Intrinsically disordered proteins and conformational noise: The hypothesis a decade later.

Phenotypic plasticity is the ability of individual genotypes to produce different phenotypes in response to environmental perturbations. We previously postulated how conformational noise emanating from conformational dynamics of intrinsically disordered proteins (IDPs) which is distinct from transcriptional noise, can contribute to phenotypic switching by rewiring the cellular protein interaction network. Since most transcription factors are IDPs, we posited that conformational noise is an integral component of transcriptional noise implying that IDPs may amplify total noise in the system either stochastically or in response to environmental changes. Here, we review progress in elucidating the details of the hypothesis. We highlight empirical evidence supporting the hypothesis, discuss conceptual advances that underscore its fundamental importance and implications, and identify areas for future investigations.

Sparse representations of high dimensional neural data.

Conventional Vector Autoregressive (VAR) modelling methods applied to high dimensional neural time series data result in noisy solutions that are dense or have a large number of spurious coefficients. This reduces the speed and accuracy of auxiliary computations downstream and inflates the time required to compute functional connectivity networks by a factor that is at least inversely proportional to the true network density. As these noisy solutions have distorted coefficients, thresholding them as per some criterion, statistical or otherwise, does not alleviate the problem. Thus obtaining a sparse representation of such data is important since it provides an efficient representation of the data and facilitates its further analysis. We propose a fast Sparse Vector Autoregressive Greedy Search (SVARGS) method that works well for high dimensional data, even when the number of time points is relatively low, by incorporating only statistically significant coefficients. In numerical experiments, our methods show high accuracy in recovering the true sparse model. The relative absence of spurious coefficients permits accurate, stable and fast evaluation of derived quantities such as power spectrum, coherence and Granger causality. Consequently, sparse functional connectivity networks can be computed, in a reasonable time, from data comprising tens of thousands of channels/voxels. This enables a much higher resolution analysis of functional connectivity patterns and community structures in such large networks than is possible using existing time series methods. We apply our method to EEG data where computed network measures and community structures are used to distinguish emotional states as well as to ADHD fMRI data where it is used to distinguish children with ADHD from typically developing children.

Intrinsically Disordered Proteins: Critical Components of the Wetware.

Despite the wealth of knowledge gained about intrinsically disordered proteins (IDPs) since their discovery, there are several aspects that remain unexplored and, hence, poorly understood. A living cell is a complex adaptive system that can be described as a wetware─a metaphor used to describe the cell as a computer comprising both hardware and software and attuned to logic gates─capable of "making" decisions. In this focused Review, we discuss how IDPs, as critical components of the wetware, influence cell-fate decisions by wiring protein interaction networks to keep them minimally frustrated. Because IDPs lie between order and chaos, we explore the possibility that they can be modeled as attractors. Further, we discuss how the conformational dynamics of IDPs manifests itself as conformational noise, which can potentially amplify transcriptional noise to stochastically switch cellular phenotypes. Finally, we explore the potential role of IDPs in prebiotic evolution, in forming proteinaceous membrane-less organelles, in the origin of multicellularity, and in protein conformation-based transgenerational inheritance of acquired characteristics. Together, these ideas provide a new conceptual framework to discern how IDPs may perform critical biological functions despite their lack of structure.

Intrinsically disordered proteins: Ensembles at the limits of Anfinsen's dogma.

Intrinsically disordered proteins (IDPs) are proteins that lack rigid 3D structure. Hence, they are often misconceived to present a challenge to Anfinsen's dogma. However, IDPs exist as ensembles that sample a quasi-continuum of rapidly interconverting conformations and, as such, may represent proteins at the extreme limit of the Anfinsen postulate. IDPs play important biological roles and are key components of the cellular protein interaction network (PIN). Many IDPs can interconvert between disordered and ordered states as they bind to appropriate partners. Conformational dynamics of IDPs contribute to conformational noise in the cell. Thus, the dysregulation of IDPs contributes to increased noise and "promiscuous" interactions. This leads to PIN rewiring to output an appropriate response underscoring the critical role of IDPs in cellular decision making. Nonetheless, IDPs are not easily tractable experimentally. Furthermore, in the absence of a reference conformation, discerning the energy landscape representation of the weakly funneled IDPs in terms of reaction coordinates is challenging. To understand conformational dynamics in real time and decipher how IDPs recognize multiple binding partners with high specificity, several sophisticated knowledge-based and physics-based in silico sampling techniques have been developed. Here, using specific examples, we highlight recent advances in energy landscape visualization and molecular dynamics simulations to discern conformational dynamics and discuss how the conformational preferences of IDPs modulate their function, especially in phenotypic switching. Finally, we discuss recent progress in identifying small molecules targeting IDPs underscoring the potential therapeutic value of IDPs. Understanding structure and function of IDPs can not only provide new insight on cellular decision making but may also help to refine and extend Anfinsen's structure/function paradigm.

Dynamic Phenotypic Switching and Group Behavior Help Non-Small Cell Lung Cancer Cells Evade Chemotherapy.

Drug resistance, a major challenge in cancer therapy, is typically attributed to mutations and genetic heterogeneity. Emerging evidence suggests that dynamic cellular interactions and group behavior also contribute to drug resistance. However, the underlying mechanisms remain poorly understood. Here, we present a new mathematical approach with game theoretical underpinnings that we developed to model real-time growth data of non-small cell lung cancer (NSCLC) cells and discern patterns in response to treatment with cisplatin. We show that the cisplatin-sensitive and cisplatin-tolerant NSCLC cells, when co-cultured in the absence or presence of the drug, display dynamic group behavior strategies. Tolerant cells exhibit a 'persister-like' behavior and are attenuated by sensitive cells; they also appear to 'educate' sensitive cells to evade chemotherapy. Further, tolerant cells can switch phenotypes to become sensitive, especially at low cisplatin concentrations. Finally, switching treatment from continuous to an intermittent regimen can attenuate the emergence of tolerant cells, suggesting that intermittent chemotherapy may improve outcomes in lung cancer.

Protein conformational dynamics and phenotypic switching.

Intrinsically disordered proteins (IDPs) are proteins that lack rigid 3D structure but exist as conformational ensembles. Because of their structural plasticity, they can interact with multiple partners. The protein interactions between IDPs and their partners form scale-free protein interaction networks (PINs) that facilitate information flow in the cell. Because of their plasticity, IDPs typically occupy hub positions in cellular PINs. Furthermore, their conformational dynamics and propensity for post-translational modifications contribute to "conformational" noise which is distinct from the well-recognized transcriptional noise. Therefore, upregulation of IDPs in response to a specific input, such as stress, contributes to increased noise and, hence, an increase in stochastic, "promiscuous" interactions. These interactions lead to activation of latent pathways or can induce "rewiring" of the PIN to yield an optimal output underscoring the critical role of IDPs in regulating information flow. We have used PAGE4, a highly intrinsically disordered stress-response protein as a paradigm. Employing a variety of experimental and computational techniques, we have elucidated the role of PAGE4 in phenotypic switching of prostate cancer cells at a systems level. These cumulative studies over the past decade provide a conceptual framework to better understand how IDP conformational dynamics and conformational noise might facilitate cellular decision-making.

Structural and Dynamical Order of a Disordered Protein: Molecular Insights into Conformational Switching of PAGE4 at the Systems Level.

Folded proteins show a high degree of structural order and undergo (fairly constrained) collective motions related to their functions. On the other hand, intrinsically disordered proteins (IDPs), while lacking a well-defined three-dimensional structure, do exhibit some structural and dynamical ordering, but are less constrained in their motions than folded proteins. The larger structural plasticity of IDPs emphasizes the importance of entropically driven motions. Many IDPs undergo function-related disorder-to-order transitions driven by their interaction with specific binding partners. As experimental techniques become more sensitive and become better integrated with computational simulations, we are beginning to see how the modest structural ordering and large amplitude collective motions of IDPs endow them with an ability to mediate multiple interactions with different partners in the cell. To illustrate these points, here, we use Prostate-associated gene 4 (PAGE4), an IDP implicated in prostate cancer (PCa) as an example. We first review our previous efforts using molecular dynamics simulations based on atomistic AWSEM to study the conformational dynamics of PAGE4 and how its motions change in its different physiologically relevant phosphorylated forms. Our simulations quantitatively reproduced experimental observations and revealed how structural and dynamical ordering are encoded in the sequence of PAGE4 and can be modulated by different extents of phosphorylation by the kinases HIPK1 and CLK2. This ordering is reflected in changing populations of certain secondary structural elements as well as in the regularity of its collective motions. These ordered features are directly correlated with the functional interactions of WT-PAGE4, HIPK1-PAGE4 and CLK2-PAGE4 with the AP-1 signaling axis. These interactions give rise to repeated transitions between (high HIPK1-PAGE4, low CLK2-PAGE4) and (low HIPK1-PAGE4, high CLK2-PAGE4) cell phenotypes, which possess differing sensitivities to the standard PCa therapies, such as androgen deprivation therapy (ADT). We argue that, although the structural plasticity of an IDP is important in promoting promiscuous interactions, the modulation of the structural ordering is important for sculpting its interactions so as to rewire with agility biomolecular interaction networks with significant functional consequences.

On the role of four-wave mixing effect in the interactions between nonlinear modes of coupled generalized nonlinear Schrödinger equation.

In this paper, we investigate the effect of four-wave mixing in the interactions among nonlinear waves such as solitons, breathers, and rogue waves of a coupled generalized nonlinear Schrödinger equation. We explore several interesting results including superposition of breather pulses, increment in the number of breather pulses and in amplitudes of breathers, and rogue waves. By strengthening the four-wave mixing parameter, we observe different transformations that occur between different localized structures. For instance, we visualize a transformation from bright soliton to breather form, bright and dark rogue wave to four-petaled rogue wave structures, four-petaled rogue wave to other rogue wave forms, and so on. Another important observation that we report here is that the interaction of a bright soliton with a rogue wave in the presence of the four-wave mixing effect provides interaction between a dark oscillatory soliton and a rogue wave.

PAGE4 and Conformational Switching: Insights from Molecular Dynamics Simulations and Implications for Prostate Cancer.

Prostate-associated gene 4 (PAGE4) is an intrinsically disordered protein implicated in prostate cancer. Thestress-response kinase homeodomain-interacting protein kinase 1 (HIPK1) phosphorylates two residues in PAGE4, serine 9 and threonine 51. Phosphorylation of these two residues facilitates the interaction of PAGE4 with activator protein-1 (AP-1) transcription factor complex to potentiate AP-1's activity. In contrast, hyperphosphorylation of PAGE4 by CDC-like kinase 2 (CLK2) attenuates this interaction with AP-1. Small-angleX-ray scattering and single-molecule fluorescence resonance energy transfer measurements have shown that PAGE4 expands upon hyperphosphorylation and that this expansion is localized to its N-terminal half. To understand the interactions underlying this structural transition, we performed molecular dynamics simulations using Atomistic AWSEM, a multi-scale molecular model that combines atomistic and coarse-grained simulation approaches. Our simulations show that electrostatic interactions drive transient formation of an N-terminal loop, the destabilization of which accounts for the dramatic change in size upon hyperphosphorylation. Phosphorylation also changes the preference of secondary structure formation of the PAGE4 ensemble, which leads to a transition between states that display different degrees of disorder. Finally, we construct a mechanism-based mathematical model that allows us to capture the interactions ofdifferent phosphoforms of PAGE4 with AP-1 and its downstream target, the androgen receptor (AR)-a key therapeutic target in prostate cancer. Our model predicts intracellular oscillatory dynamics of HIPK1-PAGE4, CLK2-PAGE4, and AR activity, indicating phenotypic heterogeneity in an isogenic cell population. Thus, conformational switching of PAGE4 may potentially affect the efficiency of therapeutically targeting AR activity.

A Mathematical Model for Storage and Recall of Images using Targeted Synchronization of Coupled Maps.

We propose a mathematical model for storage and recall of images using coupled maps. We start by theoretically investigating targeted synchronization in coupled map systems wherein only a desired (partial) subset of the maps is made to synchronize. A simple method is introduced to specify coupling coefficients such that targeted synchronization is ensured. The principle of this method is extended to storage/recall of images using coupled Rulkov maps. The process of adjusting coupling coefficients between Rulkov maps (often used to model neurons) for the purpose of storing a desired image mimics the process of adjusting synaptic strengths between neurons to store memories. Our method uses both synchronisation and synaptic weight modification, as the human brain is thought to do. The stored image can be recalled by providing an initial random pattern to the dynamical system. The storage and recall of the standard image of Lena is explicitly demonstrated.

Disorder, oscillatory dynamics and state switching: the role of c-Myc.

In this paper, using the intrinsically disordered oncoprotein Myc as an example, we present a mathematical model to help explain how protein oscillatory dynamics can influence state switching. Earlier studies have demonstrated that, while Myc overexpression can facilitate state switching and transform a normal cell into a cancer phenotype, its downregulation can reverse state-switching. A fundamental aspect of the model is that a Myc threshold determines cell fate in cells expressing p53. We demonstrate that a non-cooperative positive feedback loop coupled with Myc sequestration at multiple binding sites can generate bistable Myc levels. Normal quiescent cells with Myc levels below the threshold can respond to mitogenic signals to activate the cyclin/cdk oscillator for limited cell divisions but the p53/Mdm2 oscillator remains nonfunctional. In response to stress, the p53/Mdm2 oscillator is activated in pulses that are critical to DNA repair. But if stress causes Myc levels to cross the threshold, Myc inactivates the p53/Mdm2 oscillator, abrogates p53 pulses, and pushes the cyclin/cdk oscillator into overdrive sustaining unchecked proliferation seen in cancer. However, if Myc is downregulated, the cyclin/cdk oscillator is inactivated and the p53/Mdm2 oscillator is reset and the cancer phenotype is reversed.

MPTP activates ASK1-p38 MAPK signaling pathway through TNF-dependent Trx1 oxidation in parkinsonism mouse model.

Activation of apoptosis signal-regulating kinase 1 (ASK1)-p38 MAPK death signaling cascade is implicated in the death of dopaminergic neurons in substantia nigra in Parkinson's disease (PD). We investigated upstream activators of ASK1 using an MPTP mouse model of parkinsonism and assessed the temporal cascade of death signaling in ventral midbrain (VMB) and striatum (ST). MPTP selectively activated ASK1 and downstream p38 MAPK in a time-dependent manner in VMB alone. This occurred through selective protein thiol oxidation of the redox-sensitive thiol disulfide oxidoreductase, thioredoxin (Trx1), resulting in release of its inhibitory association with ASK1, while glutathione-S-transferase µ 1 (GSTM1) remained in reduced form in association with ASK1. Levels of tumor necrosis factor (TNF), a known activator of ASK1, increased early after MPTP in VMB. Protein covariation network analysis (PCNA) using protein states as nodes revealed TNF to be an important node regulating the ASK1 signaling cascade. In confirmation, blocking MPTP-mediated TNF signaling through intrathecal administration of TNF-neutralizing antibody prevented Trx1 oxidation and downstream ASK1-p38 MAPK activation. Averting an early increase in TNF, which leads to protein thiol oxidation resulting in activation of ASK1-p38 signaling, may be critical for neuroprotection in PD. Importantly, network analysis can help in understanding the cause/effect relationship within protein networks in complex disease states.

A study on fear memory retrieval and REM sleep in maternal separation and isolation stressed rats.

As rapid brain development occurs during the neonatal period, environmental manipulation during this period may have a significant impact on sleep and memory functions. Moreover, rapid eye movement (REM) sleep plays an important role in integrating new information with the previously stored emotional experience. Hence, the impact of early maternal separation and isolation stress (MS) during the stress hyporesponsive period (SHRP) on fear memory retention and sleep in rats were studied. The neonatal rats were subjected to maternal separation and isolation stress during postnatal days 5-7 (6h daily/3d). Polysomnographic recordings and differential fear conditioning was carried out in two different sets of rats aged 2 months. The neuronal replay during REM sleep was analyzed using different parameters. MS rats showed increased time in REM stage and total sleep period also increased. MS rats showed fear generalization with increased fear memory retention than normal control (NC). The detailed analysis of the local field potentials across different time periods of REM sleep showed increased theta oscillations in the hippocampus, amygdala and cortical circuits. Our findings suggest that stress during SHRP has sensitized the hippocampus-amygdala-cortical loops which could be due to increased release of corticosterone that generally occurs during REM sleep. These rats when subjected to fear conditioning exhibit increased fear memory and increased fear generalization. The development of helplessness, anxiety and sleep changes in human patients, thus, could be related to the reduced thermal, tactile and social stimulation during SHRP on brain plasticity and fear memory functions.

Velocity Fluctuations in Helical Propulsion: How Small Can a Propeller Be.

Helical propulsion is at the heart of locomotion strategies utilized by various natural and artificial swimmers. We used experimental observations and a numerical model to study the various fluctuation mechanisms that determine the performance of an externally driven helical propeller as the size of the helix is reduced. From causality analysis, an overwhelming effect of orientational noise at low length scales is observed, which strongly affects the average velocity and direction of motion of a propeller. For length scales smaller than a few micrometers in aqueous media, the operational frequency for the propulsion system would have to increase as the inverse cube of the size, which can be the limiting factor for a helical propeller to achieve locomotion in the desired direction.

Effect of measurement noise on Granger causality.

Most of the signals recorded in experiments are inevitably contaminated by measurement noise. Hence, it is important to understand the effect of such noise on estimating causal relations between such signals. A primary tool for estimating causality is Granger causality. Granger causality can be computed by modeling the signal using a bivariate autoregressive (AR) process. In this paper, we greatly extend the previous analysis of the effect of noise by considering a bivariate AR process of general order p. From this analysis, we analytically obtain the dependence of Granger causality on various noise-dependent system parameters. In particular, we show that measurement noise can lead to spurious Granger causality and can suppress true Granger causality. These results are verified numerically. Finally, we show how true causality can be recovered numerically using the Kalman expectation maximization algorithm.

Frustration induced oscillator death on networks.

An array of identical maps with Ising symmetry, with both positive and negative couplings, is studied. We divide the maps into two groups, with positive intra-group couplings and negative inter-group couplings. This leads to antisynchronization between the two groups which have the same stability properties as the synchronized state. Introducing a certain degree of randomness in signs of these couplings destabilizes the anti-synchronized state. Further increasing the randomness in signs of these couplings leads to oscillator death. This is essentially a frustration induced phenomenon. We explain the observed results using the theory of random matrices with nonzero mean. We briefly discuss applications to coupled differential equations.

Multivariate Granger causality: an estimation framework based on factorization of the spectral density matrix.

Granger causality is increasingly being applied to multi-electrode neurophysiological and functional imaging data to characterize directional interactions between neurons and brain regions. For a multivariate dataset, one might be interested in different subsets of the recorded neurons or brain regions. According to the current estimation framework, for each subset, one conducts a separate autoregressive model fitting process, introducing the potential for unwanted variability and uncertainty. In this paper, we propose a multivariate framework for estimating Granger causality. It is based on spectral density matrix factorization and offers the advantage that the estimation of such a matrix needs to be done only once for the entire multivariate dataset. For any subset of recorded data, Granger causality can be calculated through factorizing the appropriate submatrix of the overall spectral density matrix.

Is Granger causality a viable technique for analyzing fMRI data?

Multivariate neural data provide the basis for assessing interactions in brain networks. Among myriad connectivity measures, Granger causality (GC) has proven to be statistically intuitive, easy to implement, and generate meaningful results. Although its application to functional MRI (fMRI) data is increasing, several factors have been identified that appear to hinder its neural interpretability: (a) latency differences in hemodynamic response function (HRF) across different brain regions, (b) low-sampling rates, and (c) noise. Recognizing that in basic and clinical neuroscience, it is often the change of a dependent variable (e.g., GC) between experimental conditions and between normal and pathology that is of interest, we address the question of whether there exist systematic relationships between GC at the fMRI level and that at the neural level. Simulated neural signals were convolved with a canonical HRF, down-sampled, and noise-added to generate simulated fMRI data. As the coupling parameters in the model were varied, fMRI GC and neural GC were calculated, and their relationship examined. Three main results were found: (1) GC following HRF convolution is a monotonically increasing function of neural GC; (2) this monotonicity can be reliably detected as a positive correlation when realistic fMRI temporal resolution and noise level were used; and (3) although the detectability of monotonicity declined due to the presence of HRF latency differences, substantial recovery of detectability occurred after correcting for latency differences. These results suggest that Granger causality is a viable technique for analyzing fMRI data when the questions are appropriately formulated.

Intrinsically disordered proteins and conformational noise: implications in cancer.

Intrinsically disordered proteins, IDPs, are proteins that lack a rigid 3D structure under physiological conditions, at least in vitro. Despite the lack of structure, IDPs play important roles in biological processes and transition from disorder to order upon binding to their targets. With multiple conformational states and rapid conformational dynamics, they engage in myriad and often "promiscuous" interactions. These stochastic interactions between IDPs and their partners, defined here as conformational noise, is an inherent characteristic of IDP interactions. The collective effect of conformational noise is an ensemble of protein network configurations, from which the most suitable can be explored in response to perturbations, conferring protein networks with remarkable flexibility and resilience. Moreover, the ubiquitous presence of IDPs as transcriptional factors and, more generally, as hubs in protein networks, is indicative of their role in propagation of transcriptional (genetic) noise. As effectors of transcriptional and conformational noise, IDPs rewire protein networks and unmask latent interactions in response to perturbations. Thus, noise-driven activation of latent pathways could underlie state-switching events such as cellular transformation in cancer. To test this hypothesis, we created a model of a protein network with the topological characteristics of a cancer protein network and tested its response to a perturbation in presence of IDP hubs and conformational noise. Because numerous IDPs are found to be epigenetic modifiers and chromatin remodelers, we hypothesize that they could further channel noise into stable, heritable genotypic changes.