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BackgroundThe aging process affects all systems of the human body, and the observed increase in inflammatory components affecting the immune system in old age can lead to the development of age-associated diseases and systemic inflammation.ResultsWe propose a small clock model SImAge based on a limited number of immunological biomarkers. To regress the chronological age from cytokine data, we first use a baseline Elastic Net model, gradient-boosted decision trees models, and several deep neural network architectures. For the full dataset of 46 immunological parameters, LightGBM, DANet, and TabNet models showed the best results. Dimensionality reduction of these 3 models with SHAP values revealed the 10 most age-associated immunological parameters, taken to construct the SImAge small immunological clock. The best result of the SImAge model shown by the DANet deep neural network model has mean absolute error of 6.28 years. Explainable artificial intelligence methods allow for explaining the model solution for each individual participant.ConclusionsWe developed an approach to construct a model of immunological age based on just 10 immunological parameters, coined SImAge, for which the DANet deep neural network model had proved to be the best choice. The model shows the highest result among all published studies on immunological profiles. Gradient-boosted decision trees and neural networks showed comparable results, and can both be recommended in the further analysis of immunological profiles.

Studies of human mitochondrial (mt) genome variation may be undertaken to investigate the human history and natural selection. By analyzing nucleotide co-occurrence over the entire human mt-genome, we have developed a network model to describe human evolutionary patterns. Using 7,424 unique polymorphic sites, we found evidence that mutation biases at second codon position and RNA genes were critical to producing continental-level heterogeneity among human subpopulations. Further, the analysis highlighted richer co-mutation regions of the mt-genome and thus provided evidence of epistasis. Specifically, a large portion of COX genes co-mutate in Asian and American populations whereas, in African, European and Oceanic populations, there was greater epistasis in hypervariable regions. Very interestingly, this study demonstrated hierarchical modularity as a crucial agent for a nucleotide co-occurrence network make-up. More profoundly, our ancestry-based nucleotide module analyses showed that nucleotide co-changes cluster preferentially in known mitochondrial haplogroups. It was also conceived that contemporary human mt-genome nucleotides most closely resembled the ancestral state and very few of them were ancestral-variants. Overall, these results demonstrated that subpopulation based factors such as intra-species evolution do exert selection on mitochondrial genes by favoring specific epistatic genetic variants.

DNA methylation variability arises due to concurrent genetic and environmental influences. Each of them is a mixture of regular and noisy sources, whose relative contribution has not been satisfactorily understood yet. We conduct a systematic assessment of the age-dependent methylation by the signal-to-noise ratio and identify a wealth of “deterministic” CpG probes (about 90%), whose methylation variability likely originates due to genetic and general environmental factors. The remaining 10% of “stochastic” CpG probes are arguably governed by the biological noise or incidental environmental factors. Investigating the mathematical functional relationship between methylation levels and variability, we find that in about 90% of the age-associated differentially methylated positions, the variability changes as the square of the methylation level, whereas in the most of the remaining cases the dependence is linear. Furthermore, we demonstrate that the methylation level itself in more than 15% cases varies nonlinearly with age (according to the power law), in contrast to the previously assumed linear changes. Our findings present ample evidence of the ubiquity of strong DNA methylation regulation, resulting in the individual age-dependent and nonlinear methylation trajectories, whose divergence explains the cross-sectional variability. It may also serve a basis for constructing novel nonlinear epigenetic clocks.

BackgroundDNA methylation has a significant effect on gene expression and can be associated with various diseases. Meta-analysis of available DNA methylation datasets requires development of a specific pipeline for joint data processing.ResultsWe propose a comprehensive approach of combined DNA methylation datasets to classify controls and patients. The solution includes data harmonization, construction of machine learning classification models, dimensionality reduction of models, imputation of missing values, and explanation of model predictions by explainable artificial intelligence (XAI) algorithms. We show that harmonization can improve classification accuracy by up to 20% when preprocessing methods of the training and test datasets are different. The best accuracy results were obtained with tree ensembles, reaching above 95% for Parkinson’s disease. Dimensionality reduction can substantially decrease the number of features, without detriment to the classification accuracy. The best imputation methods achieve almost the same classification accuracy for data with missing values as for the original data. Explainable artificial intelligence approaches have allowed us to explain model predictions from both populational and individual perspectives.ConclusionsWe propose a methodologically valid and comprehensive approach to the classification of healthy individuals and patients with various diseases based on whole blood DNA methylation data using Parkinson’s disease and schizophrenia as examples. The proposed algorithm works better for the former pathology, characterized by a complex set of symptoms. It allows to solve data harmonization problems for meta-analysis of many different datasets, impute missing values, and build classification models of small dimensionality.

In humans, females live longer than males but experience a worse longevity, as genome-wide autosomal DNA methylation differences between males and females have been reported. So far, few studies have investigated if DNA methylation is differently affected by aging in males and females. We performed a meta-analysis of 4 large whole blood datasets, comparing 4 aspects of epigenetic age-dependent remodeling between the two sexes: differential methylation, variability, epimutations and entropy. We reported that a large fraction (43%) of sex-associated probes undergoes age-associated DNA methylation changes, and that a limited number of probes shows age-by-sex interaction. We experimentally validated 2 regions mapping inFIGNandPRR4genes, and showed sex-specific deviations of their methylation patterns in models of decelerated (centenarians) and accelerated (Down syndrome) aging. While we did not find sex differences in the age-associated increase in epimutations and in entropy, we showed that the number of probes showing age-related increase in methylation variability is 15 times higher in males compared to females. Our results can offer new epigenetic tools to study the interaction between aging and sex and can pave the way to the identification of molecular triggers of sex differences in longevity and age-related diseases prevalence.

In this work, we investigate the impact of mixed coupling on synchronization in a multiplex oscillatory network. The network mimics the neural–glial systems by incorporating interacting slow (“glial”) and fast (“neural”) oscillatory layers. Connections between the “glial” elements form a regular periodic structure, in which each element is connected to the eight other neighbor elements, whereas connections among “neural” elements are represented by Watts–Strogatz networks (from regular and small-world to random Erdos–Renyi graph) with a matching mean node degree. We find that the random rewiring toward small-world topology readily yields the dynamics close to that exhibited for a completely random graph, in particular, leading to coarse-graining of dynamics, suppressing multi-stability of synchronization regimes, and the onset of Kuramoto-type synchrony in both layers. The duration of transient dynamics in the system measured by relaxation times is minimized with the increase of random connections in the neural layer, remaining substantial only close to synchronization–desynchronization transitions. “Inhibitory” interactions in the “neural” subnetwork layer undermine synchronization; however, the strong coupling with the “glial” layer overcomes this effect.

Chronic kidney disease (CKD) is defined by a reduced estimated glomerular filtration rate (eGFR). This failure can be related to a phenotype of accelerated aging. In this work, we considered 76 patients with end-stage renal disease (ESRD) and 83 healthy controls. We concomitantly evaluated for the first time two measures that can be informative of the rate of aging, i.e., whole blood DNA methylation using the Illumina Infinium EPIC array and plasma levels of a selection of inflammatory/immunological proteins using multiplex immunoassays. First of all, we demonstrated accelerated aging in terms of the most common epigenetic age estimators in CKD patients. Moreover, we developed a new clock/predictor of age based on the inflammatory/immunological profile (ipAGE) and identified the inflammatory/immunological biomarkers differentially expressed between cases and controls. IpAGE appeared to be more sensitive than epigenetic clocks in quantifying the accelerated aging phenotype of ESRD patients. Interestingly, we did not find any correlation between the age acceleration evaluated according to the epigenetic clocks and ipAGE in either the ESRD group or the control group. On the whole, our data show a consistent accelerated aging phenotype in ESRD patients, which is better appreciated by quantifying the underlying inflammatory processes (inflammaging) by ipAGE than by using epigenetic clocks. SUPPLEMENTARY INFORMATION: The online version contains supplementary material available at 10.1007/s11357-022-00540-4.

Nonlinear dynamics makes use of attactors to describe asymptotic behavior of complex systems. However, in real life can be unattainable, as achieving them might require time much exceeding all relevant timescales of a system. Therefore, there is an increasing interest in quasi-stationary states, where the system rapidly converges to and remains for a long time, before getting into an absorbing (asymptotic) state. Exemplifying in the famous Dawkins‘ Battle of the Sexes game, we demonstrate that quasi-stationary distributions can produce not simply different, but a much more complex behavior, then the asymptotic ones, that is transient self-sustained oscillations of player numbers and the corresponding non-unimodal probability distribution. We find that parameters of the quasi-stationary limit cycle depend on the population size. The work was supported by the Russian Foundation forBasic Research grant 18-32-20221.

Recently, it has been demonstrated that asymptotic states of open quantum system can undergo qualitative changes resembling pitchfork, saddle-node, and period doubling classical bifurcations. Here, making use of the periodically modulated open quantum dimer model, we report and investigate a quantum Neimark-Sacker bifurcation. Its classical counterpart is the birth of a torus (an invariant curve in the Poincaré section) due to instability of a limit cycle (fixed point of the Poincaré map). The quantum system exhibits a transition from unimodal to bagel shaped stroboscopic distributions, as for Husimi representation, as for observables. The spectral properties of Floquet map experience changes reminiscent of the classical case, a pair of complex conjugated eigenvalues approaching a unit circle. Quantum Monte-Carlo wave function unraveling of the Lindblad master equation yields dynamics of single trajectories on “quantumtorus” and allows for quantifying it by rotation number. The bifurcation is sensitive to the number of quantum particles that can also be regarded as a control parameter.

Physiological and haplogroup studies performed to understand high-altitude adaptation in humans are limited to individual genes and polymorphic sites. Due to stochastic evolutionary forces, the frequency of a polymorphism is affected by changes in the frequency of a near-by polymorphism on the same DNA sample making them connected in terms of evolution. Here, first, we provide a method to model these mitochondrial polymorphisms as “co-mutation networks” for three high-altitude populations, Tibetan, Ethiopian and Andean. Then, by transforming these co-mutation networks into weighted and undirected gene–gene interaction (GGI) networks, we were able to identify functionally enriched genetic interactions of CYB and CO3 genes in Tibetan and Andean populations, while NADH dehydrogenase genes in the Ethiopian population playing a significant role in high altitude adaptation. These co-mutation based genetic networks provide insights into the role of different set of genes in high-altitude adaptation in human sub-populations.

The use of vitamin D3 along with traditional therapy opens up new prospects for increasing the adaptive capacity of nerve cells to the effects of a wide range of stress factors, including hypoxia-ischemic processes. However, questions about prophylactic and therapeutic doses of vitamin D3 remain controversial. The purpose of our study was to analyze the effects of vitamin D3 at different concentrations on morpho-functional characteristics of neuron–glial networks in hypoxia modeling in vitro. We showed that a single administration of vitamin D3 at a high concentration (1 µM) in a normal state has no significant effect on the cell viability of primary neuronal cultures, however, it has a pronounced modulatory effect on the functional calcium activity of neuron–glial networks and causes destruction of the network response. Under hypoxia, the use of vitamin D3 (1 µM) leads to total cell death of primary neuronal cultures and complete negation of functional neural network activity. In contrast, application of lower concentrations of vitamin D3 (0.01 µM and 0.1 µM) caused a pronounced dose-dependent neuroprotective effect during the studied post-hypoxic period. While the use of vitamin D3 at a concentration of 0.1 µM maintained cell viability, preventive administration of 0.01 µM not only partially preserved the morphological integrity of primary neuronal cells but also maintained the functional structure and activity of neuron–glial networks in cultures. Possible molecular mechanisms of neuroprotective action of vitamin D3 can be associated with the increased expression level of transcription factor HIF-1α and maintaining the relationship between the levels of BDNF and TrkB expression in cells of primary neuronal cultures.

Alzheimer's disease (AD) is characterized by specific alterations of brain DNA methylation (DNAm) patterns. Age and sex, two major risk factors for AD, are also known to largely affect the epigenetic profiles in brain, but their contribution to AD-associated DNAm changes has been poorly investigated. In this study we considered publicly available DNAm datasets of four brain regions (temporal, frontal, entorhinal cortex, and cerebellum) from healthy adult subjects and AD patients, and performed a meta-analysis to identify sex-, age-, and AD-associated epigenetic profiles. In one of these datasets it was also possible to distinguish 5-methylcytosine (5mC) and 5-hydroxymethylcytosine (5hmC) profiles. We showed that DNAm differences between males and females tend to be shared between the four brain regions, while aging differently affects cortical regions compared to cerebellum. We found that the proportion of sex-dependent probes whose methylation is modified also during aging is higher than expected, but that differences between males and females tend to be maintained, with only a few probes showing age-by-sex interaction. We did not find significant overlaps between AD- and sex-associated probes, nor disease-by-sex interaction effects. On the contrary, we found that AD-related epigenetic modifications are significantly enriched in probes whose DNAm varies with age and that there is a high concordance between the direction of changes (hyper or hypo-methylation) in aging and AD, supporting accelerated epigenetic aging in the disease. In summary, our results suggest that age-associated DNAm patterns concur to the epigenetic deregulation observed in AD, providing new insights on how advanced age enables neurodegeneration.

At a recent symposium on aging biology, a debate was held as to whether or not we know what biological aging is. Most of the participants were struck not only by the lack of consensus on this core question, but also on many basic tenets of the field. Accordingly, we undertook a systematic survey of our 71 participants on key questions that were raised during the debate and symposium, eliciting 37 responses. The results confirmed the impression from the symposium: there is marked disagreement on the most fundamental questions in the field, and little consensus on anything other than the heterogeneous nature of aging processes. Areas of major disagreement included what participants viewed as the essence of aging, when it begins, whether aging is programmed or not, whether we currently have a good understanding of aging mechanisms, whether aging is or will be quantifiable, whether aging will be treatable, and whether many non-aging species exist. These disagreements lay bare the urgent need for a more unified and cross-disciplinary paradigm in the biology of aging that will clarify both areas of agreement and disagreement, allowing research to proceed more efficiently. We suggest directions to encourage the emergence of such a paradigm.

Brain aging is a complex process involving many functions of our body and described by the interplay of a sleep pattern and changes in the metabolic waste concentration regulated by the microglial function and the glymphatic system. We review the existing modelling approaches to this topic and derive a novel mathematical model to describe the crosstalk between these components within the conceptual framework of inflammaging. Analysis of the model gives insight into the dynamics of garbage concentration and linked microglial senescence process resulting from a normal or disrupted sleep pattern, hence, explaining an underlying mechanism behind healthy or unhealthy brain aging. The model incorporates accumulation and elimination of garbage, induction of glial activation by garbage, and glial senescence by over-activation, as well as the production of pro-inflammatory molecules by their senescence-associated secretory phenotype (SASP). Assuming that insufficient sleep leads to the increase of garbage concentration and promotes senescence, the model predicts that if the accumulation of senescent glia overcomes an inflammaging threshold, further progression of senescence becomes unstoppable even if a normal sleep pattern is restored. Inverting this process by "rejuvenating the brain" is only possible via a reset of concentration of senescent glia below this threshold. Our model approach enables analysis of space-time dynamics of senescence, and in this way, we show that heterogeneous patterns of inflammation will accelerate the propagation of senescence profile through a network, confirming a negative effect of heterogeneity.

Despite the fact that the cause of Down Syndrome (DS) is well established, the underlying molecular mechanisms that contribute to the syndrome and the phenotype of accelerated aging remain largely unknown. DNA methylation profiles are largely altered in DS, but it remains unclear how different methylation regions and probes are structured into a network of interactions. We develop and generalize the Parenclitic Networks approach that enables finding correlations between distant CpG probes (which are not pronounced as stand-alone biomarkers) and quantifies hidden network changes in DNA methylation. DS and a familybased cohort (including healthy siblings and mothers of persons with DS) are used as a case study. Following this approach, we constructed parenclitic networks and obtained different signatures that indicate (i) differences between individuals with DS and healthy individuals; (ii) differences between young and old healthy individuals; (iii) differences between DS individuals and their age-matched siblings, and (iv) difference between DS and the adult population (their mothers). The Gene Ontology analysis showed that the CpG network approach is more powerful than the single CpG approach in identifying biological processes related to DS phenotype. This includes the processes occurring in the central nervous system, skeletal muscles, disorders in carbohydrate metabolism, cardiopathology, and oncogenes. Our open-source software implementation is accessible to all researchers. The software includes a complete workflow, which can be used to construct Parenclitic Networks with any machine learning algorithm as a kernel to build edges. We anticipate a broad applicability of the approach to other diseases.

Evolutionary game theory is a framework to formalize the evolution of collectives (“populations”) of competing agents that are playing a game and, after every round, update their strategies to maximize individual payoffs. There are two complementary approaches to modeling evolution of player populations. The first addresses essentially finite populations by implementing the apparatus of Markov chains. The second assumes that the populations are infinite and operates with a system of mean-field deterministic differential equations. By using a model of two antagonistic populations, which are playing a game with stationary or periodically varying payoffs, we demonstrate that it exhibits metastable dynamics that is reducible neither to an immediate transition to a fixation (extinction of all but one strategy in a finite-size population) nor to the mean-field picture. In the case of stationary payoffs, this dynamics can be captured with a system of stochastic differential equations and interpreted as a stochastic Hopf bifurcation. In the case of varying payoffs, the metastable dynamics is much more complex than the dynamics of the means. The authors acknowledge support of the Russian Foundation for Basic Research (Grant Nos. 18-32-20221 and 20-32-90202) and the Lobachevsky University Center of Mathematics (M.I. and O.V.). J.T. acknowledges support by the Institute for Basic Science in Korea (No. IBS-R024-Y2). Numerical simulations were performed on the Lobachevsky supercomputer (Lobachevsky University, Nizhny Novgorod).

We present a numerical approach to calculate non-equilibrium eigenstates of a periodically time-modulated quantum system. The approach is based on the use of a chain of single-step time-independent propagating operators. Each operator is time-specific and constructed by combining the Magnus expansion of the time-dependent system Hamiltonian with the Chebyshev expansion of an operator exponent. A construction of a unitary matrix of the Floquet operator, which evolves a system state over the full modulation period, is performed by propagating the identity matrix over the period. The independence of the evolutions of basis vectors makes the propagation stage suitable for implementation on a parallel cluster. Once the propagation stage is completed, a routine diagonalization of the Floquet matrix is performed. Finally, an additional propagation round, now with the eigenvectors as the initial states, allows to resolve the time-dependence of the Floquet states and calculate their characteristics. We demonstrate the accuracy and scalability of the algorithm by applying it to calculate the Floquet states of two quantum models, namely (i) a synthesized random-matrix Hamiltonian and (ii) a many-body Bose-Hubbard dimer, both of the size up to $10^4$ states.

With their constantly increasing peak performance and memory capacity, modern supercomputers offer new perspectives on numerical studies of open many-body quantum systems. These systems are often modeled by using Markovian quantum master equations describing the evolution of the system density operators. In this paper, we address master equations of the Lindblad form, which are a popular theoretical tools in quantum optics, cavity quantum electrodynamics, and optomechanics. By using the generalized Gell&ndash, Mann matrices as a basis, any Lindblad equation can be transformed into a system of ordinary differential equations with real coefficients. Recently, we presented an implementation of the transformation with the computational complexity, scaling as O(N5logN) for dense Lindbaldians and O(N3logN) for sparse ones. However, infeasible memory costs remains a serious obstacle on the way to large models. Here, we present a parallel cluster-based implementation of the algorithm and demonstrate that it allows us to integrate a sparse Lindbladian model of the dimension N=2000 and a dense random Lindbladian model of the dimension N=200 by using 25 nodes with 64 GB RAM per node.

We report Lobachevsky University Database (LUDB) of ECG signals, an open tool for validating ECG delineation algorithms, that is superior to the existing publicly available data bases in several aspects. LUDB contains 200 recordings of 10-second 12-lead electrocardiograms (ECG) from different subjects, representative of a variety of signal morphologies. The boundaries and peaks of QRS complexes and P and T waves are manually annotated by cardiologists for all recordings and independently for each lead, and all records received an expert classification by abnormalities. We present a case study for the recently proposed wavelet-based algorithm and the broadly used ecg-kit tool, and demonstrate the advantage of multi-lead ECG data analysis. LUDB contributes to the diversity of public databases employed in developing and validating novel ECG analysis algorithms, including the most advanced based on deep learning neural networks., Comment: 11 pages, 10 figures, 6 tables

Developing computational analysis of time-lapse imaging of calcium events in astrocytes is a challenging task in neuroscience. Here we report the implementation of an algorithm that solves this task. After noise reduction with the block-matching and 3D filtering (BM3D) algorithm, calcium activity is identified as fluorescence elevation above the baseline level. Individual events are detected by sliding window approach applied to the variation of pixel intensity relative to the baseline level. The maximal projection and duration of astrocytic calcium events are then assessed. The novelty of the proposed method is an adaptive construction of the baseline level. The statistical results generated by our program are consistent with the previous algorithm reported and used by us for the reference. The software is publicly available.

Inspired by the recent interest in collective dynamics of biological neural networks immersed in the glial cell medium, we investigate the frequency and phase order, i.e., Kuramoto type of synchronization in a multiplex two-layer network of phase oscillators of different time scales and topologies. One of them has a long-range connectivity, exemplified by the Erdős-Renyi random network, and supports both kinds of synchrony. The other is a locally coupled two-dimensional lattice that can reach frequency synchronization but lacks phase order. Drastically different layer frequencies disentangle intra- and interlayer synchronization. We find that an indirect but sufficiently strong coupling through the regular layer can induce both phase order in the originally nonsynchronized random layer and global order, even when an isolated regular layer does not manifest it in principle. At the same time, the route to global synchronization is complex: an initial onset of (partial) synchrony in the regular layer, when its intra- and interlayer coupling is increased, provokes the loss of synchrony even in the originally synchronized random layer. Ultimately, a developed asynchronous dynamics in both layers is abruptly taken over by the global synchrony of both kinds.

The largest eigenvalue of a network provides understanding to various dynamical as well as stability properties of the underlying system. We investigate the interplay of inhibition and multiplexing on the largest eigenvalue statistics of networks. Using numerical experiments, we demonstrate that the presence of the inhibitory coupling may lead to a behaviour of the largest eigenvalue statistics of multiplex networks very different from that of isolated networks depending upon the network architecture of the individual layer. We demonstrate that there is a transition from the Weibull to the Gumbel or to the Frechet distribution as networks are multiplexed. Furthermore, for denser networks, there is a convergence to the Gumbel distribution as network size increases indicating higher stability of larger systems.

We investigate the asymptotic state of a periodically driven many-body quantum system which is weakly coupled to an environment. The combined action of the modulations and the environment steers the system towards a state being characterized by a time-periodic density operator. To resolve this asymptotic non-equilibrium state at stroboscopic instants of time, we introduce the dissipative Floquet map, evaluate the stroboscopic density operator as its eigen-element and elucidate how particle interactions affect properties of the density operator. We illustrate the idea with a periodically modulated Bose-Hubbard dimer and discuss the relations between the interaction-induced bifurcations in a mean-field dynamics and changes in the characteristics of the genuine quantum many-body state. We argue that Floquet maps provide insight into the system relaxation towards its asymptotic state and may help to understand whether it is possible (or not) to construct a stroboscopic time-independent generator mimicking the action of the original time-dependent one., 12 pages, 4 figures

We propose and study models of two distributed synthetic gene circuits, toggle-switch and oscillator, each split between two cell strains and coupled via quorum-sensing signals. The distributed toggle switch relies on mutual repression of the two strains, and oscillator is comprised of two strains, one of which acts as an activator for another that in turn acts as a repressor. Distributed toggle switch can exhibit mobile fronts, switching the system from the weaker to the stronger spatially homogeneous state. The circuit can also act as a biosensor, with the switching front dynamics determined by the properties of an external signal. Distributed oscillator system displays another biosensor functionality: oscillations emerge once a small amount of one cell strain appears amid the other, present in abundance. Distribution of synthetic gene circuits among multiple strains allows one to reduce crosstalk among different parts of the overall system and also decrease the energetic burden of the synthetic circuit per cell, which may allow for enhanced functionality and viability of engineered cells.

In an isolated single-particle quantum system a spatial disorder can induce Anderson localization. Being a result of interference, this phenomenon is expected to be fragile in the face of dissipation. Here we show that dissipation can drive a disordered system into a steady state with tunable localization properties. This can be achieved with a set of identical dissipative operators, each one acting non-trivially only on a pair of neighboring sites. Operators are parametrized by a uniform phase, which controls selection of Anderson modes contributing to the state. On the microscopic level, quantum trajectories of a system in a localized steady regime exhibit intermittent dynamics consisting of long-time sticking events near selected modes interrupted by jumps between them., Comment: 5 pages, 5 figures

We present computational data on the thermal conductivity of nonlinear waves in disordered chains. Disorder induces Anderson localization for linear waves and results in a vanishing conductivity. Cubic nonlinearity restores normal conductivity, but with a strongly temperature-dependent conductivity $\kappa(T)$. We find indications for an asymptotic low-temperature $\kappa \sim T^4$ and intermediate temperature $\kappa \sim T^2$ laws. These findings are in accord with theoretical studies of wave packet spreading, where a regime of strong chaos is found to be intermediate, followed by an asymptotic regime of weak chaos (EPL 91 (2010) 30001)., Comment: 8 pages, 3 figures