Transfer entropy in magnetoencephalographic data: quantifying information flow in cortical and cerebellar networks

The analysis of cortical and subcortical networks requires the identification of their nodes, and of the topology and dynamics of their interactions. Exploratory tools for the identification of nodes are available, e.g. magnetoencephalography (MEG) in combination with beamformer source analysis. Competing network topologies and interaction models can be investigated using dynamic causal modelling. However, we lack a method for the exploratory investigation of network topologies to choose from the very large number of possible network graphs. Ideally, this method should not require a pre-specified model of the interaction. Transfer entropy--an information theoretic implementation of Wiener-type causality--is a method for the investigation of causal interactions (or information flow) that is independent of a pre-specified interaction model. We analysed MEG data from an auditory short-term memory experiment to assess whether the reconfiguration of networks implied in this task can be detected using transfer entropy. Transfer entropy analysis of MEG source-level signals detected changes in the network between the different task types. These changes prominently involved the left temporal pole and cerebellum--structures that have previously been implied in auditory short-term or working memory. Thus, the analysis of information flow with transfer entropy at the source-level may be used to derive hypotheses for further model-based testing.     

Estimating Temporal Causal Interaction between Spike Trains with Permutation and Transfer Entropy

Estimating the causal interaction between neurons is very important for better understanding the functional connectivity in neuronal networks. We propose a method called normalized permutation transfer entropy (NPTE) to evaluate the temporal causal interaction between spike trains, which quantifies the fraction of ordinal information in a neuron that has presented in another one. The performance of this method is evaluated with the spike trains generated by an Izhikevich’s neuronal model. Results show that the NPTE method can effectively estimate the causal interaction between two neurons without influence of data length. Considering both the precision of time delay estimated and the robustness of information flow estimated against neuronal firing rate, the NPTE method is superior to other information theoretic method including normalized transfer entropy, symbolic transfer entropy and permutation conditional mutual information. To test the performance of NPTE on analyzing simulated biophysically realistic synapses, an Izhikevich’s cortical network that based on the neuronal model is employed. It is found that the NPTE method is able to characterize mutual interactions and identify spurious causality in a network of three neurons exactly. We conclude that the proposed method can obtain more reliable comparison of interactions between different pairs of neurons and is a promising tool to uncover more details on the neural coding.     

Thermodynamic analysis of biomolecular interactions.

Direct measurement of the thermodynamics of biomolecular interactions is now relatively easy. Interpretation of these thermodynamics in simple molecular terms is not. Recent work shows how the multiplicity of weak noncovalent interactions, and the inevitable enthalpy/entropy compensation that these interactions engender, lead to difficulties in teasing out the different components.     

Measuring Information-Transfer Delays

In complex networks such as gene networks, traffic systems or brain circuits it is important to understand how long it takes for the different parts of the network to effectively influence one another. In the brain, for example, axonal delays between brain areas can amount to several tens of milliseconds, adding an intrinsic component to any timing-based processing of information. Inferring neural interaction delays is thus needed to interpret the information transfer revealed by any analysis of directed interactions across brain structures. However, a robust estimation of interaction delays from neural activity faces several challenges if modeling assumptions on interaction mechanisms are wrong or cannot be made. Here, we propose a robust estimator for neuronal interaction delays rooted in an information-theoretic framework, which allows a model-free exploration of interactions. In particular, we extend transfer entropy to account for delayed source-target interactions, while crucially retaining the conditioning on the embedded target state at the immediately previous time step. We prove that this particular extension is indeed guaranteed to identify interaction delays between two coupled systems and is the only relevant option in keeping with Wiener’s principle of causality. We demonstrate the performance of our approach in detecting interaction delays on finite data by numerical simulations of stochastic and deterministic processes, as well as on local field potential recordings. We also show the ability of the extended transfer entropy to detect the presence of multiple delays, as well as feedback loops. While evaluated on neuroscience data, we expect the estimator to be useful in other fields dealing with network dynamics.     

Entropy of dynamical social networks

Human dynamical social networks encode information and are highly adaptive. To characterize the information encoded in the fast dynamics of social interactions, here we introduce the entropy of dynamical social networks. By analysing a large dataset of phone-call interactions we show evidence that the dynamical social network has an entropy that depends on the time of the day in a typical week-day. Moreover we show evidence for adaptability of human social behavior showing data on duration of phone-call interactions that significantly deviates from the statistics of duration of face-to-face interactions. This adaptability of behavior corresponds to a different information content of the dynamics of social human interactions. We quantify this information by the use of the entropy of dynamical networks on realistic models of social interactions.     

How do helix–helix interactions help determine the folds of membrane proteins? Perspectives from the study of homo‐oligomeric helical bundles

The final, structure-determining step in the folding of membrane proteins involves the coalescence of preformed transmembrane helices to form the native tertiary structure. Here, we review recent studies on small peptide and protein systems that are providing quantitative data on the interactions that drive this process. Gel electrophoresis, analytical ultracentrifugation, and fluorescence resonance energy transfer (FRET) are useful methods for examining the assembly of homo-oligomeric transmembrane helical proteins. These methods have been used to study the assembly of the M2 proton channel from influenza A virus, glycophorin, phospholamban, and several designed membrane proteins-all of which have a single transmembrane helix that is sufficient for association into a transmembrane helical bundle. These systems are being studied to determine the relative thermodynamic contributions of van der Waals interactions, conformational entropy, and polar interactions in the stabilization of membrane proteins. Although the database of thermodynamic information is not yet large, a few generalities are beginning to emerge concerning the energetic differences between membrane and water-soluble proteins: the packing of apolar side chains in the interior of helical membrane proteins plays a smaller, but nevertheless significant, role in stabilizing their structure. Polar, hydrogen-bonded interactions occur less frequently, but, nevertheless, they often provide a strong driving force for folding helix-helix pairs in membrane proteins. These studies are laying the groundwork for the design of sequence motifs that dictate the association of membrane helices.     

Competing interactions for antimicrobial selectivity based on charge complementarity

Antimicrobial peptides (AMPs) are an evolutionary conserved component of the innate immune system and possible templates for the development of new antibiotics. An important property of antimicrobial peptides is their ability to discriminate bacterial from eucaryotic cells which is attributed to the difference in lipid composition of the outer leaflet of the plasma membrane between the two types of cells. Whereas eucaryotic cells usually expose zwitterionic lipids, procaryotic cells expose also anionic lipids which bind the cationic antimicrobial peptides electrostatically. An example is the antimicrobial peptide NK-2 which is highly cationic and favors binding to anionic membranes. In the present study, the difference in binding affinity of NK-2 for palmitoyl-oleoyl-phosphatidyl-glycerol (POPG) and palmitoyl-oleoyl-phosphatidyl-choline (POPC) is studied using molecular dynamics simulations in conjunction with a coarse grained model and thermodynamic integration, by computing the change in free energy and its components upon the transfer of NK-2 from POPC to POPG. The transfer is indeed found to be highly favorable. Interestingly, the favorable contribution from the electrostatic interaction between the peptide and the anionic lipids is overcompensated by an unfavorable contribution from the change in lipid-cation interactions due to the release of counterions from the lipids. The increase in entropy due to the release of the cations is compensated by other entropic components. The largest favorable contribution arises from the solvation of the counterions. Overall the interaction between NK-2 and POPG is not determined by a single driving force but a subtle balance of competing interactions.     

Cell viability and probe-cell membrane interactions of XR1 glial cells imaged by atomic force microscopy.

As atomic force microscopy (AFM) imaging of live specimens becomes more commonplace, at least two important questions arise: 1) do live specimens remain viable during and after AFM, and 2) is there transfer of membrane components from the cell to the AFM probe during probe-membrane interactions? We imaged live XR1 glial cells in culture by single- or dual-pass contact or tapping-mode AFM, examined cell viability at various postimaging times, and report that AFM-imaged live XR1 cells remained viable up to 48 h postimaging and that cell death rates did not increase. To determine if nonlethal, transient interactions between the AFM probe and cell membrane led to transfer of XR1 cell membrane phospholipid components on the probe, we treated the scanned probes with the lipid-binding fluorophore FM 1-43. Confocal microscopy revealed that phospholipid membrane components did accumulate on the probe, and to a generally greater extent during contact-mode imaging than during tapping-mode imaging. Moreover, membrane accumulations on the probe were greater when live XR1 cells were damaged or perturbed, yet membrane did not accumulate in fluorescently detectable quantities during repeated "force curves" during control experiments. Taken together, our data indicate that although AFM imaging of live cells in culture does not affect long-term cell viability, there are substantial probe-membrane interactions that lead to transfer of membrane components to the probe.     

A detailed thermodynamic analysis of ras/effector complex interfaces

Many cellular functions are based on the interaction and crosstalk of various signaling proteins. Among these, members of the Ras family of small GTP-binding proteins are important for communicating signals into different pathways. In order to answer the question of how binding affinity and specificity is achieved, we analyzed binding energetics on the molecular level, with reference to the available structural data. The interaction of two members of the Ras subfamily with two different effector proteins, namely Raf and RalGDS, were investigated using isothermal titration calorimetry and a fluorescence-based method. Experiments with alanine mutants, located in the complex interfaces, yielded an energy map for the contact areas of the Ras/effector complexes, which could be differentiated into enthalpy and entropy contributions. In addition, by using double mutant cycle analysis, we probed the energetic contribution of selected pairs of amino acid residues. The resulting energy landscapes of the Ras/effector interface areas show a highly different topology when comparing the two effectors, Raf and RalGDS, demonstrating the specificity of the respective interactions. Particularly, we observe a high degree of compensating effects between enthalpy and entropy; differences between these components are much greater than the overall free energy differences. This is observed also when using the software FOLD-X to predict the effect of point mutations on the crystal structures of the different complexes. Prediction of the free energy changes shows a very good correlation with the experimentally observed energies. Furthermore, in line with experimental data, energy decomposition indicates that many different components of large magnitude counteract each other to produce a smaller change in overall free energy, illustrating the importance of long-range electrostatic forces in complex formation.     

The thermodynamics of protein-ligand interaction and solvation: insights for ligand design

Isothermal titration calorimetry is able to provide accurate information on the thermodynamic contributions of enthalpy and entropy changes to free energies of binding. The Structure/Calorimetry of Reported Protein Interactions Online database of published isothermal titration calorimetry studies and structural information on the interactions between proteins and small-molecule ligands is used here to reveal general thermodynamic properties of protein-ligand interactions and to investigate correlations with changes in solvation. The overwhelming majority of interactions are found to be enthalpically favoured. Synthetic inhibitors and biological ligands form two distinct subpopulations in the data, with the former having greater average affinity due to more favourable entropy changes on binding. The greatest correlation is found between the binding free energy and apolar surface burial upon complex formation. However, the free-energy contribution per unit area buried is only 30-50% of that expected from earlier studies of transfer free energies of small molecules. A simple probability-based estimator for the maximal affinity of a binding site in terms of its apolar surface area is proposed. Polar surface area burial also contributes substantially to affinity but is difficult to express in terms of unit area due to the small variation in the amount of polar surface buried and a tendency for cancellation of its enthalpic and entropic contributions. Conventionally, the contribution of apolar desolvation to affinity is attributed to gain of entropy due to solvent release. Although data presented here are supportive of this notion, because the correlation of entropy change with apolar surface burial is relatively weak, it cannot, on present evidence, be confidently considered to be correct. Further, thermodynamic changes arising from small differences between ligands binding to individual proteins are relatively large and, in general, uncorrelated with changes in solvation, suggesting that trends identified across widely differing proteins are of limited use in explaining or predicting the effects of ligand modifications.     

Intergroup affiliative interactions and intergroup transfer of young male Japanese macaques (Macaca fuscata)

The social interactions between young male Japanese macaques (Macaca fuscata) and members of two adjacent groups were studied. Young males usually associated with members of a single group. Although some young males occasionally interacted with members of the neighboring group, the frequency of their intergroup affiliative interactions was much lower than that of their intragroup interactions. The intergroup affiliative interactions were less symmetrical than the intragroup interactions. Three- or four-year-old males who remained in their natal group interacted with males of the neighboring group, whereas males over 5 years old did not. Young males revealed a dramatic change in their association partners from males in one group to those in another during the course of their intergroup transfer. Males who remained in their natal group did not attempt to interact with females of the neighboring group. In contrast, males who had transferred to a non-natal group interacted with females in their natal group. It is suggested that intergroup affiliative interactions and intergroup transfer of young male macaques are influenced by close associations between males. The immediate motivation for transfer of young natal macaques may be some attraction to males outside their group rather than sexual attraction to unfamiliar females.     

Equilibrium and kinetics studies of transnitrosation between S-nitrosothiols and thiols

Using UV-vis spectrometrical measurements, equilibrium constants for NO transfer between S-nitroso-N-acetyl-penicillamine (SNAP) and different thiols as well as kinetic data for NO transfer from S-nitroso bovine serum albumin (BSANO) to thiols have been obtained. NO transfer from SNAP to other primary/secondary thiols are thermodynamically favorable, whereas other S-nitrosothiols exhibit similar NO transfer potential. The obtained Gibbs free energy, enthalpy and entropy data indicated that NO transfer reactions from SNAP to four thiols are exothermic with entropy loss. The kinetic behavior of BSANO/RSH transfer can be related to both the acidity of sulfhydryl group and the electronic structure in thiol.     

The filamentous fungal pellet and forces driving its formation

Filamentous fungi play an important role not only in the bio-manufacturing of value-added products, but also in bioenergy and environmental research. The bioprocess manipulation of filamentous fungi is more difficult than that of other microbial species because of their different pellet morphologies and the presence of tangled mycelia under different cultivation conditions. Fungal pellets, which have the advantages of harvest ease, low fermentation broth viscosity and high yield of some proteins, have been used for a long time. Many attempts have been made to establish the relationship between pellet and product yield using quantitative approaches. Fungal pellet formation is attributed to the combination of electrostatic interactions, hydrophobicity and specific interactions from spore wall components. Electrostatic interactions result from van der Waals forces and negative charge repulsion from carboxyl groups in the spore wall structure. Electrostatic interactions are also affected by counter-ions (cations) and the physiologic conditions of spores that modify the carboxyl groups. Fungal aggregates are promoted by the hydrophobicity generated by hydrophobins, which form a hydrophobic coat that covers the spore. The specific interactions of spore wall components contribute to spore aggregation through salt bridging. A model of spore aggregation was proposed based on these forces. Additionally, some challenges were addressed, including the limitations of research techniques, the quantitative determination of forces and the complex information of biological systems, to clarify the mechanism of fungal pellet formation.     

Efficient Transfer Entropy Analysis of Non-Stationary Neural Time Series

Information theory allows us to investigate information processing in neural systems in terms of information transfer, storage and modification. Especially the measure of information transfer, transfer entropy, has seen a dramatic surge of interest in neuroscience. Estimating transfer entropy from two processes requires the observation of multiple realizations of these processes to estimate associated probability density functions. To obtain these necessary observations, available estimators typically assume stationarity of processes to allow pooling of observations over time. This assumption however, is a major obstacle to the application of these estimators in neuroscience as observed processes are often non-stationary. As a solution, Gomez-Herrero and colleagues theoretically showed that the stationarity assumption may be avoided by estimating transfer entropy from an ensemble of realizations. Such an ensemble of realizations is often readily available in neuroscience experiments in the form of experimental trials. Thus, in this work we combine the ensemble method with a recently proposed transfer entropy estimator to make transfer entropy estimation applicable to non-stationary time series. We present an efficient implementation of the approach that is suitable for the increased computational demand of the ensemble method''s practical application. In particular, we use a massively parallel implementation for a graphics processing unit to handle the computationally most heavy aspects of the ensemble method for transfer entropy estimation. We test the performance and robustness of our implementation on data from numerical simulations of stochastic processes. We also demonstrate the applicability of the ensemble method to magnetoencephalographic data. While we mainly evaluate the proposed method for neuroscience data, we expect it to be applicable in a variety of fields that are concerned with the analysis of information transfer in complex biological, social, and artificial systems.     

Experimental assignment of the structure of the transition state for the association of barnase and barstar

Association of a protein complex follows a two step reaction mechanism, with the first step being the formation of an encounter complex which evolves into the final complex. Here we present new experimental data for the association of the bacterial ribonuclease barnase and its polypeptide inhibitor barstar which shed light on the thermodynamics and structure of the transition state and preceding encounter complex of association at diminishing electrostatic attraction. We show that the activation entropy at the transition state is close to zero, with the activation enthalpy being equal to the free energy of binding. This observation was independent of the magnitude of the mutual electrostatic attraction, which were altered by mutagenesis or by addition of salt. The low activation entropy implies that the transition state is mostly solvated at all ionic strengths. The structure of the transition state was probed by measuring pairwise interaction energies using double-mutant-cycles. While at low ionic strength all proximal charge-pairs form contacts, at high salt only a subset of these interactions are maintained. More specifically, charge-charge interactions between partially buried residues are lost, while exposed charged residues maintain their ability to form specific interactions even at the highest salt concentration. Uncharged residues do not interact at any ionic strength. The results presented here suggest that the barnase-barstar binding sites are correctly aligned during the transition state even at diminishing electrostatic attraction, although specific short range interactions of uncharged residues are not yet formed. Furthermore, most of the interface desolvation (which contributes to the entropy of the system) has not yet occurred. This picture seems to be valid at low and high salt. However, at high salt, interactions of the activated complex are limited to a more restricted set of residues which are easier approached during diffusion, prior to final docking. This suggest that the steering region at high salt is more limited, albeit maintaining its specificity.     

Multiple molecular recognition mechanisms. Cytochrome P450--a case study

Biomolecular recognition is complex. The balance between the different molecular properties that contribute to molecular recognition, such as shape, electrostatics, dynamics and entropy, varies from case to case. This, along with the extent of experimental characterization, influences the choice of appropriate computational approaches to study biomolecular interactions. Here, we present computational studies of cytochrome P450 enzymes and their interactions with small molecules and with other proteins. These interactions exemplify some of the diversity of molecular determinants of binding affinity and specificity observed for proteins and we discuss some of the challenges that they pose for molecular modelling and simulation.     

Binding energy and the information content of some elementary biological processes

Protein interactions within a multimolecular complex can result in information and energy transfer between proteins. This can lead in turn to the emergence of novel functions of some proteins of the complex. Various examples of this situation can be found in the scientific literature. This is probably the case for prion protein, chloroplast phosphoribulokinase bound to glyceraldehyde phosphate dehydrogenase, Ras system, and pancreatic lipase bound to biomembranes, to cite but a few. Any enzyme reaction, or enzyme reaction network, carries Shannon entropy and information. On contrary to genome entropy, the entropy of enzyme reactions and metabolic sequences is sensitive to 'external' signals, such as substrate, effector and proton concentrations. Complex structural organization of the cell is associated with a higher entropy content, and one can calculate the gain of entropy and information due to integration and complexity. One may conclude from this brief analysis that the informational content of a living cell is much larger than that of its genome.     

Elucidating protein thermodynamics from the three-dimensional structure of the native state using network rigidity

Given the three-dimensional structure of a protein, its thermodynamic properties are calculated using a recently introduced distance constraint model (DCM) within a mean-field treatment. The DCM is constructed from a free energy decomposition that partitions microscopic interactions into a variety of constraint types, i.e., covalent bonds, salt-bridges, hydrogen-bonds, and torsional-forces, each associated with an enthalpy and entropy contribution. A Gibbs ensemble of accessible microstates is defined by a set of topologically distinct mechanical frameworks generated by perturbing away from the native constraint topology. The total enthalpy of a given framework is calculated as a linear sum of enthalpy components over all constraints present. Total entropy is generally a nonadditive property of free energy decompositions. Here, we calculate total entropy as a linear sum of entropy components over a set of independent constraints determined by a graph algorithm that builds up a mechanical framework one constraint at a time, placing constraints with lower entropy before those with greater entropy. This procedure provides a natural mechanism for enthalpy-entropy compensation. A minimal DCM with five phenomenological parameters is found to capture the essential physics relating thermodynamic response to network rigidity. Moreover, two parameters are fixed by simultaneously fitting to heat capacity curves for histidine binding protein and ubiquitin at five different pH conditions. The three free parameter DCM provides a quantitative characterization of conformational flexibility consistent with thermodynamic stability. It is found that native hydrogen bond topology provides a key signature in governing molecular cooperativity and the folding-unfolding transition.     

Competition between Different Variegating Rearrangements for Limited Heterochromatic Factors in Drosophila Melanogaster

Position effect variegation (PEV) results from the juxtaposition of a euchromatic gene to heterochromatin. In its new position the gene is inactivated in some cells and not in others. This mosaic expression is consistent with variability in the spread of heterochromatin from cell to cell. As many components of heterochromatin are likely to be produced in limited amounts, the spread of heterochromatin into a normally euchromatic region should be accompanied by a concomitant loss or redistribution of the protein components from other heterochromatic regions. We have shown that this is the case by simultaneously monitoring variegation of a euchromatic and a heterochromatic gene associated with a single chromosome rearrangement. Secondly, if several heterochromatic regions of the genome share limited components of heterochromatin, then some variegating rearrangements should compete for these components. We have examined this hypothesis by testing flies with combinations of two or more different variegating rearrangements. Of the nine combinations of pairs of variegating rearrangements we studied, seven showed nonreciprocal interactions. These results imply that many components of heterochromatin are both shared and present in limited amounts and that they can transfer between chromosomal sites. Consequently, even nonvariegation portions of the genome will be disrupted by re-allocation of heterochromatic proteins associated with PEV. These results have implications for models of PEV.     

Flexibility and conformational entropy in protein-protein binding

To better understand the interplay between protein-protein binding and protein dynamics, we analyzed molecular dynamics simulations of 17 protein-protein complexes and their unbound components. Complex formation does not restrict the conformational freedom of the partner proteins as a whole, but, rather, it leads to a redistribution of dynamics. We calculate the change in conformational entropy for seven complexes with quasiharmonic analysis. We see significant loss, but also increased or unchanged conformational entropy. Where comparison is possible, the results are consistent with experimental data. However, stringent error estimates based on multiple independent simulations reveal large uncertainties that are usually overlooked. We observe substantial gains of pseudo entropy in individual partner proteins, and we observe that all complexes retain residual stabilizing intermolecular motions. Consequently, protein flexibility has an important influence on the thermodynamics of binding and may disfavor as well as favor association. These results support a recently proposed unified model for flexible protein-protein association.