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## Introduction

Cellular respiration is perhaps the most important biochemical reaction by which life forms obtain energy to harvest and store in molecules, such as ATP, to be further utilized1,2,3,4,5,6,7. Usually, the overall reaction of respiration to turn glucose and oxygen into carbon dioxide and water is expressed by Eq. (1)8. Our recent mass spectroscopic study to accurately determine the total rate constant of the respiration reaction of Camellia Japonica leaves has confirmed the exact 1:1 correspondence between O2 consumption and CO2 emission suggested by Eq. (1)8.

$${\text{C}}_{{6}} {\text{H}}_{{{12}}} {\text{O}}_{{6}} + {\text{ 6O}}_{{2}} \to {\text{6CO}}_{{2}} + {\text{ 6H}}_{{2}} {\text{O}}$$
(1)

However, mass spectroscopic measurement using a heavy 18O2 marker revealed that H2O indispensably is involved in the reaction; thus, Eq. (1) has to be revised into the form shown in Eq. (2)8,9,10,11. At present, we do not exactly understand the explicit role of water as the mechanism in Eq. (2); i.e., the transport of water can play a critical role as an “assistant” in systematic enzyme catalysis.

$${\text{C}}_{{6}} {\text{H}}_{{{12}}} {\text{O}}_{{6}} + {\text{ 6H}}_{{2}} {\text{O }} + { 6}^{{{18}}} {\text{O}}_{{2}} \to {\text{6CO}}_{{2}} + {\text{12H}}_{{2}}^{{{18}}} {\text{O}}$$
(2)

We carried out measurements to obtain the temperature dependence of the respiration rate9, as revealed in Fig. 1. It clearly indicates non-Arrhenius behavior as the nonlinear function of the natural logarithm of the rate coefficient vs. reciprocal of the absolute temperature (1/T)12, the so-called super-Arrhenius behavior (the “convex” type deviation), which is (Fig. 1). Ea is no more than a constant of activation energy but decreases with increasing temperature. By assuming that the activation energy varies with temperature Ea(T), we were able to obtain its physical meaning, that the super-Arrhenius behavior would be related to the diffusion kinetics term9,13. Therefore, it is worthwhile to investigate how this diffusion kinetics term works at the molecular level14,15,16. There has been an increasing number of studies measuring the temperature dependence of enzyme reactions, showing super- Arrhenius behaviors12,17,18,19,20,21,22,23. Although theoretical model studies have been advanced for the interpretation of such super-Arrhenius behaviors12,17,18,20, space remains for more specific interpretations, such as enzyme complex dynamics in conjunction with a temperature-dependent structural reorganizations and reaction processes12. The diffusion in supercooled liquids and glasses have also been observed from super-Arrhenius plots17,18,20, which have yielded to detailed analysis with temporal relaxation of supercooled liquids approaching the glass transition by multiple steps. In the present paper, we mention the case of plant cell respiration process5,6,7,24 consisting of multiple reactions, where a super-Arrhenius plot (Fig. 1) has been observed8,9,13, the elementary steps have not been completed or cannot be unified12,17,18. We draw attention to an analogy with water diffusion in metastable fluids taking place in the relevant respiration enzymes. We try to point out that enzyme complex places a severe constraint on the fluid diffusion for the involved chemical steps toward a super-Arrhenius behavior12.

### Scheme of respiration reaction

Cellular respiration involves a long list of metabolic processes2, including major portals (Fig. 2) for cytoplasm enzymes towards (1) glycolysis as producers of ATP and mitochondria with (2) the citric acid cycle, and (3) electron transport chain/oxidative phosphorylation25; however, the precise roles of each in respiration remain unknown26. Glycolysis is the initial step of aerobic respiration, which leads to the generation of pyruvate/lactate, NADH and ATP in anaerobic conditions and finally glucose is oxidized to carbon dioxide27. Global metabolic profiling of bacteriostatic antibiotic treatment revealed that the accumulation of metabolites was linked to the buildup of energy metabolites that feed the electron transport chain28 and a marked increase in glucose uptake, triggering the activity of phosphofructokinase-1 (PFK-1), a key rate-limiting enzyme of the glycolysis pathway27,29. Regarding the “committed” step of glycolysis at PFK-1, it catalyzes the conversion of fructose 6-phosphate (F6P) and adenosine triphosphate (ATP) to fructose 1,6-bisphosphate (F1,6BP) and adenosine diphosphate (ADP), as explicitly shown in Fig. 2. That figure reveals the dominant metabolic regulation, yet it presents a puzzling regulatory mechanism of the allosteric effect between the active and effector sites14,30.

Analysis of each step of glycolysis shows that, of those rate-limiting enzymes of the pathway31,32, isoforms of PFK-1 may afford the greatest opportunity33,34 as targets to explain the complex oligomeric process coupled water dynamics to interpret the super-Arrhenius behavior20. In particular, we can go beyond the interpretation that a super-Arrhenius plot can be interpreted as a reciprocal of the diffusion states in between specific configuration ‘entities’ upon temperature, to account for transport behavior to the occurrence of long series of collective events12. The super-Arrhenius behavior often manifests because of such as rates of enzymatic catalysis-promoted processes and temperatures, and we can say, under certain conditions, that the microenvironment affects water fluid diffusion in supplying the respiratory chain. First, intracellular water transport through the enzyme might direct the chain and react with oxygen to generate a messenger to sequential reactions. Second, due to its metastable transition for changing the hydrogen-bonding network and diffusible nature, water fluid might act as a messenger because of its diffusion among the specific enzyme matrix, which confers properties to systematic protein conformational change and communicates as a signaling molecule35. Third, these bio-water fluids might have hierarchical regulation corroborated enzyme matrix reactions; therefore, the increases in plant cell respiration rate are somehow buffered at temperatures of 285–305 K9,36 (Fig. 1) and has been found by recording the oxygen consumption of 283–313 K23, to abide super-Arrhenius behavior, possibly to prevent an overshoot of metabolism at high temperatures37.

Hence, to examine the contributions of various modes of enzyme regulation of water fluid, type-1 phosphofructokinase (PFK-1) was chosen (Fig. 1), for this enzyme is known to be a pacemaker (a rate-limiting enzyme) in the glycolysis process of respiration31,32. In this research, we attempted to develop a more microscopically-detailed interpretation resting on the intricate respiration enzyme of PFK-1 complex structures and employ molecular dynamics simulations to investigate the hydration water diffusive motions through the tetrameric phosphofructokinase conformation dynamics over a physiological temperature range of 270–330 K, where experimental data are available. The PFK-1 tetramer structure places a severe constraint on water diffusion at low temperature by crossing appreciable potential barriers, whereas at high-temperature, water diffusive motions are almost free1,38,39. Our model recognizes that as the temperature rises, thermal energies become larger than barrier heights; what is required for a super-Arrhenius plot is that the average energy of water molecules making transitions (in PFK-1 tetramer, diffusive water hops from one local minimum to another through the channels) rises more slowly than the average energy of all hydration water12.

## Discussion

### Water influx tetramer PFK-1 enzymatic regulation

Protein-water interactions are crucial factors in PFK-1 conformational reorganization, so molecular dynamics (MD) simulation were carried out to investigate the conformational dynamics of the reactant and the product states in the active site in aqueous solution40,41,42,43. Remarkably, it is known that PFK-1 undergoes an intricate oligomerization process, shifting among monomers, dimers, tetramers, and more complex oligomeric structures1,31. The monomer PFK-1 has one active and one effector site but shows no activity. The equilibrium between PFK-1 dimers and tetramers is directly correlated with activity regulation26, in which the activity remains very low up to a dimer level and then increases sharply with a tetramer form in the fully active state16. Although evidence for oligomerization between members of PFK-1s to affect the allosteric regulation of glycolysis is increasing, not much is known about the underlying mechanism. PFK-1s have been shown to exist as tetramers via noncovalent interactions, and tetramerization favors the agonist-mediated reaction coupling efficiency44, suggesting a role for tetramerization in the internalization properties of coupling effector and active sites33.

Although a growing number of crystallographic PFK-1 structures has been resolved33, it has been possible to identify the interdomain folding feature with findings on the regulatory aspects of PFK-1-water interactions and their relation to the control of metabolism. Figure 3 shows the PFK-1 crystallographic structure (PDB entry 1PFK) of a tetramer assembly45, an enlarged view of the ‘Enzyme–Substrate’ site, and one of the ‘Enzyme-Product’ structures at an active site. An enlarged view of an effector site is also shown in the figure. The present analysis has revealed that the magnesium ion (Mg2+) coordinates strongly with the relevant water molecules, phosphate ligand and the amino acid of D103. Structural analyses show that the hydration environments are quite different among the active site, effector site and protein interface. Moreover, MD simulation has allowed us to identify the main folding channels and potential site connecting channels in PFK-1 by using the internal structural mining software MOLEonline46, as shown in Fig. 4a. The tetramer assembly folded PFK-1 has revealed the peculiarity of harboring multiple channels in connection with each active and effector site, and the protein interface. With channel search and channel classification, we first tentatively identified the water molecules of the PFK-1 solution into three categories: the internal waters bound in (1) the interface, (2) the porous channel, and (3) the transient channel of PFK-1 enzyme, respectively.

### Features of tetramer PFK-1s with interior site connection water channels

The PFK-1 tetramer assembly is exhibited in Fig. 4a, where each monomer (α or β) has an effector and an active site, the pair-oriented monomer of α and β forms a dimer, and then follows a complex tetramer. The interface channel connects between either an active site (designated as A in the figure) or an effector site (designated as E), referred to as A–A or E–E, respectively. Detailed analyses were conducted on the interface channel (cleft) with PFK-1 tetramer crystallography structures, involving residues crossing two monomers1,38. Mutation of the interface amino residues indeed affects PFK-1 activities. Moreover, we find that these PFK-1 monomer complexed interfaces can form multiple different types of interior cavities and/or channels, such as porous and transient channels, for the accommodation of substrate and water molecules. The porous channel connects any active site (or an effector site) and the outside of the enzyme (the bulk water area). The layout of such water channels is schematically drawn for convenience in the bottom panels in Fig. 4a, where the connections to each of the effector and active sites of the PFK-1 tetramer structure are indicated by Greek alphabet symbols, α1, α2, β1, and β2, and the interconnections are given with arrows. The drawing is not precise in scale, but it reflects the mutual symmetry within the tetramer enzyme. To be notice of the transient channel is not stable in time, in contrast to the other two channels. However, we can identify the transient channel as the rest of the water channels of the enzyme, which are clearly different from either the interface channel or the porous channel. Each PFK-1 monomer has an active site and an effector site, and these sites are connected by channels in the tetramer complex. In the right side of Fig. 4b, we use MOLEonline46 to calculate and analyze the channels for the crystal structure (PDB entry 1PFK). The interface, porous and transient channel are about 90, 98 and 72 Å long and have bottlenecks of about 4.5, 3.7 and 3.7 Å wide, respectively. The interface channel is rather hydrophilic, for its hydrophobicity is − 0.79 to − 0.08. By comparison, the hydropathy values of the porous and transient channels range from − 1.07 to 0.80 and from − 1.07 to 0.63, respectively, suggesting that these channels are more variable than the interface channel. In terms of time, the transient channel changes more significantly than does the porous channel. In addition, these channels have different abilities to restrict water. The interface channel traps water with hydrophilic amino acids, whereas the porous channel space uses candy-shaped spaces to restrict water movement. Compared with the previous two, the transient channel has a very unstable water volume (note the larger error bar on the left side of Fig. 4b). These reasons make the function of the transient channel in regulating water diffusion very different from those of the other two.

Nevertheless, understanding for the PFK-1 three types of channels their trapped water motions gives specific support to the kinetic models, whose validity should be examined by further experiments and detailed simulations12. The present study does specifically invoke these channel water diffusive effects associated with the super-Arrhenius behavior. Such as for uptake of the protons required in water formation on reduction of O2 and uptake and release of the protons pumped across the membrane need the restricted channel waters for the proton transfer pathways. Control by temperature that the channels make a close and open form by the amino acid to influence the water-protein interactions in the channel. The channel water diffusion may make a key role in controlling the respiration reaction for the proton coupled electron transfer26,47,48,49 and the O2 reduction. It has been found that the protons released need requires a proton-transferring pathway (a “proton well”) in the relevant protein. Due to this, the electrical potential difference is created entirely, or nearly so, by transmembrane proton transfer across the “well”, involving amino acid residues, two of which have been as carboxylates, and perhaps water molecules26.

For a quantitative description of hydration water behavior coupled with the PFK-1 enzyme, we performed temperature-dependent MD simulations with explicit solvent water molecules to analyze the water-coupled PFK-1 dynamics. Each channel holds water molecules in a constricted channel diameter and length to permit the trapped water motion to connect between each effector and active site for the interface, and the porous channels and the transient channel function to link the channel water with the bulk area outside of the enzyme. The average numbers of water molecules in each channel are 11 ± 4 for the interface channel, 17 ± 3 for the porous channel, and 15 ± 9 for the transient channel, respectively. The standard deviation of the number is 270–340 over time, on average. Figure 4b displays the number of water molecules in each channel as a function of temperature in units of K and also its reciprocal in units of K−1, obtained by MD simulation. Three channel diameters as a function of channel length are also displayed in Fig. 4b, where the channel length is measured from the center of the enzyme. The broad widths along the plots indicate a time-dependent distribution of water in each channel. The channel widths are barely wide enough for a few water molecules to move around under the interaction with the channel wall. Thus, it is expected that the trapped channel water molecules are exchanged dynamically with the PFK-1 enzyme surface water and that the penetration of water molecules into channels significantly varies with temperature. However, the space inside each channel permits the trapped water to move and interconnects each effector and active site at the end of the channel.

### Channel water influx temperature-dependent diffusion kinetics

We also carried out MD simulations to obtain the water diffusion dynamics in those three types of folded channels in the tetramer PFK-1. The mean squared displacement (MSD), Eq. (3), is measured over time to determine if water molecules are spreading solely due to diffusion. Single-particle tracking (SPT) enabled us to resolve the modes of motion of the diffusion of individual water molecules50. In the MD ensemble average, the time dependence of MSD for water motion in pure water is quite different from those of PFK-1 enzyme waters. Comparing Fig. 5a and Fig. 5b shows the dissimilar water motion behaviors of the enzyme-confined water and that of pure water without enzyme solute. As a result, the self-water diffusion coefficients D can be derived readily by Eq. (4)50. The self-diffusion coefficients D as a function of temperature are plotted in the corresponding figure inset, revealing an apparent difference in temperature-dependent water diffusion kinetics between the channel waters and pure water without the enzyme-confined matrix. The value of D for the channel water varies significantly with temperature in a non-monotonic way, while the pure water follows a monotonic linear change with temperature. The temperature dependence of channel water diffusion is described by Eq. (7), where the logarithm of estimated D against the reciprocal of the accessible temperature turns out to be a curve, rather than the straight line expected according to the Arrhenius law, Eq. (6).

We say that the reaction abides by a super-Arrhenius dependence, being associated with a convex curve, where plots are accounted for by an additional quadratic term in 1/RT and the parameters elaborated in Table 1. For simplicity, we first discuss the results in terms of their activation energy Ea, which is no longer a constant but depends on 1/T, as compared with the cellular reparation barrier observation (Table 1). The trapped-water motion barrier does not remain constant, instead depending on the temperature as well as the channel confinement to which it belongs. From comparing these parameters and derived barriers, we observe that at low temperature, water molecules diffuse by crossing appreciable potential barriers, whereas at high temperature, the diffusive motions are barrier-less; the transient channel water motions seem significantly sensitive to ambient temperature, revealing 31.5 kJ mol−1 by 290 K and dropping to 23.0 kJ mol−1 by 310 K. These are in contrast to those of porous and interface water motions, which are revealed to be more steadily confined to a barrier region. The data analysis shows that in PFK-1 relevant water dynamics, the motion barriers are not just cumulative as respiration barriers; they primarily share a feature similar to super-Arrhenius behavior.

The water self-diffusion coefficient is directly associated with the tracer mutual diffusion of conjunction. Therefore, it is a fundamental transport property, essential for an accurate description of mass transfer relevant in the cellular respiration process, which is environmentally related51. This implies that respiration reactions are dominated by the entropy term because of the required water vehicle in facilitating the proton coupled electron transfer. Figure 6 presents a comparison of the temperature-dependent diffusion plots of the pure water and those for the transient channel water, the interface channel water, and the porous channel water of the PFK-1 tetramer structure, respectively. The axis of the reciprocal temperature T is presented at the bottom varying in the temperature range from 270 to 330 K. The temperature dependence for the bulk water diffusion looks almost monotonic against 1/T, and the value of the diffusion coefficients for the pure water by temperature is ca. 8.2 × 10–9–1.1 × 10–8 m2 s−1 as a monotonic linear dependence. Compared with the channel water diffusions of the PFK-1 matrix, it clearly shows the temperature-dependent water kinetics in the significant super-Arrhenius behavior, with a primary similarity to that of the respiration rates k experimentally observed in the profile8. Moreover, Fig. 6 shows that increasing the temperature makes the channel waters move much more slowly than the tenfold magnitude of the diffusion coefficient quantity from that of the non-enzyme-confined pure water. Knowing the diffusion, we can also compare all the quantities of interest, such as the modulus of the diffusion coefficient. The results also show that the self-diffusion coefficients for the interface channel water at temperatures of 270–330 K change from 1.0 × 10–9 to 1.1 × 10–8 m2 s−1 for the transient channel, 4.6 × 10–10 to 5.2 × 10–9 m2 s−1 for the interface channel water, and 4.4 × 10–10 to 2.6 × 10–9 m2 s−1 for the porous channel.

Figure 6 reveals the case of a markedly convex curve feature in the nano-channel confinements as compared to that of pure water. These super-Arrhenius-behavior-like profiles exhibit the constraint confinement to microscopic interpretations12. For such a PFK tetramer complex, we suppose that the reaction configuration space may be partitioned into two types of conformations, one of which (called the open state) is reactive, with a diffusion coefficient D(T), and the other of which (called the closed state) is nonreactive. This provides a mechanism for D(T) to decrease with T; i.e., the water density of the state decreases with T faster than D(T) increases, see Fig. 4(b). This could occur because higher-diffusion water molecules visit a wider region of the phase space and hence spend a smaller fraction of their time in the constrained region, referring to the transient channel water behavior in Fig. 4(b) and Fig. 6. Such more sensitive water number variation in the transient channel by temperature is reflected in its diffusion rate. The more limited transient channel spaces compared to those of the interface channel and porous channel (see Fig. 4b) lend the corresponding water dynamics an apparent feature of super-Arrhenius behavior in Fig. 6. This interpretation may be of significant importance for enzymology. In recent years, several studies have focused on understanding the role of conformational sampling in enzyme catalysis12. There is still room for arguing that the active configurations sampled by PFK-1 enzymatic complex are not of higher energy than the general population of conformers and that populating these conformations of specific water/substrate channel network may determine the reaction rate38.

Under a certain temperature range, the long-term slowing of self-diffusivity is featuring a super-Arrhenius trend. Our results suggest smaller channel spacings the continuous changing of energy barriers due to the continuous variation of water configurational excess entropy with temperature. It appears that the Arrhenius to super-Arrhenius behavior could be a more generic feature arising from the interplay of water confinement motions and temperature effects rather than the details of the chemical reaction steps15. Note also that such a scenario agrees with the findings of Ref.14, where it is demonstrated that the Arrhenius to super-Arrhenius crossover takes place for smaller spacing confinement of argon gas diffusion at lower temperatures. We discuss the significance of this observation, and we provide an interpretation of this finding by combining experiments and simulations. The accuracy of these simulated water diffusion coefficient quantities is considered and compared to the earlier experimental measurements22,52,53 in Table 2, since data for water diffusion coefficients over temperature in multiple condition systems have been reported for pure water, isotope water, nanocarbon tube water, and aquaporin’s channel water, respectively22,52,53. The most significant difference seems to be that for the maximum water diffusion coefficient appearing in bulk water. Moreover, our results are consistent with the measurements for systems that show maxima at about 10–8 m2 s−1, depending on the examined temperature, and provide evidence of the sensitivity to the temperature effect of protein matrix channel bio-water diffusion.

As manifested above, the self-diffusion of water in the bulk phase and under confinement, which is very relevant to characterizing the dynamic bottlenecks of the water-utilizing allosteric control between multiple effector and active sites in PFK-1 tetramer. We demonstrate that the water density of the state changes by temperature, quantitatively describing the accumulation network caught in between the active and effector sites. As the temperature increases, water molecules have higher kinetic energy to escape the channel confinement; from channel water numbers estimated by temperature (Fig. 4b), we present three aspects—residue distribution, trapped water number, and time trajectory variation—to distinguish the channel water behaviors. The residue distribution and the total water number are complementary, for in the interface channel, the hydrophilic residues are evenly distributed, which leads to the smaller water number and causes the water network to form a two- or even one-dimensional water network. In the long porous channels, more hydrophobic residues are occupied and cause more water molecules to be trapped, inhibiting the hopping or transfer along a one-dimensional water chain alignment. In the transient channel, the good distribution of alternative hydrophilic and hydrophobic residues keeps the water number appropriate, and the one-dimensional water chain may help the communication of sites. For example, two water molecules suspended in a narrow channel tend to orient in one-dimensional contact because of the water alignment induced by the hydrogen bonding network.

Our study recognizes that as the temperature rises, thermal energies will become larger than barrier heights20. What is required for a super-Arrhenius plot is that the average energy of water molecules making transitions by diffusive hops between sites along the channel rises more slowly than the average energy of those bulk (or pure) water molecules. This is due to the water diffusion kinetics in the PFK-1 tetramer matrix, wherein the water molecules interact strongly with the protein amino residues; those bound water molecules in the protein surface and in the interior channels, which occupy internal cavities and deep clefts, interact with the amino residues with different chemical characteristics and are extensively involved in the water hydration H-bonding, in conjunction with potential sites, and thus exhibit assorted dynamics.

Nevertheless, the respiration process with water transport mentioned above may be complicated by multiple parallel pathways in cellular organisms. This study highlights water motions in the PFK-1 tetramer complexed channel, which have yielded to detailed analysis compared to those water-related systems. The super-Arrhenius behavior manifests the basic features of PFK-1 channel water dynamics in collective phenomena such as enzymatic catalysis-promoted respiration processes. The temperature dependence of rates of enzymatic catalysis reactions has been identified, such as the super-Arrhenius kinetics, where transport phenomena accelerate as the temperature increases17. Although such an examination is expedient and appropriate for the complicated respiration processes, the entropy of the activation term should never be overlooked because some reactions, in particular those involving strongly-solvated ions, may be dominated by entropy. As illustrated in Fig. 2, which presents the magnesium ion coordination, this domination is due to water solvation effects.

It appears that one can acquire a more fundamental understanding of the very complicated respiration processes by applying the super-Arrhenius kinetic and reaction modeling on a molecular level. An important remark regarding the MD simulations is that these dynamics of structural modulation involve a locally-constrained environment of protein and the trapped-water motion within the channel54. The trapped-water motion takes place in different ways, and our interest lies in how such water molecules in the active or effector site could be regulated in different channels with different structural states in terms of water diffusion in the channels. It seems likely that the hydrophilic nature of the active/effector site and channel make it favorable for water molecules to remain inside, and they can move in and out of these sites through the channels and those interfaces, for the channel structures are dynamical. A water network in conjunction with potential sites between the crystallographic structure and the MD simulation is thus conceivable. Presumably, the channel water might also be associated with a water hydration network in precise control of the communication of different sites and conformational change.

Therefore, we conclude that channel water diffusion must play a key role in controlling the respiration reaction of the Camellia plant, assuming that the enzymatic catalyzed reaction at PFK-1 is the rate-determining step in glycolysis. This result on channel water diffusion also suggests that the interaction of water in the channels changes due to a stronger perturbation of seclusion by binding sites associated with the channel wall conformation in the network, and such interaction would vary with temperature. Accordingly, we expect that the present MD simulation clarifies the key point that the temperature dependence of the water diffusion in the channels of PFK-1 should be able to provide us an important clue about the interconnection mechanism with water molecules and how they exchange controlling signals between allosteric sites of the PFK-1 tetramer matrix in the cellular respiration reaction. We provide an interpretation of this finding that places it in the broader perspective of water transport as well as biochemical kinetics.

## Summary

We report in this work the temperature dependence of channel water diffusion in the PFK-1 tetramer matrix, determined by MD simulations, thereby providing a molecular understanding of the water transport role in the super-Arrhenius behavior of the respiration rate of the Camellia plant obtained in our previous experiment. We found two stable types of water channels of PFK-1, classified as (1) the interface channel and (2) the porous channel, and a third water channel, the so-called (3) transient channel, which is structurally rather less stable. These channels serve as interconnecting pathways with confined water diffusion in channels between active sites and effector sites of the enzyme, as well as linking to the outside bulk water. The temperature dependence of PFK-1 channel water diffusion is revealed by super-Arrhenius dependence, being associated with a convex curve, and the activation energy Ea is no longer a constant but depends on 1/T, which suggests a surprising similarity to the cellular reparation barrier observation. Our results suggest that the water molecules in the PFK-1 tetramer matrix protein surface and in the interior channels are extensively involved in the water hydration H-bonding in conjunction with potential effector and active sites and thus exhibit the assorted dynamics. Such super-Arrhenius temperature dependence of the channel water diffusion provides the important information that the water in the scheme of the respiration reaction, proposed as C6H12O6 + 6H2O + 6O2 → 6CO2 + 12H2O, can be interpreted as a transport medium linking allosteric sites of the PFK-1 enzyme through diffusion in the channels, but not as a simple reactant molecule, as usually understood. It is thus important to point out that we should be careful in clarifying the role of water, depending upon its location in the enzyme system. The present MD simulation results demonstrate that a comparative study provides insight for further clarification of the water-utilizing allosteric control between multiple effector and active sites in enzymes.

## Methods

### Molecular dynamics simulation

Because the PFK-1 A subunit has more similar reaction conformations than the B subunit, molecular dynamics simulations were performed on a globular protein constructed of a PFK-1 (PDB entry 1PFK) A subunit, referred to as a many species structure1,15,45,55,56, in of dilute aqueous solution. MD simulations were performed with the AMBER 14 package57 using a 0.1 ps integration time step and periodic boundary conditions with the 124 × 115 × 131 Å3 box. The proteins were modeled with the Amber ff14SB force field58 and solvated with the TIP4P-Ew water model, and a set of GAFF59 parameters was adopted for the description of fructose-1,6-bisphosphate (F-1,6-bP) and fructose-6-phosphate (F-6-P). The fully solvated and ionized system contained 238 thousand atoms with about 56,000 TIP4P water molecules.

For the optimization cycles (3000 steps for the steepest descent and total max 20,000 steps for the conjugate gradient), the system density and box lengths were converged to reach equilibration; the system density was stabilized at ~ 1.06 g/cm3. Then the system underwent the each 1 ns annealing NPT ensemble with equilibrated steps from 0 to 340 K, which temperature gap is 10 K, at every target temperature and a target pressure of 1.0 bar at a collision frequency of 1 ps−1 by using a Langevin-thermostat. Form 0 to 150 K do the onion restraint with the center of active site by the force of 500 kcal mol−1 Å2 and release every 2 Å form the outer to the active site residues at every 10 K till release all the residues. The MD sampling of temperature 260—340 K is during the heating and density dynamics and further equilibrated the systems for 10 ns to get well-settled pressure and temperature for conformational, chemical analyses and take the final 300 ps of the trajectory for the RMSD and diffusion calculation. All structural analyses were performed with Visual Molecular Dynamics (VMD)60.

### Channel search and channel classification

The channels in PFK-1 were computed in MOLEonline46. Any cavities were identified and their start and end points linked. Various channel properties and scale were computed and tabulated according to the amino acids. In the submitted version, we count the amount of water at around 5 Å of the channel amino acid under cumulative time. At the prompting of the reviewer, we noticed that this definition counted more water molecules trapped inside the protein and interfered with our observations, so we adjusted the statistical definition and calculated the number of water molecules at 3 Å around the center of the channel. The result is shown on the right side of Fig. 4b.

We employed MOLEonline software46 to search for the channels of the PFK-1 tetramer enzyme from the protein data bank with Phosphofructokinase-1 (PDB entry 1PFK)45. Starting points for the channel search were set at the center of the enzyme, and also at four active sites and four effector sites of PFK-1, to clarify the channels (or tunnels) that interconnect possible buried protein cavities within the protein surface. The probe radius was set at 20 Å. The current version of MOLEonline uses the Dijkstra’s Shortest Path Algorithm to find the channels between starting and ending points, and also employs Voronoi diagrams for constructing the approximated structure, in which individual atoms are replaced by 4, 8, 12 or 20 spheres with a radius of the smallest atom of a given protein of PFK-146. Under this approximation, the Voronoi diagram made it possible to construct correct channel structures and identify water paths. By algorithm search and careful examination, we are able to classify the channels into three types, one less-stable transient channel and two constantly stable channels, namely, the interface and the porous channels, as shown in Fig. 3.

### Water diffusion coefficient calculation

To distinguish whether the dispersion behavior of particles is caused by diffusion, it can be determined by the mean-squared displacement (MSD) ,and using Langevin equation to try to describe the diffusion of Brown particles3, where MSD is defined as

$${\text{MSD}} = \left( {x - x_{0} } \right)^{2} = \frac{1}{N}\mathop \sum \limits_{n = 1}^{N} \left( {x_{n} \left( t \right) - x_{n} \left( 0 \right)} \right)^{2}$$
(3)

where N is the number of particles to be averaged, $$x_{n} \left( 0 \right) = x_{0}$$ is the reference position of each particle, and $$x_{n} \left( t \right)$$ is the position of each particle in the determined time t. Thus, the directed motion of diffusion is given by Eq. (3).

$$r^{2} = 4Dt + \left( {Vt} \right)^{2}$$
(4)

where $$r^{2}$$ is the mean-square displacement, D is the simultaneous diffusion, t is the time, and V is velocity.

$${\text{MSD}} = x^{2} = 4Dt + \left( {Vt} \right)^{2}$$
(5)

By comparing Eqs. (4) and (5)50, we obtain the MSD, which can be used to derive the water diffusion coefficients as a function of time in the logarithmic scale in ps for pure water without the enzyme as the reference, the corresponding channel waters. Then the corresponding D can be plotted as the function of the inverse temperature in the logarithm scale, 1/T, respectively.

### Approach to temperature dependence of water diffusion kinetics

The more literature refers that the temperature-dependent rate is to consider deviations from the Arrhenius law so that it can more accurately simulate the diverse phenomena that modern chemical kinetics need to solve. It is outlined how such deviations manifest as concave or convex Arrhenius plots and can be referred to as sub- and super-Arrhenius behavior, respectively. In Fig. 2, the logarithm of the observed respiration rate against the reciprocal of the absolute temperature T seems to deviate from a straight line, as expected according to the Arrhenius law,

$$k\left( T \right) = A \exp \left( { - \frac{{E_{a} }}{RT}} \right)\quad {\text{or}}\quad \ln k\left( T \right) = \ln A - \frac{{E_{a} }}{RT}$$
(6)

where A is the pre-exponential factor and Ea as the activation energy. Ea is assumed to be independent of temperature in Eq. (6). However, in most situations of solution biochemistry interest, the accessible temperature rate measurement turns out to be a curve, rather than the straight line expected according to the Arrhenius law13. In Ref.8, deviations from linearity in an Arrhenius plot are accounted for by an additional quadratic term in 1/RT according to Eq. (7).

$$\ln k\left( T \right) = \ln A + \frac{B}{RT} + \frac{C}{{\left( {RT} \right)^{2} }}$$
(7)

where C dictates whether the plot is ‘concave’ or ‘convex’12,13,18, and the reaction abides by a sub- or super-Arrhenius dependence, such that the activation energy Ea, being connected to the slope of the Arrhenius plot, as Eq. (6) shows, is no longer a constant but depends linearly on 1/T, by Eq. (8).

$$E_{a} \left( T \right) = - \frac{d \ln k}{{d \left( \frac{1}{RT} \right)}} = - B - \frac{2C}{{RT}}$$
(8)

here the C coefficient, which is given by Eq. (9), is negative or positive for the sub- or super-Arrhenius cases, respectively.

$$C = \frac{1}{2}\frac{{d^{2} \ln k}}{{d \left( \frac{1}{RT} \right)^{2} }}$$
(9)

Aquilanti et al. provide a molecular level examination to endorse the reciprocal-activation-reciprocal-temperature relationship12,17, by extending the Tolman theorem12,17, and following the interpretation by Truhlar et. al.12,17. The function Ea, is written as the logarithmic derivative of the rate constants with respect to 1/RT (Eq. 8). The specific transitivity can be interpreted by defining the reciprocal as a measure of the propensity of the reaction progress12,17 to proceed the process. Since we demonstrate a description of the rate departures upon temperature to the McLaurin and Taylor series, considered as a differential equation for k(1/RT), is of first order in the variable (1/RT) and is easily integrated, specifying the lower limit of the integration range, yielding the deformed Arrhenius. This can be described from the thermodynamic limit as due to the interruption of a discrete temporal sequence of events12,17. One can understand this considering that to accumulate the respiration reaction process needs a sequence of favorable collisions and particle diffusion transports.

Let’s elaborate further on the extension to the deformed exponential rate equation (Eq. 7) for the connection between super-Arrhenius and water fluid diffusion kinetics, which adequately describes the relationship between the diffusion coefficient and temperature. In general, for particles in an ideal gas the diffusion coefficient can be predicted from first principles by taking the values for the mean free path and average velocity from the Maxwell–Boltzmann distribution; for molecules in a viscous fluid (or a liquid solution), they obey Stokes–Einstein equation. Note that for solvent molecules under a specific confinement the viscosity itself strongly depends on temperature, and the diffusion coefficient D(T) normally obeys the relevant Arrhenius relation, Eq. (10)

$$D\left( T \right) = D_{0} \exp \left( { - \frac{{E_{a} }}{RT}} \right)$$
(10)

Here, Ea is an "activation energy of diffusion"; in a super Arrhenius-like case assume that diffusion coefficients in the solution phase grow faster than exponentally with temperature and a law Eq. (11)

$$\ln D\left( T \right) = \ln D_{0} + \frac{B}{RT} + \frac{C}{{\left( {RT} \right)^{2} }},\quad E_{a} \left( T \right) = - \frac{d \ln D}{{d \left( \frac{1}{RT} \right)}} = - B - \frac{2C}{{RT}}$$
(11)

similar to Eq. (7): the fact is here emphasized that the transmissibility may not be constant, and the temperature-dependent diffusion coefficient D(T) can be described according to a deformed Arrhenius type of formula, so that low order derivatives of $$\ln D$$ with respect to 1/RT can be extracted by molecular dynamics simulation and compare to experiments when available12,17,18,19. Thus the parameters measure deviations between equilibrium and non-equilibrium situations leading to the continuous generalization of the diffusion states as series of replicas of microscopic ‘entities’, the specific configurations depending on temperature, The coefficient B and C account for transport behavior attributed to the occurrence of long series of events. Collective phenomena such as the rate of the enzymatic catalysis process, food preservation process, and the basic features of the dynamics of complex or glass-forming liquids and solids have exhibited Super Arrhenius behavior12,17,18,19: a significant number of studies on the temperature dependence of rates of biophysical reactions have been analyzed and found to manifest the super-Arrhenius kinetics, where transport phenomena accelerate as the temperature increases faster than according to the Arrhenius law. The role of transitivity12,17,18,19 is a promising target of further studies.


text:
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text:
# to study the extraction of fe from its ore

## Extraction of metals from saprolitic laterite ore through ...

2015-10-1 · In this study, pressure hydrochloric-acid selective leaching was used to treat saprolitic laterite ore for extracting Ni and Co. A flowchart of the treatment process is shown in Fig. 1.Ni and Co were leached from laterite, while the majority of Fe remained in the residue in the form of FeOOH/Fe 2 O 3.We studied various factors affecting the leaching of nickel, cobalt, manganese, iron, and ...

## Extraction of Iron - QS Study

2021-9-8 · Extraction of Iron . Iron is the most widely used metal in the world, mainly in the form of steel. The main. ore of iron is called haematite. It is a mineral composed of iron(III) oxide, Fe 2 O 3.. About 4.12% (or 4.15%) of earth’s crust is Iron.

## EXTRACTION OF IRON- EXTRACTION OF IRON FROM

2012-7-24 · Extraction of iron is based on the reduction of HAEMATITE (Fe 2 O 3) with carbon.. The process of the extraction of iron is carried out by the following steps: In this metallurgical operation, the ore is concentrated by removing impurities like soil etc. The. process involves the crushing and washing of ore.

## Extraction of Metals - Methods of Extraction of Metals ...

The extraction of metals from ores is what allows us to use the minerals in the ground! The ores are very different from the finished metals that we see in buildings and bridges. Ores consist of the desired metal compound and the impurities and earthly substances called Gangue. The extraction of metals and its

## Extraction of Iron & Concentration of Ore - An Overview

2019-9-23 · The extraction of iron begins with concentration through calcination which is a process in a limited supply of oxygen applied to ores to bring thermal decomposition. An ore

## Solved: A Key Step In The Extraction Of Iron From Its Ore ...

A key step in the extraction of iron from its ore is shown below. FeO (s) + CO (g) Fe (s) + CO2 (g) Kp = 0.403 at 1000°C This step occurs in the 700° to 1200°C zone within a blast furnace. What are the equilibrium partial pressures of CO (g) and CO2 (g) when 2.23 atm CO (g) and excess FeO (s) react in a sealed container at 1000°C?

## Notes On Extraction Of Iron From Its Oxides - CBSE Class ...

Detailed explanation with examples on extraction-of-iron-from-its-oxides helps you to understand easily , designed as per NCERT. QnA , Notes & Videos ... the ore from which iron to be extracted, is concentrated by roasting or calcination impurities can be eliminated. ... (FeO, Fe), and downwards for the oxidation of carbon to carbon monoxide(C,CO).

## UNIT METALLURGY 1

2019-4-9 · The extraction of crude metals from the concentrated ores is carried out in two steps namely, (i) conversion of the ore into oxides of the metal of interest and (ii) reduction of the metal oxides to elemental metals. In the concentrated ore, the metal exists in positive oxidation state and hence it is to be reduced to its elemental state.

## Solved: A Key Step In The Extraction Of Iron From Its Ore ...

A key step in the extraction of iron from its ore is shown below. FeO(s) + CO(g) Fe(s) + CO 2 (g) K p = 0.403 at 1000°CThis step occurs in the 700° to 1200°C zone within a blastfurnace. What are the equilibrium partial pressures ofCO(g) and CO 2 (g) when 1.38 atmCO(g) and excess FeO(s) react in a sealedcontainer at 1000°C?

## A GCSE/IGCSE random multiple choice science

2021-3-9 · Stages in the extraction of iron in a blast furnace are described below after iron ore, coke and limestone have been added at the top, but not in the correct order. (1) carbon dioxide reacts with coke (carbon) to form carbon monoxide (2) limestone combines with

## Extraction of Metals - Methods of Extraction of Metals ...

The extraction of metals from ores is what allows us to use the minerals in the ground! The ores are very different from the finished metals that we see in buildings and bridges. Ores consist of the desired metal compound and the impurities and earthly substances called Gangue. The extraction of metals and its

## Extraction of Iron & Concentration of Ore - An Overview

2019-9-23 · The extraction of iron begins with concentration through calcination which is a process in a limited supply of oxygen applied to ores to bring thermal decomposition. An ore of a metal is an impure source. It consists of many impurities other than metal

## Solved: A Key Step In The Extraction Of Iron From Its Ore ...

A key step in the extraction of iron from its ore is shown below. FeO (s) + CO (g) Fe (s) + CO2 (g) Kp = 0.403 at 1000°C This step occurs in the 700° to 1200°C zone within a blast furnace. What are the equilibrium partial pressures of CO (g) and CO2 (g) when 2.23 atm CO (g) and excess FeO (s) react in a sealed container at 1000°C?

## CBSE NCERT Notes Class 12 Chemistry Isolation of Elements

Calcination: - It is the process of converting an ore into its oxides by heating it strongly, below its melting point in a limited supply of air or in absence of air. During calcination, volatile impurities as well as organic matter and moisture are removed. Fe 2 O 3 x H 2 O (∆)à Fe

## Solved: A Key Step In The Extraction Of Iron From Its Ore ...

A key step in the extraction of iron from its ore is FeO (s) + CO (g) ⇌ Fe (s) + CO2 (g) Kp = 0.403 at 1,000°C This step occurs in the 700°C to 1,200°C zone within a blast furnace. What are the equilibrium partial pressures of CO (g) and CO2 (g) when 1.1100 atm of CO (g) and excess FeO (s) react in a sealed container at 1000°C?

## Extraction Of Metals: Metallurgy | Notes, Videos, QA and ...

Overview Introduction to Extraction Of Metals: Metallurgy,The process of extracting a metal in pure form from its ore is known as metallurgy. The process of treatment depends on upon the nature of the ore, impurities, and the metal.It is reacts to Concentration or Benefaction of the Ore,includes the Crushing and grinding of the ore,Gravity separation or Levigation, Magnetic Separation, Froth ...

## Extraction of Metal from Concentrated Ore | Definition ...

Concentration of ore is a step involved in the process of extraction of metal from its ore. Unwanted rocks and grit are eliminated from mineral ore in this process. Hydraulic washing, froth floatation process, magnetic separation and chemical separation are some of the techniques used for concentrating or purification of ores.

## Extraction of metals | Metals Quiz - Quizizz

Fe 2 O 3 + 3C -> 2Fe + 3CO. C + O 2 -> CO 2. Tags: Question 9 . SURVEY . ... Which statement is correct about extraction of iron from its ore? answer choices . Iron is more difficult to extract than copper. Iron is the easiest metal to extract. Iron cannot be extracted from its ore.

## extraction of titanium

2015-7-17 · Titanium extraction. The overall process. Titanium is extracted from its ore, rutile - TiO 2. It is first converted into titanium(IV) chloride, which is then reduced to titanium using either magnesium or sodium. Conversion of TiO 2 into TiCl 4. The ore rutile (impure titanium(IV) oxide) is heated with chlorine and coke at a temperature of about ...

## A GCSE/IGCSE random multiple choice science

2021-3-9 · Stages in the extraction of iron in a blast furnace are described below after iron ore, coke and limestone have been added at the top, but not in the correct order. (1) carbon dioxide reacts with coke (carbon) to form carbon monoxide (2) limestone combines with

## Extraction of Metals from Ores – Different Processes Involved

The basic extraction of metals from ores has the following steps. 1. Grinding and Crushing. The bigger chunks of the ore extracted are crushed and ground in ball mills and crushers. It helps to increase the surface area of the chunks for better chemical actions later. In technical terms, it is called pulverization.

## Solved: A Key Step In The Extraction Of Iron From Its Ore ...

A key step in the extraction of iron from its ore is shown below. FeO (s) + CO (g) Fe (s) + CO2 (g) Kp = 0.403 at 1000°C This step occurs in the 700° to 1200°C zone within a blast furnace. What are the equilibrium partial pressures of CO (g) and CO2 (g) when 2.23 atm CO (g) and excess FeO (s) react in a sealed container at 1000°C?

## CBSE NCERT Notes Class 12 Chemistry Isolation of Elements

Calcination: - It is the process of converting an ore into its oxides by heating it strongly, below its melting point in a limited supply of air or in absence of air. During calcination, volatile impurities as well as organic matter and moisture are removed. Fe 2 O 3 x H 2 O (∆)à Fe

## Solved: A Key Step In The Extraction Of Iron From Its Ore ...

A key step in the extraction of iron from its ore is FeO (s) + CO (g) ⇌ Fe (s) + CO2 (g) Kp = 0.403 at 1,000°C This step occurs in the 700°C to 1,200°C zone within a blast furnace. What are the equilibrium partial pressures of CO (g) and CO2 (g) when 1.1100 atm of CO (g) and excess FeO (s) react in a sealed container at 1000°C?

## Chemistry of Elements

2013-2-26 · Iron Iron ore reserves in the country are estimated at 1750 crore tonnes. Major iron ore mining is done in Goa, Madhya Pradesh, Bihar, Karnataka, Orissa and Maharashtra. Aluminium Its chief ore, bauxite is available in India in abundance. The annual production of bauxite is estimated to be more than 2 million tonnes.

## A key step in the extraction of iron from its ore is ...

Find step-by-step Chemistry solutions and your answer to the following textbook question: A key step in the extraction of iron from its ore is $\mathrm { FeO } ( \mathrm { s } ) + \mathrm { CO } ( \mathrm { g } ) \rightleftharpoons \mathrm { Fe } ( \mathrm { s } ) + \mathrm { CO } 2 ( \mathrm { g } ) \mathrm { Kp } = 0.403$ at $1000 ^ { \circ } \mathrm { C }$ This step occurs in the \$700 ...

## Extraction of metals | Metals Quiz - Quizizz

Fe 2 O 3 + 3C -> 2Fe + 3CO. C + O 2 -> CO 2. Tags: Question 9 . SURVEY . ... Which statement is correct about extraction of iron from its ore? answer choices . Iron is more difficult to extract than copper. Iron is the easiest metal to extract. Iron cannot be extracted from its ore.

## extraction of titanium

2015-7-17 · Titanium extraction. The overall process. Titanium is extracted from its ore, rutile - TiO 2. It is first converted into titanium(IV) chloride, which is then reduced to titanium using either magnesium or sodium. Conversion of TiO 2 into TiCl 4. The ore rutile (impure titanium(IV) oxide) is heated with chlorine and coke at a temperature of about ...

## A GCSE/IGCSE random multiple choice science

2021-3-9 · Stages in the extraction of iron in a blast furnace are described below after iron ore, coke and limestone have been added at the top, but not in the correct order. (1) carbon dioxide reacts with coke (carbon) to form carbon monoxide (2) limestone combines with

## explain the term metallurgy what are the steps involved in ...

2015-10-29 · The extraction of a metal from its ore depends on: 1. The type of ore being used. 2. The nature of the impurities present in the ore. 3. The degree of the reactivity of the metal that is to be extracted. Steps involved in Extraction Crushing and Grinding:


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# Is there any way to measure of Gaussian-ness?

I have some sampled data that has $1/f$ noise in it, with departures from the mean. These are long term departures. I could use something like a median filter but the window length would be longer than I would like.

• Is there any way to measure its departure from a Gaussian distribution?
• Or measure how 'Gaussian-like' a time series statstical sample is?
• The Gaussian distribution has some properties that are useful. You can start here. – Envidia Apr 3 '17 at 21:44
• I have troubles understanding was you really are looking for: a measure of gaussianity? Characterization of $1/f$ noise? Why is your original data Gaussian? Why does a median filter come into play? – Laurent Duval Apr 3 '17 at 21:54
• I have taken a few statistical courses and I'm aware of the math. I want to know what other people use – Voltage Spike Apr 4 '17 at 4:57

A classical measure of "gaussianity" is the kurtosis of your random variable (RV). Kurtosis is the forth order cumulant of a RV. Say $y$ is your RV with zero mean, the kurtosis can be defined as: $$kurt(y)=E[y^4] - 3(E[y^2])^2$$ If $y$ is gaussian, $E[y^4]=3(E[y^2])^2$ and therefore $$kurt(y)=0$$


text:
96-404 Erdos L., Solovej J.P.
Semiclassical eigenvalue estimates for the Pauli operator with strong non-homogeneous magnetic fields: II. Leading order asymptotic estimates (175K, LaTeX) Sep 9, 96
Abstract , Paper (src), View paper (auto. generated ps), Index of related papers

Abstract. We give the leading order semiclassical asymptotics for the sum of the negative eigenvalues of the Pauli operator (in dimension two and three) with a strong non-homogeneous magnetic field. As in \cite{LSY-II} for homogeneous field, this result can be used to prove that the magnetic Thomas-Fermi theory gives the leading order ground state energy of large atoms. We develop a new localization scheme well suited to the anisotropic character of the strong magnetic field. We also use the basic Lieb-Thirring estimate obtained in our companion paper \cite{ES-I}.

Files: 96-404.tex


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?

Free Version
Easy

# Closed Operations on R

ABSALG-JAXKNG

Identify which of the following binary operations are closed on $\mathbb{R}$. Note that the minimum binary operation outputs the minimum of two numbers.

For example, $\textrm{min}(-2, 3)=-2.$

Select ALL that apply.

A

B

Subtraction

C

Multiplication

D

Division

E

Minimum

F

None of the above operations are closed on $\mathbb{R}$.


text:
# 5.3 Time dilation

 Page 1 / 8
By the end of this section, you will be able to:
• Explain how time intervals can be measured differently in different reference frames.
• Describe how to distinguish a proper time interval from a dilated time interval.
• Describe the significance of the muon experiment.
• Calculate time dilation given the speed of an object in a given frame.

The analysis of simultaneity shows that Einstein’s postulates imply an important effect: Time intervals have different values when measured in different inertial frames. Suppose, for example, an astronaut measures the time it takes for a pulse of light to travel a distance perpendicular to the direction of his ship’s motion (relative to an earthbound observer), bounce off a mirror, and return ( [link] ). How does the elapsed time that the astronaut measures in the spacecraft compare with the elapsed time that an earthbound observer measures by observing what is happening in the spacecraft?

Examining this question leads to a profound result. The elapsed time for a process depends on which observer is measuring it. In this case, the time measured by the astronaut (within the spaceship where the astronaut is at rest) is smaller than the time measured by the earthbound observer (to whom the astronaut is moving). The time elapsed for the same process is different for the observers, because the distance the light pulse travels in the astronaut’s frame is smaller than in the earthbound frame, as seen in [link] . Light travels at the same speed in each frame, so it takes more time to travel the greater distance in the earthbound frame.

## Time dilation

Time dilation is the lengthening of the time interval between two events for an observer in an inertial frame that is moving with respect to the rest frame of the events (in which the events occur at the same location).

To quantitatively compare the time measurements in the two inertial frames, we can relate the distances in [link] to each other, then express each distance in terms of the time of travel (respectively either $\text{Δ}t$ or $\text{Δ}\tau$ ) of the pulse in the corresponding reference frame. The resulting equation can then be solved for $\text{Δ}t$ in terms of $\text{Δ}\tau .$

The lengths D and L in [link] are the sides of a right triangle with hypotenuse s . From the Pythagorean theorem,

${s}^{2}={D}^{2}+{L}^{2}.$

The lengths 2 s and 2 L are, respectively, the distances that the pulse of light and the spacecraft travel in time $\text{Δ}t$ in the earthbound observer’s frame. The length D is the distance that the light pulse travels in time $\text{Δ}\tau$ in the astronaut’s frame. This gives us three equations:

$2s=c\text{Δ}t;2L=v\text{Δ}t;2D=c\text{Δ}\tau .$

Note that we used Einstein’s second postulate by taking the speed of light to be c in both inertial frames. We substitute these results into the previous expression from the Pythagorean theorem:

عند قذف جسم إلى أعلى بسرعة إبتدائية فإنه سيصل إلى ارتفاع معين (أقصى ارتفاع) ثم يعود هابطاً نحو سطح الأرض .   إذا قُذِفَ جسم إلى أعلى ووجد أن سرعته 18 م / ث عندما قطع 1/4 المسافة التي تمثل أقصى ارتفاع سيصله فالمطلوب إيجاد السرعة التي قُذِف بها بالمتر / ث . إن هذه السرعة هي واحدة من الإجابات التالية
what is light
light is a kind of radiation That stimulates sight brightness a source of illumination.
kenneth
Electromagnet radiation creates space 7th, 8th, and 9th dimensions at the rate of c.
John
That is the reason that the speed of light is constant.
John
This creation of new space is "Dark Energy".
John
The first two sets of three dimensions, 1 through 6, are "Dark Matter".
John
As matter decays into luminous matter, a proton, a neutron, and an electron creat deuterium.
John
There are three sets of three protons, 9.
John
There are three sets of three neutrons, 9.
John
A free neutron decays into a proton, an electron, and a neutrino.
John
There are three sets of five neutrinoes, 15.
John
Neutrinoes are two dimensional.
John
A positron is composed of the first three dimensions.
John
An electron is composed of the second three dimensions.
John
What is photoelectric
light energy (photons) through semiconduction of N-P junction into electrical via excitation of silicon purified and cristalized into wafers with partially contaminated silicon to allow this N-P function to operate.
Michael
i.e. Solar pannel.
Michael
If you lie on a beach looking at the water with your head tipped slightly sideways, your polarized sunglasses do not work very well.Why not?
it has everything to do with the angle the UV sunlight strikes your sunglasses.
Jallal
this is known as optical physics. it describes how visible light, ultraviolet light and infrared light interact when they come into contact with physical matter. usually the photons or light upon interaction result in either reflection refraction diffraction or interference of the light.
Jallal
I hope I'm clear if I'm not please tell me to clarify further or rephrase
Jallal
what is bohrs model for hydrogen atom
hi
Tr
Hello
Youte
Hi
Nwangwu-ike
hi
Siddiquee
hi
Omar
helo
Mcjoi
what is the value of speed of light
1.79×10_¹⁹ km per hour
Swagatika
what r dwarf planet
what is energy
কাজের একক কী
Jasim
কাজের একক কী
Jasim
Energy is ability so capacity to do work.
kenneth
friction ka direction Kaise pata karte hai
friction is always in the opposite of the direction of moving object
Punia
A twin paradox in the special theory of relativity arises due to.....? a) asymmetric of time only b) symmetric of time only c) only time
b) symmetric of time only
Swagatika
fundamental note of a vibrating string
every matter made up of particles and particles are also subdivided which are themselves subdivided and so on ,and the basic and smallest smallest smallest division is energy which vibrates to become particles and thats why particles have wave nature
Alvin
what are matter waves? Give some examples
according to de Broglie any matter particles by attaining the higher velocity as compared to light'ill show the wave nature and equation of wave will applicable on it but in practical life people see it is impossible however it is practicaly true and possible while looking at the earth matter at far
Manikant
a centeral part of theory of quantum mechanics example:just like a beam of light or a water wave
Swagatika
Mathematical expression of principle of relativity
given that the velocity v of wave depends on the tension f in the spring, it's length 'I' and it's mass 'm'. derive using dimension the equation of the wave
What is the importance of de-broglie's wavelength?
he related wave to matter
Zahid
at subatomic level wave and matter are associated. this refering to mass energy equivalence
Zahid
it is key of quantum
Manikant


text:
ghc-9.2.5: The GHC API

GHC.Tc.Solver

Synopsis

# Documentation

data InferMode Source #

How should we choose which constraints to quantify over?

Constructors

 ApplyMR Apply the monomorphism restriction, never quantifying over any constraints EagerDefaulting See Note [TcRnExprMode] in GHC.Tc.Module, the :type +d case; this mode refuses to quantify over any defaultable constraint NoRestrictions Quantify over any constraint that satisfies pickQuantifiablePreds

#### Instances

Instances details
 Source # Instance detailsDefined in GHC.Tc.Solver Methods

solveEqualities :: String -> TcM a -> TcM a Source #

Type-check a thing that emits only equality constraints, solving any constraints we can and re-emitting constraints that we can't. Use this variant only when we'll get another crack at it later See Note [Failure in local type signatures]

Panics if we solve any non-equality constraints. (In runTCSEqualities we use an error thunk for the evidence bindings.)

Return (Just new_inerts) if the Givens are satisfiable, Nothing if definitely contradictory

Return True if the Wanteds are soluble, False if not

Normalise a type as much as possible using the given constraints. See Note [tcNormalise].

Simplify top-level constraints, but without reporting any unsolved constraints nor unsafe overlapping.

runTcSDeriveds :: TcS a -> TcM a Source #

This variant of runTcS will keep solving, even when only Deriveds are left around. It also doesn't return any evidence, as callers won't need it.


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After many discussions scientists from AstraZeneca’s knowledge graph and target prioritization platform (BIKG) about the PyKEEN knowledge graph embedding model package, I joined them in writing a review on biomedical knowledge graphs. I’m giving a talk in their group tomorrow - this blog post is a longer form of some ideas I’ll be presenting there. Here are the slides.

### Who is the Arbiter of Benchmarks?

The publication of each new knowledge graph embedding model is usually accompanied by an evaluation of the link prediction task on common benchmark datasets and a comparison to previous models’ performances on those datasets. There’s a variety of ways authors report this, either by copying the original results from the original papers describing the competitor models, by re-running the author’s original code, by reimplementing the code themselves, or by running an implementation from a larger knowledge graph embedding model package.

Who can be trusted? The original authors of each competitor model were/are motivated to show that theirs was better than others, so their paper might have inflated metrics. The authors of the current paper are motivated to show that previous models are worse, so they may deflate metrics for old models (and inflate theirs). Knowledge graph embedding model packages are motivated to inflate metrics to show that using their package is better – I’ve indeed been asked questions about why other packages are “performing better” than PyKEEN and had to answer that they’re just evaluating differently that is less conservative, and sometimes misleading. Even in a world without these competing interests, it’s very hard to understand each other’s code, especially since it’s not just about the model itself but also the training and evaluation pipeline.

This motivates large benchmarking studies that use unified frameworks and re-implementations of models to investigate the relative performances of models (and can for a brief moment, let you throw away your metric fixation). The PyKEEN benchmark is the largest and most comprehensive to date, covering many models, datasets, and configurations. I was proudly a part of this work under the leadership of Mehdi Ali @mali-git and with an amazing team. There’s also the smaller LibKGE benchmark as well as several others that we’re maintaining in a curated list in this meta-review repository.

The Open Graph Benchmark (OGB) provides a potential solution to these issues by externalizing and standardizing the code that evaluates the results. There are still sneaky ways to game this that I won’t describe in detail, but I like the concept. However, the framework does not cover some of the more popular datasets and the ones that it does introduce in the link prediction task, it does not do so with any provenance on how they’re created. During my talk, Benedek Rozemberczki (@benedekrozemberczki) brought up another interesting point - many of the benchmarks in OGB have surpassed the size that can be addressed by the physical infrastructure or cloud compute budgets of many small academic institutions. As machine learning continues to rely on larger, more exotic models, we will soon have to reckon with this divide.

### Testing/Validation Leakage

Leakage is when triples in the testing/validation sets can be trivially inferred from triples in the training set. This leads to an over-estimation in the performance of the model.

The most common form of leakage occurs when a knowledge graph contains inverse triples. An example:

• A and B are entities in a knowledge graph
• part of and has part are inverse relations in the knowledge graph
• A part of B is in the training set
• B has part A is in the testing set

This scenario occurs in two datasets widely used in knowledge graph embedding model benchmarking: the FreeBase 15K (FB15k) dataset proposed by Bordes et al. (2013) alongside the TransE model and the WordNet-18 ( WN18) dataset proposed by Bordes et al. (2014) alongside the Unstructured Model. Toutanova and Chen (2015) outlined an algorithm for removing leakage due to inverse triples and applied it to respectively derive two new datasets: the Freebase 15K-237 (FB15k-237) dataset and the WordNet-18 (RR; WN-18RR) dataset.

Dataset Entities Relations Triples
WN-18 40,943 18 151,442
WN-18RR 40,559 11 92,583
FB15k 14,951 1,345 592,213
FB15k-237 14,505 237 310,079

These datasets are of roughly the same size and are widely available. Therefore, if I read a paper that presents results on only FB15k and WN-18 but not FB15k-237 and WN-18RR, I am concerned that the authors either do not know about this problem, or are intentionally misleading the reader.

An alternative kind of testing leakage can occur in chemoinformatic applications. The following three chemical structures are from a structural class called statins (CHEBI:87631) that inhibit the HMG-CoA reductase enzyme and are often used to lower blood pressure to reduce the risk of heart disease.

All three share a stereospecific cyclic lactone motif (top right part of the molecule) that result in their functional activity. If you’re building a classifier for activity against HMG-CoA reductase, or more generally, a link prediction model that includes chemical-protein or chemical-enzyme class predictions, including some of these molecules in the training set and others in the test set would enable trivial inference and ultimately result in an over-estimation of the performance of your model (barring discussion about how this could be considered trivial is a huge triumph of modern pharmacology and chemoinformatics). The solution often referenced in proteochemometrics papers is to use scaffold-based splitting like Bemis-Murcko scaffold splitting or to pre-cluster molecules based on structural similarity and to split based on clusters. More information on these methods is reviewed in Bongers et al. (2019) and Parks et al. (2020).

### Redundant Information in Graphs

Transcriptomics experiments are often analyzed with statistical over-representation analysis or gene set enrichment analysis (GSEA) with respect to gene sets derived from pathway databases like KEGG, Reactome, WikiPathways, the Pathway Interaction Database, and BioCyc, functional annotations such as those from the Gene Ontology, and biological signatures such as those listed in MSigDB. In addition to p-values, the high number of statistical tests also motivates the reporting of adjusted p-values to address multiple hypothesis testing. However, these adjustments do not consider the high overlap in information due to the redundancy of pathway databases or potential correlations between expert-defined pathways and certain functional or biological signatures. Therefore, several results from the same theme or biological phenomena often co-occur as enriched which often leads to an inflated sense of concordance of results and thus higher confirmation bias. Further, Mubeen et al. (2019) showed that the variability of definitions in each pathway database for the “same” pathway (e.g., apoptosis) caused significantly different results in many downstream tasks.

Many benchmark biological knowledge graphs such as Hetionet and OpenBioLink include multiple pathway databases. Therefore, link prediction tasks between genes and pathways (such as the case scenario presented in Ali et al., 2019) could be skewed both during the training and evaluation of link prediction models, and ultimately in the interpretation of predicted results. While methodological improvements like SetRank (Simillion et al., 2017) have been proposed to mitigate the redundancies in the over-representation analysis setting, there have not yet been methodological improvements for link prediction models that consider these kinds redundancies. Further, these kinds of redundancies do not fit as neatly into the “leakage” issue.

Tools like ComPath can be used to enrich KGs with links between functionally equivalent pathways, but this kind of information is not readily used by typical knowledge graph embedding models. Alternative rule-based and symbolic reasoning systems might present solutions in this area.

### The Effect of Non-Determinism

There are several meaningful non-deterministic factors that affect the performance of a model:

1. The effect of random initialization. The commonly used pytorch.nn.Embedding initializes its weightings by default with the vanilla pytorch.nn.init.normal_ function. Using alternative parameters for the distribution (e.g., Xavier/Glorot) have been shown to improve performance in a variety of settings.
2. The effect of non-determinism during training that occurs during batch shuffling. This has shown to quicken convergence and mitigate overfitting based on the ordering of training data.
3. The effect of random negative sampling when training under the stochastic local closed world (sLCWA) assumption.

To compensate for the non-determinism, several trials using the same settings should be run and the distribution of each metric (or at minimum, summary statistics over their means and standard deviations) should be presented. Below, the distribution of mean ranks resulting from 10 trials of training of several models on the Hetionet dataset are presented (credit to Stephen Bonner @sbonner for running this trial and generating the chart).

From this chart, it is obvious that the DistMult model has the potential to perform both very well and very poorly based on the random initialization. There are two reasons that this might be the case for this model/dataset in particular:

1. The interaction for DistMult function uses the Hadamard operators, which commute, so f(h,r,t) = f(t,r,h). With the Hetionet dataset, where there is meaningful directionality and also the existence of inverse edges, this could be a liability to both meaningful training and accurate evaluation
2. The L_1 norm formulation of DistMult makes it much more susceptible to random initialization than if the L_2 norm were specified. Because the original formulation of the DistMult model uses the idiom of matrix factorization, the norm’s p is not configurable within PyKEEN (the tool used to train these models and generate these charts).

While there are a few insights into the formulations of models and the properties of datasets that could inform users to avoid using models like DistMult with datasets like Hetionet, the more general case is likely more bleak - even small changes in hyper-parameters could potentially affect the robustness of a model to non-determinism. However, there does not yet exist a comprehensive benchmarking study (even over small KGs) to confirm this nor elucidate general insight.

As an end note - this is not meant to say that non-determinism should be mitigated nor removed. It is a vital part to creating meaningful results and fixing the random seed is certainly not a viable solution.

### Splits Aren’t Gospel

Benchmark datasets for knowledge graph embedding models typically consist of a pre-stratified training, testing, and validation set. The FB15k, FB15k-237, WN18, WN18-RR, and YAGO3-10 are the most commonly used to assess the performance of new models or compare the performance of old models due to their relatively small sizes. More specifically, the same exact pre-stratification is used in all papers.

While this potentially could make results more comparable across different experiments, it presents a huge liability in that it is another “random” effect that is completely discarded. Rather than present results on a single split, results should be presented as a distribution over several splits to show if there are certain properties of the split that cause some models to perform better than others. Perhaps this kind of evaluation would have informed Bordes et al. about the issues with the FB15k and WN18 datasets before they were later identified by other authors. This also presents a practical problem since there are potentially many ways to generate dataset splits, and typical knowledge graph embedding model researchers likely aren’t interested in adding another level of cross-validation to their already complex pipelines.

This issue is more apparent when applying knowledge graph embedding models to datasets that are not pre-stratified such as the Common Sense Knowledge Graph, NELL, or any biomedical knowledge graph. It can again be demonstrated with another image from Stephen Bonner. Below, the distribution of mean ranks resulting from 10 random splits of the Hetionet dataset and using the same training configuration for several models.

The confidence intervals support an even more startling conclusion - the effect of random splitting of the dataset could cause a model like DistMult to outperform models like ComplEx due to its huge variance. Further, datasets like Hetionet definitely are susceptible to the leakage problem. Note: there’s a prototype implementation of the Toutanova and Chen algorithm for removing leakages in PyKEEN, but it’s neither finished nor integrated in the PyKEEN pipeline. Combine with DistMult’s penchant for modeling inverse relations the same as the canonical ones, this could wreak havoc on its performance depending on how bad the leakage was by chance. Like with random initialization, there has not yet been a comprehensive benchmarking that could give more general insight into the liabilities of not reporting distributions of results over several random splits. Even further, combine with the previous section’s prescription to report distribution over multiple random initializations, doing a “proper” investigating of robustness seems to be getting more high dimensional.

## Do KGEMs Actually Learn?

The formulation of the link prediction task on knowledge graphs as a binary classification task over a dataset of true positives and randomly sampled negatives (under the local closed world assumption) is already a concession to many more obvious machine learning tasks (like classifying Mendel’s irises). Even further, many machine learning techniques have the concept of a null model against which they can compare if their results are meaningful, or just good by chance. In classification tasks, this is often done with y-scrambling which compares the evaluation metric obtained from the real dataset to the evaluation metrics obtained by repetitively shuffling the response variable and generating a null distribution of evaluation metrics against which the real one can be compared. Ultimately, the p-value of a one-tailed test against this distribution tells you if you actually learned something, or if your results are the same as what you’d get with a random dataset.

While this is a pretty typical test that reviewers ask for in other machine learning tasks, it has not (yet) penetrated the knowledge graph embedding model domain. I think this likely limited by the difficulty in writing performant research code. While tools like PyTorch BigGraph and Amazon’s DGL are able to scale across many machines or GPUs, they are not accessible to anyone without significant dev-ops abilities nor are they frameworks that could support researchers. PyKEEN itself has the opposite problem (for now, get hyped for new improvements in 2021). A solution might lie in Marius, a new knowledge graph embedding model library that could be the best of both worlds. Assuming these weren’t considerations, there are two themes through which I could see the question “did I learn anything?”: knowledge graph reorganization and knowledge graph deterioration.

The idea of reorganization is a bit more classic, coming from previous approaches. There are a few concepts that could be used to generate derived knowledge graphs based on a given one that could be use to generate null networks:

1. Degree-preserving network shuffle (pmc:PMC3307026; pmc:PMC5168876)
2. Xswap (paper; code)
3. Relabeling of nodes, similarly to the y-scrambling technique

Himmelstein et al (2017) provides a gold standard on how these techniques could be used to report on how meaningful results are from a link prediction model on a knowledge graph.

The idea of deterioration encompasses randomly removing edges from the knowledge graph and training. It could determine how much of knowledge graph is actually needed to train the model to make meaningful prediction. This would be done for multiple increments (e.g., remove 10% of the training set, remove 20% of the training set) with multiple trials on each.

### Are KGEMs King Now?

The COVID-19 pandemic has not only been a battle for public health, but it has also resulted in the largest wave of low quality scientific work in recent memory. Lots of this low-quality work included half-baked usage of network representation learning, matrix completion, and for the first time, knowledge graph embedding models in the mainstream to predict drug-disease links. It was lacking on many fronts: the formulations of knowledge graphs, the evaluation of training and link prediction, and the communication of results.

As a critique, I will share with you my own drug repositioning algorithm for COVID-19: take the list of all drugs in DrugBank, and randomly select 100 of them. You’d be surprised how competitive this algorithm is with others.

The more upsetting examples was from a very prominent network science group that used the RotatE (and other standard) knowledge graph embedding models for the link prediction task for drug repositioning for COVID-19. They evaluated the task on its ability to recover drugs that had already gone into clinical trials for COVID-19, and claimed victory because they were able to recapitulate enough of the list. This was a bad evaluation because the drugs in the clinic were there based on inference made on the same data used to train and evaluate the model. More importantly, it was bad because it was a famous group that misrepresented the way science is done - link prediction in biomedicine is a way to generate hypotheses that can be tested in the lab. Focusing communication around its ability to recapitulate what’s in the training set (i.e., overfitting) is one of the first lessons that students are taught to avoid in their introduction to machine learning classes.

Does this mean that knowledge graph embedding models are king now? No - there’s still a long way to go in biomedicine before they are accepted as canonical tools, that must include some more high-profile papers in that include experimental evaluation of the results. This will most likely happen in the chemistry or target prioritization domains, where the predictions of the model can motivate experiments of actual value.

As a few of the vignettes in this post mentioned, there are still problems with model formulation that can cause huge performance deficits on biomedical knowledge graphs. A different class of models based on random walks, while less complex than knowledge graph embedding models, continue to perform well on tasks in biomedical knowledge graphs because their formulation is oriented towards extracting local community structure - a feature that is very important in biology. Several attempts have been made to extend these to better take into account directionality and edge type. Some of those generalizations even fit into the philosophy and architecture of graph neural networks (GNNs). Further, there are lots of quantitative data such as chemical-enzyme IC50 values, chemical-receptor EC50 values, GWAS p-values, etc. that all have widely different scales and dynamic ranges that are difficult to simultaneously incorporate in any of these methods in a meaningful way.

### Willkommen in der Zukunft (The Future)

Random parting thoughts:

Biomedical knowledge graph benchmarks are consistently topping out around 5 million edges. If I had to guess, that’s because that’s the number where if you go higher, you have to start using some packages that are not meant for users like Amazon’s DGL or PyTorch’s BigGraph. We’ll need more accessible, scalable systems if we’re ever going to move on from drug repositioning, where there are only on the scale of thousands of chemicals against hundreds of proven targets, to bona fide drug discovery, where there are on the scale of billions of chemicals (or more if you want to get fancy with virtual screening) against potentially the whole genome of ~20K protein coding genes, the products of other non-protein-coding genes, and other stuff floating around in and around the cell.

Negative knowledge should be emphasized. Formats like the Biological Expression Language (BEL) should support its curation, natural language processing systems and tools like INDRA should start extracting in bulk, and the concept of the open world assumption, local closed world assumption, and closed world assumption need to be amended to better reflect the reality that we know some negative knowledge, but it’s distributed across different relations in different densities.

Genes, RNAs, and proteins are not the same! Even worse, they can have several variants due to differential splicing, and they can have all sorts of pre- and post-translational modifications. BEL and related formats like BioPAX can represent these, but the knowledge graph approach to biology tends to throw this away completely. On the other hand, high granular information on this level of specificity is likely too low to make dense knowledge graphs over which typical tasks like link prediction can be successful. This motivates a greater focus on modeling the hierarchies of information (a task at which INDRA excels) to generate knowledge graphs that balance these competing qualities.

The entire semantic infrastructure for building knowledge graphs is improving (I’d like to say in some small part due to the tools I’ve built) that enables mappings between entities, such as Inspector Javert’s Xref database and the new SSSOM format to make mapping data more standardized.