Structure and function of the S. pombe III-IV-cyt c supercomplex.

The respiratory chain in aerobic organisms is composed of a number of membrane-bound protein complexes that link electron transfer to proton translocation across the membrane. In mitochondria, the final electron acceptor, complex IV (CIV), receives electrons from dimeric complex III (CIII2), via a mobile electron carrier, cytochrome c. In the present study, we isolated the CIII2CIV supercomplex from the fission yeast Schizosaccharomyces pombe and determined its structure with bound cyt. c using single-particle electron cryomicroscopy. A respiratory supercomplex factor 2 was found to be bound at CIV distally positioned in the supercomplex. In addition to the redox-active metal sites, we found a metal ion, presumably Zn2+, coordinated in the CIII subunit Cor1, which is encoded by the same gene (qcr1) as the mitochondrial-processing peptidase subunit ß. Our data show that the isolated CIII2CIV supercomplex displays proteolytic activity suggesting a dual role of CIII2 in S. pombe. As in the supercomplex from S. cerevisiae, subunit Cox5 of CIV faces towards one CIII monomer, but in S. pombe, the two complexes are rotated relative to each other by ~45°. This orientation yields equal distances between the cyt. c binding sites at CIV and at each of the two CIII monomers. The structure shows cyt. c bound at four positions, but only along one of the two symmetrical branches. Overall, this combined structural and functional study reveals the integration of peptidase activity with the CIII2 respiratory system and indicates a two-dimensional cyt. c diffusion mechanism within the CIII2-CIV supercomplex.

Structure of mycobacterial respiratory complex I.

Oxidative phosphorylation, the combined activity of the electron transport chain (ETC) and adenosine triphosphate synthase, has emerged as a valuable target for the treatment of infection by Mycobacterium tuberculosis and other mycobacteria. The mycobacterial ETC is highly branched with multiple dehydrogenases transferring electrons to a membrane-bound pool of menaquinone and multiple oxidases transferring electrons from the pool. The proton-pumping type I nicotinamide adenine dinucleotide (NADH) dehydrogenase (Complex I) is found in low abundance in the plasma membranes of mycobacteria in typical in vitro culture conditions and is often considered dispensable. We found that growth of Mycobacterium smegmatis in carbon-limited conditions greatly increased the abundance of Complex I and allowed isolation of a rotenone-sensitive preparation of the enzyme. Determination of the structure of the complex by cryoEM revealed the "orphan" two-component response regulator protein MSMEG_2064 as a subunit of the assembly. MSMEG_2064 in the complex occupies a site similar to the proposed redox-sensing subunit NDUFA9 in eukaryotic Complex I. An apparent purine nucleoside triphosphate within the NuoG subunit resembles the GTP-derived molybdenum cofactor in homologous formate dehydrogenase enzymes. The membrane region of the complex binds acyl phosphatidylinositol dimannoside, a characteristic three-tailed lipid from the mycobacterial membrane. The structure also shows menaquinone, which is preferentially used over ubiquinone by gram-positive bacteria, in two different positions along the quinone channel, comparable to ubiquinone in other structures and suggesting a conserved quinone binding mechanism.

Structure of the bc1-cbb3 respiratory supercomplex from Pseudomonas aeruginosa.

Energy conversion by electron transport chains occurs through the sequential transfer of electrons between protein complexes and intermediate electron carriers, creating the proton motive force that enables ATP synthesis and membrane transport. These protein complexes can also form higher order assemblies known as respiratory supercomplexes (SCs). The electron transport chain of the opportunistic pathogen Pseudomonas aeruginosa is closely linked with its ability to invade host tissue, tolerate harsh conditions, and resist antibiotics but is poorly characterized. Here, we determine the structure of a P. aeruginosa SC that forms between the quinol:cytochrome c oxidoreductase (cytochrome bc1) and one of the organism's terminal oxidases, cytochrome cbb3, which is found only in some bacteria. Remarkably, the SC structure also includes two intermediate electron carriers: a diheme cytochrome c4 and a single heme cytochrome c5. Together, these proteins allow electron transfer from ubiquinol in cytochrome bc1 to oxygen in cytochrome cbb3. We also present evidence that different isoforms of cytochrome cbb3 can participate in formation of this SC without changing the overall SC architecture. Incorporating these different subunit isoforms into the SC would allow the bacterium to adapt to different environmental conditions. Bioinformatic analysis focusing on structural motifs in the SC suggests that cytochrome bc1-cbb3 SCs also exist in other bacterial pathogens.

Structural basis of mammalian complex IV inhibition by steroids.

The mitochondrial electron transport chain maintains the proton motive force that powers adenosine triphosphate (ATP) synthesis. The energy for this process comes from oxidation of reduced nicotinamide adenine dinucleotide (NADH) and succinate, with the electrons from this oxidation passed via intermediate carriers to oxygen. Complex IV (CIV), the terminal oxidase, transfers electrons from the intermediate electron carrier cytochrome c to oxygen, contributing to the proton motive force in the process. Within CIV, protons move through the K and D pathways during turnover. The former is responsible for transferring two protons to the enzyme's catalytic site upon its reduction, where they eventually combine with oxygen and electrons to form water. CIV is the main site for respiratory regulation, and although previous studies showed that steroid binding can regulate CIV activity, little is known about how this regulation occurs. Here, we characterize the interaction between CIV and steroids using a combination of kinetic experiments, structure determination, and molecular simulations. We show that molecules with a sterol moiety, such as glyco-diosgenin and cholesteryl hemisuccinate, reversibly inhibit CIV. Flash photolysis experiments probing the rapid equilibration of electrons within CIV demonstrate that binding of these molecules inhibits proton uptake through the K pathway. Single particle cryogenic electron microscopy (cryo-EM) of CIV with glyco-diosgenin reveals a previously undescribed steroid binding site adjacent to the K pathway, and molecular simulations suggest that the steroid binding modulates the conformational dynamics of key residues and proton transfer kinetics within this pathway. The binding pose of the sterol group sheds light on possible structural gating mechanisms in the CIV catalytic cycle.

New Structures Reveal Interaction Dynamics in Respiratory Supercomplexes.

Mitochondrial energy conversion involves a chain of membrane-bound proteins that are wired to conduct an electron current, which drives transmembrane proton translocation. These enzymes associate to form supercomplexes, but the functional relevance of the higher-order structures is unknown. A recent study by Letts et al. presents structures of a supercomplex, which suggest how the interaction choreography may control overall functionality.

Cryo-EM structure and kinetics reveal electron transfer by 2D diffusion of cytochrome c in the yeast III-IV respiratory supercomplex.

Energy conversion in aerobic organisms involves an electron current from low-potential donors, such as NADH and succinate, to dioxygen through the membrane-bound respiratory chain. Electron transfer is coupled to transmembrane proton transport, which maintains the electrochemical proton gradient used to produce ATP and drive other cellular processes. Electrons are transferred from respiratory complexes III to IV (CIII and CIV) by water-soluble cytochrome (cyt.) c In Saccharomyces cerevisiae and some other organisms, these complexes assemble into larger CIII2CIV1/2 supercomplexes, the functional significance of which has remained enigmatic. In this work, we measured the kinetics of the S. cerevisiae supercomplex cyt. c-mediated QH2:O2 oxidoreductase activity under various conditions. The data indicate that the electronic link between CIII and CIV is confined to the surface of the supercomplex. Single-particle electron cryomicroscopy (cryo-EM) structures of the supercomplex with cyt. c show the positively charged cyt. c bound to either CIII or CIV or along a continuum of intermediate positions. Collectively, the structural and kinetic data indicate that cyt. c travels along a negatively charged patch on the supercomplex surface. Thus, rather than enhancing electron transfer rates by decreasing the distance that cyt. c must diffuse in three dimensions, formation of the CIII2CIV1/2 supercomplex facilitates electron transfer by two-dimensional (2D) diffusion of cyt. c This mechanism enables the CIII2CIV1/2 supercomplex to increase QH2:O2 oxidoreductase activity and suggests a possible regulatory role for supercomplex formation in the respiratory chain.

Structure and Mechanism of Respiratory III-IV Supercomplexes in Bioenergetic Membranes.

In the final steps of energy conservation in aerobic organisms, free energy from electron transfer through the respiratory chain is transduced into a proton electrochemical gradient across a membrane. In mitochondria and many bacteria, reduction of the dioxygen electron acceptor is catalyzed by cytochrome c oxidase (complex IV), which receives electrons from cytochrome bc1 (complex III), via membrane-bound or water-soluble cytochrome c. These complexes function independently, but in many organisms they associate to form supercomplexes. Here, we review the structural features and the functional significance of the nonobligate III2IV1/2 Saccharomyces cerevisiae mitochondrial supercomplex as well as the obligate III2IV2 supercomplex from actinobacteria. The analysis is centered around the Q-cycle of complex III, proton uptake by CytcO, as well as mechanistic and structural solutions to the electronic link between complexes III and IV.

NMR structural analysis of the yeast cytochrome c oxidase subunit Cox13 and its interaction with ATP.

BACKGROUND: Mitochondrial respiration is organized in a series of enzyme complexes in turn forming dynamic supercomplexes. In Saccharomyces cerevisiae (baker's yeast), Cox13 (CoxVIa in mammals) is a conserved peripheral subunit of Complex IV (cytochrome c oxidase, CytcO), localized at the interface of dimeric bovine CytcO, which has been implicated in the regulation of the complex. RESULTS: Here, we report the solution NMR structure of Cox13, which forms a dimer in detergent micelles. Each Cox13 monomer has three short helices (SH), corresponding to disordered regions in X-ray or cryo-EM structures of homologous proteins. Dimer formation is mainly induced by hydrophobic interactions between the transmembrane (TM) helix of each monomer. Furthermore, an analysis of chemical shift changes upon addition of ATP revealed that ATP binds at a conserved region of the C terminus with considerable conformational flexibility. CONCLUSIONS: Together with functional analysis of purified CytcO, we suggest that this ATP interaction is inhibitory of catalytic activity. Our results shed light on the structural flexibility of an important subunit of yeast CytcO and provide structure-based insight into how ATP could regulate mitochondrial respiration.

Solution NMR structure of yeast Rcf1, a protein involved in respiratory supercomplex formation.

The Saccharomyces cerevisiae respiratory supercomplex factor 1 (Rcf1) protein is located in the mitochondrial inner membrane where it is involved in formation of supercomplexes composed of respiratory complexes III and IV. We report the solution structure of Rcf1, which forms a dimer in dodecylphosphocholine (DPC) micelles, where each monomer consists of a bundle of five transmembrane (TM) helices and a short flexible soluble helix (SH). Three TM helices are unusually charged and provide the dimerization interface consisting of 10 putative salt bridges, defining a "charge zipper" motif. The dimer structure is supported by molecular dynamics (MD) simulations in DPC, although the simulations show a more dynamic dimer interface than the NMR data. Furthermore, CD and NMR data indicate that Rcf1 undergoes a structural change when reconstituted in liposomes, which is supported by MD data, suggesting that the dimer structure is unstable in a planar membrane environment. Collectively, these data indicate a dynamic monomer-dimer equilibrium. Furthermore, the Rcf1 dimer interacts with cytochrome c, suggesting a role as an electron-transfer bridge between complexes III and IV. The Rcf1 structure will help in understanding its functional roles at a molecular level.

Scavenging of superoxide by a membrane-bound superoxide oxidase.

Superoxide is a reactive oxygen species produced during aerobic metabolism in mitochondria and prokaryotes. It causes damage to lipids, proteins and DNA and is implicated in cancer, cardiovascular disease, neurodegenerative disorders and aging. As protection, cells express soluble superoxide dismutases, disproportionating superoxide to oxygen and hydrogen peroxide. Here, we describe a membrane-bound enzyme that directly oxidizes superoxide and funnels the sequestered electrons to ubiquinone in a diffusion-limited reaction. Experiments in proteoliposomes and inverted membranes show that the protein is capable of efficiently quenching superoxide generated at the membrane in vitro. The 2.0 Å crystal structure shows an integral membrane di-heme cytochrome b poised for electron transfer from the P-side and proton uptake from the N-side. This suggests that the reaction is electrogenic and contributes to the membrane potential while also conserving energy by reducing the quinone pool. Based on this enzymatic activity, we propose that the enzyme family be denoted superoxide oxidase (SOO).

Lipid Composition Affects the Efficiency in the Functional Reconstitution of the Cytochrome c Oxidase.

The transmembrane protein cytochrome c oxidase (CcO) is the terminal oxidase in the respiratory chain of many aerobic organisms and catalyzes the reduction of dioxygen to water. This process maintains an electrochemical proton gradient across the membrane hosting the oxidase. CcO is a well-established model enzyme in bioenergetics to study the proton-coupled electron transfer reactions and protonation dynamics involved in these processes. Its catalytic mechanism is subject to ongoing intense research. Previous research, however, was mainly focused on the turnover of oxygen and electrons in CcO, while studies reporting proton turnover rates of CcO, that is the rate of proton uptake by the enzyme, are scarce. Here, we reconstitute CcO from R. sphaeroides into liposomes containing a pH sensitive dye and probe changes of the pH value inside single proteoliposomes using fluorescence microscopy. CcO proton turnover rates are quantified at the single-enzyme level. In addition, we recorded the distribution of the number of functionally reconstituted CcOs across the proteoliposome population. Studies are performed using proteoliposomes made of native lipid sources, such as a crude extract of soybean lipids and the polar lipid extract of E. coli, as well as purified lipid fractions, such as phosphatidylcholine extracted from soybean lipids. It is shown that these lipid compositions have only minor effects on the CcO proton turnover rate, but can have a strong impact on the reconstitution efficiency of functionally active CcOs. In particular, our experiments indicate that efficient functional reconstitution of CcO is strongly promoted by the addition of anionic lipids like phosphatidylglycerol and cardiolipin.

Regulatory role of the respiratory supercomplex factors in Saccharomyces cerevisiae.

The respiratory supercomplex factors (Rcf) 1 and 2 mediate supramolecular interactions between mitochondrial complexes III (ubiquinol-cytochrome c reductase; cyt. bc1) and IV (cytochrome c oxidase; CytcO). In addition, removal of these polypeptides results in decreased activity of CytcO, but not of cyt. bc1 In the present study, we have investigated the kinetics of ligand binding, the single-turnover reaction of CytcO with O2, and the linked cyt. bc1-CytcO quinol oxidation-oxygen-reduction activities in mitochondria in which Rcf1 or Rcf2 were removed genetically (strains rcf1Δ and rcf2Δ, respectively). The data show that in the rcf1Δ and rcf2Δ strains, in a significant fraction of the population, ligand binding occurs over a time scale that is ∼100-fold faster (τ ≅ 100 µs) than observed with the wild-type mitochondria (τ ≅ 10 ms), indicating structural changes. This effect is specific to removal of Rcf and not dissociation of the cyt. bc1-CytcO supercomplex. Furthermore, in the rcf1Δ and rcf2Δ strains, the single-turnover reaction of CytcO with O2 was incomplete. This observation indicates that the lower activity of CytcO is caused by a fraction of inactive CytcO rather than decreased CytcO activity of the entire population. Furthermore, the data suggest that the Rcf1 polypeptide mediates formation of an electron-transfer bridge from cyt. bc1 to CytcO via a tightly bound cyt. c We discuss the significance of the proposed regulatory mechanism of Rcf1 and Rcf2 in the context of supramolecular interactions between cyt. bc1 and CytcO.

Structural and functional heterogeneity of cytochrome c oxidase in S. cerevisiae.

Respiration in Saccharomyces cerevisiae is regulated by small proteins such as the respiratory supercomplex factors (Rcf). One of these factors (Rcf1) has been shown to interact with complexes III (cyt. bc1) and IV (cytochrome c oxidase, CytcO) of the respiratory chain and to modulate the activity of the latter. Here, we investigated the effect of deleting Rcf1 on the functionality of CytcO, purified using a protein C-tag on core subunit 1 (Cox1). Specifically, we measured the kinetics of ligand binding to the CytcO catalytic site, the O2-reduction activity and changes in light absorption spectra. We found that upon removal of Rcf1 a fraction of the CytcO is incorrectly assembled with structural changes at the catalytic site. The data indicate that Rcf1 modulates the assembly and activity of CytcO by shifting the equilibrium of structural sub-states toward the fully active, intact form.

NMR Study of Rcf2 Reveals an Unusual Dimeric Topology in Detergent Micelles.

The Saccharomyces cerevisiae mitochondrial respiratory supercomplex factor 2 (Rcf2) plays a role in assembly of supercomplexes composed of cytochrome bc1 (complex III) and cytochrome c oxidase (complex IV). We expressed the Rcf2 protein in Escherichia coli, refolded it, and reconstituted it into dodecylphosphocholine (DPC) micelles. The structural properties of Rcf2 were studied by solution NMR, and near complete backbone assignment of Rcf2 was achieved. The secondary structure of Rcf2 contains seven helices, of which five are putative transmembrane (TM) helices, including, unexpectedly, a region formed by a charged 20-residue helix at the C terminus. Further studies demonstrated that Rcf2 forms a dimer, and the charged TM helix is involved in this dimer formation. Our results provide a basis for understanding the role of this assembly/regulatory factor in supercomplex formation and function.

Mutation of a single residue in the ba3 oxidase specifically impairs protonation of the pump site.

The ba3-type cytochrome c oxidase from Thermus thermophilus is a membrane-bound protein complex that couples electron transfer to O2 to proton translocation across the membrane. To elucidate the mechanism of the redox-driven proton pumping, we investigated the kinetics of electron and proton transfer in a structural variant of the ba3 oxidase where a putative "pump site" was modified by replacement of Asp372 by Ile. In this structural variant, proton pumping was uncoupled from internal electron transfer and O2 reduction. The results from our studies show that proton uptake to the pump site (time constant ∼65 µs in the wild-type cytochrome c oxidase) was impaired in the Asp372Ile variant. Furthermore, a reaction step that in the wild-type cytochrome c oxidase is linked to simultaneous proton uptake and release with a time constant of ∼1.2 ms was slowed to ∼8.4 ms, and in Asp372Ile was only associated with proton uptake to the catalytic site. These data identify reaction steps that are associated with protonation and deprotonation of the pump site, and point to the area around Asp372 as the location of this site in the ba3 cytochrome c oxidase.

Reaction of S. cerevisiae mitochondria with ligands: Kinetics of CO and O2 binding to flavohemoglobin and cytochrome c oxidase.

Kinetic methods used to investigate electron and proton transfer within cytochrome c oxidase (CytcO) are often based on the use of light to dissociate small ligands, such as CO, thereby initiating the reaction. Studies of intact mitochondria using these methods require identification of proteins that may bind CO and determination of the ligand-binding kinetics. In the present study we have investigated the kinetics of CO-ligand binding to S. cerevisiae mitochondria and cellular extracts. The data indicate that CO binds to two proteins, CytcO and a (yeast) flavohemoglobin (yHb). The latter has been shown previously to reside in both the cell cytosol and the mitochondrial matrix. Here, we found that yHb resides also in the intermembrane space and binds CO in its reduced state. As observed previously, we found that the yHb population in the mitochondrial matrix binds CO, but only after removal of the inner membrane. The mitochondrial yHb (in both the intermembrane space and the matrix) recombines with CO with τ≅270ms, which is significantly slower than observed with the cytosolic yHb (main component τ≅1.3ms). The data indicate that the yHb populations in the different cell compartments differ in structure.

Mimicking respiratory phosphorylation using purified enzymes.

The enzymes of oxidative phosphorylation is a striking example of the functional association of multiple enzyme complexes, working together to form ATP from cellular reducing equivalents. These complexes, such as cytochrome c oxidase or the ATP synthase, are typically investigated individually and therefore, their functional interplay is not well understood. Here, we present methodology that allows the co-reconstitution of purified terminal oxidases and ATP synthases in synthetic liposomes. The enzymes are functionally coupled via proton translocation where upon addition of reducing equivalents the oxidase creates and maintains a transmembrane electrochemical proton gradient that energizes the synthesis of ATP by the F1F0 ATP synthase. The method has been tested with the ATP synthases from Escherichia coli and spinach chloroplasts, and with the quinol and cytochrome c oxidases from E. coli and Rhodobacter sphaeroides, respectively. Unlike in experiments with the ATP synthase reconstituted alone, the setup allows in vitro ATP synthesis under steady state conditions, with rates up to 90 ATP×s(-1)×enzyme(-1). We have also used the novel system to study the phenomenon of "mild uncoupling" as observed in mitochondria upon addition of low concentrations of ionophores (e.g. FCCP, SF6847) and the recoupling effect of 6-ketocholestanol. While we could reproduce the described effects, our data with the in vitro system does not support the idea of a direct interaction between a mitochondrial protein and the uncoupling agents as proposed earlier.

Isolation of yeast complex IV in native lipid nanodiscs.

We used the amphipathic styrene maleic acid (SMA) co-polymer to extract cytochrome c oxidase (CytcO) in its native lipid environment from S. cerevisiae mitochondria. Native nanodiscs containing one CytcO per disc were purified using affinity chromatography. The longest cross-sections of the native nanodiscs were 11nm×14nm. Based on this size we estimated that each CytcO was surrounded by ~100 phospholipids. The native nanodiscs contained the same major phospholipids as those found in the mitochondrial inner membrane. Even though CytcO forms a supercomplex with cytochrome bc1 in the mitochondrial membrane, cyt. bc1 was not found in the native nanodiscs. Yet, the loosely-bound Respiratory SuperComplex factors were found to associate with the isolated CytcO. The native nanodiscs displayed an O2-reduction activity of ~130 electrons CytcO-1s-1 and the kinetics of the reaction of the fully reduced CytcO with O2 was essentially the same as that observed with CytcO in mitochondrial membranes. The kinetics of CO-ligand binding to the CytcO catalytic site was similar in the native nanodiscs and the mitochondrial membranes. We also found that excess SMA reversibly inhibited the catalytic activity of the mitochondrial CytcO, presumably by interfering with cyt. c binding. These data point to the importance of removing excess SMA after extraction of the membrane protein. Taken together, our data shows the high potential of using SMA-extracted CytcO for functional and structural studies.

Modeling gating charge and voltage changes in response to charge separation in membrane proteins.

Measurements of voltage changes in response to charge separation within membrane proteins can offer fundamental information on mechanisms of charge transport and displacement processes. A recent example is provided by studies of cytochrome c oxidase. However, the interpretation of the observed voltage changes in terms of the number of charge equivalents and transfer distances is far from being trivial or unique. Using continuum approaches to describe the voltage generation may involve significant uncertainties and reliable microscopic simulations are not yet available. Here, we attempt to solve this problem by using a coarse-grained model of membrane proteins, which includes an explicit description of the membrane, the electrolytes, and the electrodes. The model evaluates the gating charges and the electrode potentials (c.f. measured voltage) upon charge transfer within the protein. The accuracy of the model is evaluated by a comparison of measured voltage changes associated with electron and proton transfer in bacterial photosynthetic reaction centers to those calculated using our coarse-grained model. The calculations reproduce the experimental observations and thus indicate that the method is of general use. Interestingly, it is found that charge-separation processes with different spatial directions (but the same distance perpendicular to the membrane) can give similar observed voltage changes, which indicates that caution should be exercised when using simplified interpretation of the relationship between charge displacement and voltage changes.

Protonation Dynamics on Lipid Nanodiscs: Influence of the Membrane Surface Area and External Buffers.

Lipid membrane surfaces can act as proton-collecting antennae, accelerating proton uptake by membrane-bound proton transporters. We investigated this phenomenon in lipid nanodiscs (NDs) at equilibrium on a local scale, analyzing fluorescence fluctuations of individual pH-sensitive fluorophores at the membrane surface by fluorescence correlation spectroscopy (FCS). The protonation rate of the fluorophores was ∼100-fold higher when located at 9- and 12-nm diameter NDs, compared to when in solution, indicating that the proton-collecting antenna effect is maximal already for a membrane area of ∼60 nm(2). Fluorophore-labeled cytochrome c oxidase displayed a similar increase when reconstituted in 12 nm NDs, but not in 9 nm NDs, i.e., an acceleration of the protonation rate at the surface of cytochrome c oxidase is found when the lipid area surrounding the protein is larger than 80 nm(2), but not when below 30 nm(2). We also investigated the effect of external buffers on the fluorophore proton exchange rates at the ND membrane-water interfaces. With increasing buffer concentrations, the proton exchange rates were found to first decrease and then, at millimolar buffer concentrations, to increase. Monte Carlo simulations, based on a simple kinetic model of the proton exchange at the membrane-water interface, and using rate parameter values determined in our FCS experiments, could reconstruct both the observed membrane-size and the external buffer dependence. The FCS data in combination with the simulations indicate that the local proton diffusion coefficient along a membrane is ∼100 times slower than that observed over submillimeter distances by proton-pulse experiments (Ds ∼ 10(-5)cm(2)/s), and support recent theoretical studies showing that proton diffusion along membrane surfaces is time- and length-scale dependent.