Non-Canonical Phenotypic Stability Factor for a Hybrid Epithelial/Mesenchymal Phenotype.

Metastasis remains the cause of over 90% of cancer-related deaths. Cells undergoing metastasis use phenotypic plasticity to adapt to their changing environmental conditions and avoid therapy and immune response. Reversible transitions between epithelial and mesenchymal phenotypes - epithelial-mesenchymal transition (EMT) and its reverse mesenchymal-epithelial transition (MET) - form a key axis of phenotypic plasticity during metastasis and therapy resistance. Recent studies have shown that the cells undergoing EMT/MET can attain one or more hybrid epithelial/mesenchymal (E/M) phenotypes, the process of which is termed as partial EMT/MET. Cells in hybrid E/M phenotype(s) can be more aggressive than those in either epithelial or mesenchymal state. Thus, it is crucial to identify the factors and regulatory networks enabling such hybrid E/M phenotypes. Here, employing an integrated computational-experimental approach, we show that the transcription factor nuclear factor of activated T-cell (NFATc) can inhibit the process of complete EMT, thus stabilizing the hybrid E/M phenotype. It increases the range of parameters enabling the existence of a hybrid E/M phenotype, thus behaving as a phenotypic stability factor (PSF). However, unlike previously identified PSFs, it does not increase the mean residence time of the cells in hybrid E/M phenotypes, as shown by stochastic simulations; rather it enables the co-existence of epithelial, mesenchymal and hybrid E/M phenotypes and transitions among them. Clinical data suggests the effect of NFATc on patient survival in a tissue-specific or context-dependent manner. Together, our results indicate that NFATc behaves as a non-canonical PSF for a hybrid E/M phenotype.

Timing effciency in small-RNA-regulated post-transcriptional processes.

Gene regulation in a cellular environment is a stochastic phenomenon leading to a large variability in mRNAs and protein numbers that are often produced in bursts. The regulation leading to varied protein dynamics can be ascribed to transcriptional or post-transcriptional mechanisms. In transcriptional regulation, the gene dynamically switches between an active and an inactive state, while in the post-transcriptional regulation small RNAs tune the activity of mRNAs. In either scenario, it is possible to calculate the time-dependent probability distribution of proteins and address the interesting question pertaining to their first passage time statistics. The coefficient of variation of first passage time can be considered to be an indicator of efficiency in controlling regulatory pathways and we show that post-transcriptional regulation performs better than simple transcriptional regulation for comparable protein yields.

Integrated Regulation of HuR by Translation Repression and Protein Degradation Determines Pulsatile Expression of p53 Under DNA Damage.

Expression of tumor suppressor p53 is regulated at multiple levels, disruption of which often leads to cancer. We have adopted an approach combining computational systems modeling with experimental validation to elucidate the translation regulatory network that controls p53 expression post DNA damage. The RNA-binding protein HuR activates p53 mRNA translation in response to UVC-induced DNA damage in breast carcinoma cells. p53 and HuR levels show pulsatile change post UV irradiation. The computed model fitted with the observed pulse of p53 and HuR only when hypothetical regulators of synthesis and degradation of HuR were incorporated. miR-125b, a UV-responsive microRNA, was found to represses the translation of HuR mRNA. Furthermore, UV irradiation triggered proteasomal degradation of HuR mediated by an E3-ubiquitin ligase tripartite motif-containing 21 (TRIM21). The integrated action of miR-125b and TRIM21 constitutes an intricate control system that regulates pulsatile expression of HuR and p53 and determines cell viability in response to DNA damage.

First-passage time statistics of stochastic transcription process for time-dependent reaction rates.

Transcription in gene expression is an intrinsically noisy process which involves production and degradation of mRNAs. An important quantity to describe this stochastic process is the first-passage time (FPT), i.e., the time taken by mRNAs to reach a particular threshold. The process of transcription can be modelled as a simple birth-death process, assuming that the promoter is always in an active state and to encode the stochastic environment we consider the transcription rate to be time dependent. This generalization is suitable to capture bursty mRNA dynamics usually modelled as an ON-Off model and simplifies the calculation of FPT statistics for a cell population. We study the role of periodic modulation of the transcription rate on different moments of FPT distribution of a population of cells. Our calculation shows that for sinusoidal modulation there exists an extremal value of mean FPT as a function of the time period and phase of the transcription signal. However, for the square wave modulation of transcription rates simulation results show that the extremal value of the MFPT behaves monotonically with the variation of the phase.

Stability and mean residence times for hybrid epithelial/mesenchymal phenotype.

Cancer metastasis and drug resistance remain unsolved clinical challenges. A phenotypic transition that is often implicated in both these processes is epithelial-mesenchymal transition (EMT) during which epithelial cells weaken their cell-cell adhesion and gain traits of migration and invasion, typical of mesenchymal cells. However, recent studies indicate that apart from these two states, cells can also exist in one or more hybrid E/M state(s), which plays an aggressive role in progression of the disease. Furthermore, computational and experimental studies have identified a variety of phenotypic stability factors (PSFs) that stabilize the hybrid E/M state(s) and can increase disease aggressiveness. In this work, we study EMT regulatory networks, in the presence of different PSFs, as dynamical systems subjected to random fluctuations. The cells thus explore different stable E, M, E/M states in the potential landscape and our aim is to quantify the residence time in each of these states. Our stochastic simulations indicate an universal feature that the mean residence time (MRT) in the hybrid E/M state is enhanced in the presence of PSFs. We demonstrate that the feature is consistent for a variety of PSFs, namely, GRHL2, OVOL, ΔNp63α, miR-145/OCT4, participating in the core EMT regulatory network. Our results reveal potential targets for pushing cells out of a hybrid E/M state and thus halting metastatic progression.

Estimation of mean first passage time for bursty gene expression.

Gene expression is an intrinsically noisy process, typically, producing mRNAs and proteins in bursts. An important description of such stochastic processes can be done in terms of the mean first passage time (MFPT), i.e., the time taken by mRNAs/proteins to reach a particular threshold. We study the role of burstiness on MFPT and obtain an analytical expression for different models of transcriptional and translational bursts. Our analytical results and numerical simulations confirm that MFPT monotonically decreases with burstiness.

Statistics of Lyapunov exponent spectrum in randomly coupled Kuramoto oscillators.

Characterization of spatiotemporal dynamics of coupled oscillatory systems can be done by computing the Lyapunov exponents. We study the spatiotemporal dynamics of randomly coupled network of Kuramoto oscillators and find that the spectral statistics obtained from the Lyapunov exponent spectrum show interesting sensitivity to the coupling matrix. Our results indicate that in the weak coupling limit the gap distribution of the Lyapunov spectrum is Poissonian, while in the limit of strong coupling the gap distribution shows level repulsion. Moreover, the oscillators settle to an inhomogeneous oscillatory state, and it is also possible to infer the random network properties from the Lyapunov exponent spectrum.

Transcriptional dynamics with time-dependent reaction rates.

Transcription is the first step in the process of gene regulation that controls cell response to varying environmental conditions. Transcription is a stochastic process, involving synthesis and degradation of mRNAs, that can be modeled as a birth-death process. We consider a generic stochastic model, where the fluctuating environment is encoded in the time-dependent reaction rates. We obtain an exact analytical expression for the mRNA probability distribution and are able to analyze the response for arbitrary time-dependent protocols. Our analytical results and stochastic simulations confirm that the transcriptional machinery primarily act as a low-pass filter. We also show that depending on the system parameters, the mRNA levels in a cell population can show synchronous/asynchronous fluctuations and can deviate from Poisson statistics.

Non-equilibrium dynamics of stochastic gene regulation.

The process of gene regulation is comprised of intrinsically random events resulting in large cell-to-cell variability in mRNA and protein numbers. With gene expression being the central dogma of molecular biology, it is essential to understand the origin and role of these fluctuations. An intriguing observation is that the number of mRNA present in a cell are not only random and small but also that they are produced in bursts. The gene switches between an active and an inactive state, and the active gene transcribes mRNA in bursts. Transcriptional noise being bursty, so are the number of proteins and the subsequent gene expression levels. It is natural to ask the question: what is the reason for the bursty mRNA dynamics? And can the bursty dynamics be shown to be entropically favorable by studying the reaction kinetics underlying the gene regulation mechanism? The dynamics being an out-of-equilibrium process, the fluctuation theorem for entropy production in the reversible reaction channel is discussed. We compute the entropy production rate for varying degrees of burstiness. We find that the reaction parameters that maximize the burstiness simultaneously maximize the entropy production rate.

Adaptation to changes in higher-order stimulus statistics in the salamander retina.

Adaptation in the retina is thought to optimize the encoding of natural light signals into sequences of spikes sent to the brain. While adaptive changes in retinal processing to the variations of the mean luminance level and second-order stimulus statistics have been documented before, no such measurements have been performed when higher-order moments of the light distribution change. We therefore measured the ganglion cell responses in the tiger salamander retina to controlled changes in the second (contrast), third (skew) and fourth (kurtosis) moments of the light intensity distribution of spatially uniform temporally independent stimuli. The skew and kurtosis of the stimuli were chosen to cover the range observed in natural scenes. We quantified adaptation in ganglion cells by studying linear-nonlinear models that capture well the retinal encoding properties across all stimuli. We found that the encoding properties of retinal ganglion cells change only marginally when higher-order statistics change, compared to the changes observed in response to the variation in contrast. By analyzing optimal coding in LN-type models, we showed that neurons can maintain a high information rate without large dynamic adaptation to changes in skew or kurtosis. This is because, for uncorrelated stimuli, spatio-temporal summation within the receptive field averages away non-gaussian aspects of the light intensity distribution.

Phase description of spiking neuron networks with global electric and synaptic coupling.

Phase models are among the simplest neuron models reproducing spiking behavior, excitability, and bifurcations toward periodic firing. However, coupling among neurons has been considered only using generic arguments valid close to the bifurcation point, and the differentiation between electric and synaptic coupling remains an open question. In this work we aim to address this question and derive a mathematical formulation for the various forms of coupling. We construct a mathematical model based on a planar simplification of the Morris-Lecar model. Based on geometric arguments we then derive a phase description of a network of the above oscillators with biologically realistic electric coupling and subsequently with chemical coupling under fast synapse approximation. We demonstrate analytically that electric and synaptic coupling are differently expressed on the level of the network's phase description with qualitatively different dynamics. Our mathematical analysis shows that a breaking of the translational symmetry in the phase flows is responsible for the different bifurcations paths of electric and synaptic coupling. Our numerical investigations confirm these findings and show excellent correspondence between the dynamics of the full network and the network's phase description.

General properties of transcriptional time series in Escherichia coli.

Gene activity is described by the time series of discrete, stochastic mRNA production events. This transcriptional time series shows intermittent, bursty behavior. One consequence of this temporal intricacy is that gene expression can be tuned by varying different features of the time series. Here we quantify copy-number statistics of mRNA from 20 Escherichia coli promoters using single-molecule fluorescence in situ hybridization in order to characterize the general properties of these transcriptional time series. We find that the degree of burstiness is correlated with gene expression level but is largely independent of other parameters of gene regulation. The observed behavior can be explained by the underlying variation in the duration of bursting events. Using Shannon's mutual information function, we estimate the mutual information transmitted between an outside stimulus, such as the extracellular concentration of inducer molecules, and intracellular levels of mRNA. This suggests that the outside stimulus transmits information reflected in the properties of transcriptional time series.

Simple model for bursting dynamics of neurons.

Neuronal cells in isolation or as an assembly exhibit bursting behavior on two different time scales. We introduce a simple one-dimensional model which requires only one phase variable to describe the phenomenon of parabolic bursting. The analysis in the continuum limit reveals that for any unimodal distribution of frequencies, the qualitative properties of the full and the reduced model are identical. Further, we derive analytically an exact low-dimensional description of a globally coupled network of bursting oscillators for our model. Study of the stability for this low-dimensional model reveals different dynamical signatures in the parameter space. We demonstrate that the structure of the parameter space remains independent of the number of spikes per burst.

Noise during rest enables the exploration of the brain's dynamic repertoire.

Traditionally brain function is studied through measuring physiological responses in controlled sensory, motor, and cognitive paradigms. However, even at rest, in the absence of overt goal-directed behavior, collections of cortical regions consistently show temporally coherent activity. In humans, these resting state networks have been shown to greatly overlap with functional architectures present during consciously directed activity, which motivates the interpretation of rest activity as day dreaming, free association, stream of consciousness, and inner rehearsal. In monkeys, it has been shown though that similar coherent fluctuations are present during deep anesthesia when there is no consciousness. Here, we show that comparable resting state networks emerge from a stability analysis of the network dynamics using biologically realistic primate brain connectivity, although anatomical information alone does not identify the network. We specifically demonstrate that noise and time delays via propagation along connecting fibres are essential for the emergence of the coherent fluctuations of the default network. The spatiotemporal network dynamics evolves on multiple temporal scales and displays the intermittent neuroelectric oscillations in the fast frequency regimes, 1-100 Hz, commonly observed in electroencephalographic and magnetoencephalographic recordings, as well as the hemodynamic oscillations in the ultraslow regimes, <0.1 Hz, observed in functional magnetic resonance imaging. The combination of anatomical structure and time delays creates a space-time structure in which the neural noise enables the brain to explore various functional configurations representing its dynamic repertoire.