Browsing by Author "Chu, Jhih-Wei"

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An incoherent feed-forward loop switches the Arabidopsis clock rapidly between two hysteretic states
(2018-09-17)
Joanito, Ignasius
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Chu, Jhih-Wei
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Wu, Shu-Hsing
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Hsu, Chao-Ping
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生物科技學系
In higher plants (e.g., Arabidopsis thaliana), the core structure of the circadian clock is mostly governed by a repression process with very few direct activators. With a series of simplified models, we studied the underlying mechanism and found that the Arabidopsis clock consists of type-2 incoherent feedforward loops (IFFLs), one of them creating a pulse-like expression in PRR9/7. The double-negative feedback loop between CCA1/LHY and PRR5/TOC1 generates a bistable, hysteretic behavior in the Arabidopsis circadian clock. We found that the IFFL involving PRR9/7 breaks the bistability and moves the system forward with a rapid pulse in the daytime, and the evening complex (EC) breaks it in the evening. With this illustration, we can intuitively explain the behavior of the clock under mutant conditions. Thus, our results provide new insights into the underlying network structures of the Arabidopsis core oscillator.
Analysis of Trajectory Entropy for Continuous Stochastic Processes at Equilibrium
(2014-07-17)
Haas, Kevin R.
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Yang, Haw
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Chu, Jhih-Wei
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生物科技學系
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生物資訊及系統生物研究所
The analytical expression for the trajectory entropy of the overdamped Langevin equation is derived via two approaches. The first route goes through the Fokker-Planck equation that governs the propagation of the conditional probability density, while the second method goes through the path integral of the Onsager-Machlup action. The agreement of these two approaches in the continuum limit underscores the equivalence between the partial differential equation and the path integral formulations for stochastic processes in the context of trajectory entropy. The values obtained using the analytical expression are also compared with those calculated with numerical solutions for arbitrary time resolutions of the trajectory. Quantitative agreement is clearly observed consistently across different models as the time interval between snapshots in the trajectories decreases. Furthermore, analysis of different scenarios illustrates how the deterministic and stochastic forces in the Langevin equation contribute to the variation in dynamics measured by the trajectory entropy.
Analysis of Trajectory Entropy for Continuous Stochastic Processes at Equilibrium
(2014-07-17)
Haas, Kevin R.
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Yang, Haw
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Chu, Jhih-Wei
;
生物科技學系
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生物資訊及系統生物研究所
The analytical expression for the trajectory entropy of the overdamped Langevin equation is derived via two approaches. The first route goes through the Fokker-Planck equation that governs the propagation of the conditional probability density, while the second method goes through the path integral of the Onsager-Machlup action. The agreement of these two approaches in the continuum limit underscores the equivalence between the partial differential equation and the path integral formulations for stochastic processes in the context of trajectory entropy. The values obtained using the analytical expression are also compared with those calculated with numerical solutions for arbitrary time resolutions of the trajectory. Quantitative agreement is clearly observed consistently across different models as the time interval between snapshots in the trajectories decreases. Furthermore, analysis of different scenarios illustrates how the deterministic and stochastic forces in the Langevin equation contribute to the variation in dynamics measured by the trajectory entropy.
Compound Molecular Logic in Accessing the Active Site of Mycobacterium tuberculosis Protein Tyrosine Phosphatase B
(2018-11-07)
Morrell, Thomas E.
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Rafalska-Metcalf, Ilona U.
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Yang, Haw
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Chu, Jhih-Wei
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生物資訊及系統生物研究所
Protein tyrosine phosphatase B (PtpB) from Mycobacterium tuberculosis (Mtb) extends the bacteria's survival in hosts and hence is a potential target for Mtb-specific drugs. To study how Mtb-specific sequence insertions in PtpB may regulate access to its active site through large-amplitude conformational changes, we performed free-energy calculations using an all-atom explicit solvent model. Corroborated by biochemical assays, the results show that PtpB's active site is controlled via an "either/or" compound conformational gating mechanism, an unexpected discovery that Mtb has evolved to bestow a single enzyme with such intricate logical operations. In addition to providing unprecedented insights for its active-site surroundings, the findings also suggest new ways of inactivating PtpB.
Coupling between Protein Conformation and Local Unfolding Highlights the Role of Disorder in Protein Function and Suggests a New Target for Tuberculosis Treatment
(2014-01-28)
Morrell, Thomas E.
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Rafalska-Metcalf, Ilona U.
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Chu, Jhih-Wei
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Yang, Haw
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交大名義發表
Dynamic mesoscale model of dipolar fluids via fluctuating hydrodynamics
(2014-11-07)
Persson, Rasmus A. X.
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Voulgarakis, Nikolaos K.
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Chu, Jhih-Wei
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生物科技學系
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生物資訊及系統生物研究所
Fluctuating hydrodynamics (FHD) is a general framework of mesoscopic modeling and simulation based on conservational laws and constitutive equations of linear and nonlinear responses. However, explicit representation of electrical forces in FHD has yet to appear. In this work, we devised an Ansatz for the dynamics of dipole moment densities that is linked with the Poisson equation of the electrical potential phi in coupling to the other equations of FHD. The resulting phi-FHD equations then serve as a platform for integrating the essential forces, including electrostatics in addition to hydrodynamics, pressure-volume equation of state, surface tension, and solvent-particle interactions that govern the emergent behaviors of molecular systems at an intermediate scale. This unique merit of phi-FHD is illustrated by showing that the water dielectric function and ion hydration free energies in homogeneous and heterogenous systems can be captured accurately via the mesoscopic simulation. Furthermore, we show that the field variables of phi-FHD can be mapped from the trajectory of an all-atom molecular dynamics simulation such that model development and parametrization can be based on the information obtained at a finer-grained scale. With the aforementioned multiscale capabilities and a spatial resolution as high as 5 angstrom, the phi-FHD equations represent a useful semi-explicit solvent model for the modeling and simulation of complex systems, such as biomolecular machines and nanofluidics. (C) 2014 AIP Publishing LLC.
Endoglucanase Peripheral Loops Facilitate Complexation of Glucan Chains on Cellulose via Adaptive Coupling to the Emergent Substrate Structures
(2013-09-19)
Lin, Yuchun
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Beckham, Gregg T.
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Himmel, Michael E.
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Crowley, Michael F.
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Chu, Jhih-Wei
We examine how the catalytic domain of a glycoside hydrolase family 7 endoglucanase catalytic domain (Cel7B CD) facilitates complexation of cellulose chains from a crystal surface. With direct relevance to the science of biofuel production, this problem also represents a model system of biopolymer processing by proteins in Nature. Interactions of Cel7B CD with a cellulose microfibril along different paths of complexation are characterized by mapping the atomistic fluctuations recorded in free-energy simulations onto the parameters of a coarse-grain model. The resulting patterns of protein-biopolymer couplings also uncover the sequence signatures of the enzyme in peeling off glucan chains from the microfibril substrate. We show that the semiopen active site of Cel7B CD exhibits similar barriers and free energies of complexation over two distinct routes; namely, scooping of a chain into the active-site cleft and threading from the chain end into the channel. On the other hand, the complexation energetics strongly depends on the surface packing of the targeted chain and the resulting interaction sites with the enzyme. A revealed principle is that Cel7B CD facilitates cellulose deconstruction via adaptive coupling to the emergent substrate. The flexible, peripheral segments of the protein outside of the active-site cleft are able to accommodate the varying features of cellulose along the simulated paths of complexation. The general strategy of linking physics-based molecular interactions to protein sequence could also be helpful in elucidating how other protein machines process biopolymers.
Expectation-Maximization of the Potential of Mean Force and Diffusion Coefficient in Langevin Dynamics from Single Molecule FRET Data Photon by Photon
(2013-12-12)
Haas, Kevin R.
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Yang, Haw
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Chu, Jhih-Wei
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生物科技學系
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生物資訊及系統生物研究所
The dynamics of a protein along a well-defined coordinate can be formally projected onto the form of an overdamped Lagevin equation. Here, we present a comprehensive statistical-learning framework for simultaneously quantifying the deterministic force (the potential of mean force, PMF) and the stochastic force (characterized by the diffusion coefficient, D) from single-molecule Forster-type resonance energy transfer (smFRET) experiments. The likelihood functional of the Langevin parameters, PMF and D, is expressed by a path integral of the latent smFRET distance that follows Langevin dynamics and realized by the donor and the acceptor photon emissions. The solution is made possible by an eigen decomposition of the time-symmetrized form of the corresponding Fokker-Planck equation coupled with photon statistics. To extract the Langevin parameters from photon arrival time data, we advance the expectation-maximization algorithm in statistical learning, originally developed for and mostly used in discrete-state systems, to a general form in the continuous space that allows for a variational calculus on the continuous PMF function. We also introduce the regularization of the solution space in this Bayesian inference based on a maximum trajectory-entropy principle. We use a highly nontrivial example with realistically simulated smFRET data to illustrate the application of this new method.
Fisher information metric for the Langevin equation and least informative models of continuous stochastic dynamics
(2013-09-28)
Haas, Kevin R.
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Yang, Haw
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Chu, Jhih-Wei
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生物科技學系
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生物資訊及系統生物研究所
The evaluation of the Fisher information matrix for the probability density of trajectories generated by the over-damped Langevin dynamics at equilibrium is presented. The framework we developed is general and applicable to any arbitrary potential of mean force where the parameter set is now the full space dependent function. Leveraging an innovative Hermitian form of the corresponding Fokker-Planck equation allows for an eigenbasis decomposition of the time propagation probability density. This formulation motivates the use of the square root of the equilibrium probability density as the basis for evaluating the Fisher information of trajectories with the essential advantage that the Fisher information matrix in the specified parameter space is constant. This outcome greatly eases the calculation of information content in the parameter space via a line integral. In the continuum limit, a simple analytical form can be derived to explicitly reveal the physical origin of the information content in equilibrium trajectories. This methodology also allows deduction of least informative dynamics models from known or available observables that are either dynamical or static in nature. The minimum information optimization of dynamics is performed for a set of different constraints to illustrate the generality of the proposed methodology. (C) 2013 AIP Publishing LLC.
Identifying the structural and kinetic elements in protein large-amplitude conformational motions
(2017-01-01)
Chu, Jhih-Wei
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Yang, Haw
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生物科技學系
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生物資訊及系統生物研究所
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分子醫學與生物工程研究所
The importance of how a protein reconfigures its structure to achieve its function has long been appreciated; yet, the progress in our fundamental understanding of protein dynamics does not seem to be commensurate with the rapid advances in experimental techniques and ever increasing computational prowess. In this review, we attempt to look at this issue based on quantitative characterisations that go beyond simply determining the kinetics rates or only allowing qualitative statements about conformational states. We summarise the theoretical basis for determining from experimental data the kinetics and the structural elements of protein conformational dynamics. The two kinetics elements include the apparent potential of mean force and the intra-molecular diffusion coefficient along a coordinate defined by the pair of single-molecule Forster-type resonance energy transfer reporters that are chemically attached to the protein. We show that it is now possible to resolve the relative contributions of these two kinetics elements when discussing the physical origin of the protein's conformation-reconfiguration rate changes due to mutation or interaction with chemical effectors or with other proteins. The structural element refers to the orthogonal conformational modes that give rise to the intrinsic conformational motions of the protein, and could allow a comparative study among proteins from different families. To achieve these, it is essential that experimental data be rigorously analysed and integrated with molecular simulations - which include molecular dynamics simulations, coarse-grained modelling, and enhanced sampling. In turn, the close interplay between computation and experiment through this new direction could accelerate the discovery of predictive models.
Importance of Internal Porosity for Glucan Adsorption in Mesoporous Carbon Materials
(2015-07-07)
Chung, Po-Wen
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Charmot, Alexandre
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Click, Timothy
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Lin, Yuchun
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Bae, YounJue
To better understand the adsorption of long-chain poly(1 -> 4)beta-D-glucans on carbon surfaces as well as interactions responsible for this adsorption, we use a comparative study involving mesoporous carbon silica composite materials that have been etched to varying degrees and all-atom molecular dynamics simulations. The materials synthesized as part of this etching study consist of an as-synthesized composite material (MCN-MSN), MCN-MSN-0.5 (composite materials consisting of 50% carbon by mass), MCN-MSN-0.3 (composite materials consisting of 70% carbon by mass), and MCN, in which silica etching was conducted using an aqueous ethanolic solution of either NaOH or HF. Data for the adsorption of long-chain glucans to these materials from concentrated aqueous HCl (37 wt %) solution demonstrate a direct relationship between the amount of beta-glu adsorption and the magnitude of exposed carbon mesopore surface area, which systematically increases and is also accompanied by an increase in the mesopore size during silica etching. This demonstrates beta-glu adsorption as occurring on internal carbon mesopores rather than exclusively on the external carbon surface. These experimental data on adsorption were corroborated by molecular dynamics (MD) simulations of beta-glu adsorption to a graphene bilayer separated by a distance of 3.2 nm, chosen to correspond to the carbon mesopore diameter of the experimental system. Simulation results using a variety of beta-glu solvent systems demonstrate the rapid adsorption of a beta-glu strand on the graphitic carbon surface via axial coupling and are consistent with experimentally observed trends in fast adsorption kinetics. Solvent-mediated effects such as small-scale hydrophobicity and preferential interactions with ions are shown to play important roles in modulating glucan adsorption to carbon surfaces, whereas experimental data on hydrophobically modified silica demonstrate that hydrophobicity in and of itself is insufficient to cause beta-glu adsorption from concentrated aqueous HCl solution.
Kinetic Modeling at Single-Molecule Resolution Elucidates the Mechanisms of Cellulase Synergy
(2014-07-01)
Shang, Barry Z.
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Chu, Jhih-Wei
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生物科技學系
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生物資訊及系統生物研究所
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Department of Biological Science and Technology
The mechanism of synergistic cooperation between celluloses in decomposing cellulose is revealed by resolving the molecular structures of enzymes and substrates via kinetic modeling. The emergence of endo-exo and exo-exo synergy through enzyme-enzyme and enzyme-substrate couplings was investigated with the main exoglucanases (TrCel7A and TrCel6A) and endoglucanase (TrCel7B) of the Trichoderma reesei fungus. The degree of synergy was found to depend on the interplay between two competing effects: (1) enhancement of the complexation rates of exoglucanases with the chain ends created by endoglucanases and (2) surface inhibition of processive exoglucanases by uneven layers of glucan chains on cellulose. We show that the sole effect of TrCel7B in creating more free ends for TrCel7A is insufficient to cause synergistic activity. The combined actions of TrCel7A and TrCel7B produce a rougher substrate surface that subsequently promotes blocking of processing TrCel7A enzymes. This anti-synergy can be counteracted by enhancing the rates of complexation of TrCel7A with the TrCel7B-created chain ends, allowing their synergistic cooperation. Similarly, kinetic simulations show that exo-exo synergy does not occur if TrCel6A and TrCel7A have only opposite specificities in targeting the two ends of glucan chains. Incorporating endo activity into TrCel6A with complexation rate enhancement, however, can lead to synergy with TrCel7A. Therefore, we find that endo-exo and exo-exo synergies may share the same mechanistic origin. The results of work also highlight that resolving molecular configurations in kinetic modeling allows systematic analysis for elucidating the mechanism of interfacial biocatalysis.
Kinetic Modeling at Single-Molecule Resolution Elucidates the Mechanisms of Cellulase Synergy
(2014-07-01)
Shang, Barry Z.
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Chu, Jhih-Wei
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生物科技學系
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生物資訊及系統生物研究所
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Department of Biological Science and Technology
The mechanism of synergistic cooperation between celluloses in decomposing cellulose is revealed by resolving the molecular structures of enzymes and substrates via kinetic modeling. The emergence of endo-exo and exo-exo synergy through enzyme-enzyme and enzyme-substrate couplings was investigated with the main exoglucanases (TrCel7A and TrCel6A) and endoglucanase (TrCel7B) of the Trichoderma reesei fungus. The degree of synergy was found to depend on the interplay between two competing effects: (1) enhancement of the complexation rates of exoglucanases with the chain ends created by endoglucanases and (2) surface inhibition of processive exoglucanases by uneven layers of glucan chains on cellulose. We show that the sole effect of TrCel7B in creating more free ends for TrCel7A is insufficient to cause synergistic activity. The combined actions of TrCel7A and TrCel7B produce a rougher substrate surface that subsequently promotes blocking of processing TrCel7A enzymes. This anti-synergy can be counteracted by enhancing the rates of complexation of TrCel7A with the TrCel7B-created chain ends, allowing their synergistic cooperation. Similarly, kinetic simulations show that exo-exo synergy does not occur if TrCel6A and TrCel7A have only opposite specificities in targeting the two ends of glucan chains. Incorporating endo activity into TrCel6A with complexation rate enhancement, however, can lead to synergy with TrCel7A. Therefore, we find that endo-exo and exo-exo synergies may share the same mechanistic origin. The results of work also highlight that resolving molecular configurations in kinetic modeling allows systematic analysis for elucidating the mechanism of interfacial biocatalysis.
Linking hydrophobicity and hydrodynamics by the hybrid fluctuating hydrodynamics and molecular dynamics methodologies
(2013-08-27)
Voulgarakis, Nikolaos K.
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Shang, Barry Z.
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Chu, Jhih-Wei
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生物科技學系
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生物資訊及系統生物研究所
The development of a hybrid fluctuating hydrodynamics (FHD) and molecular dynamics (MD) simulation method that combines the molecular dynamics of moving particles with the fluctuating hydrodynamics of solvent fields on Eulerian grid cells is presented. This method allows resolution of solute-solvent interfaces and realization of excluded volumes of particles in the presence of hydrodynamic coupling. With these capabilities, we show that the ubiquitous forces mediated by the solvent, hydrophobicity and hydrodynamics, can be linked in a mesoscopic simulation. The strategies we devise to overcome the numerical issues of mixing variables in the Eulerian and Lagrangian coordinate systems, i.e., using a pair of auxiliary fluids to realize the excluded volumes of particles and assigning collocating gridding systems on solutes to interface with solvent fields, are also presented. Simulation results show that the hybrid FHD and MD method can reproduce the solvation free energies and scaling laws of particles dynamics for hydrophobes of different sizes. The collapse of two hydrophobic particles was also simulated to illustrate that the hybrid FHD and MD method has the potential to be generally applied to study nanoscale self-assembly and dynamics-structure-function relationships of biomolecules.
LWD-TCP complex activates the morning gene CCA1 in Arabidopsis
(2016-10-13)
Wu, Jing-Fen
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Tsai, Huang-Lung
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Joanito, Ignasius
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Wu, Yi-Chen
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Chang, Chin-Wen
A double-negative feedback loop formed by the morning genes CIRCADIAN CLOCK ASSOCIATED1 (CCA1)/LATE ELONGATED HYPOCOTYL (LHY) and the evening gene TIMING OF CAB EXPRESSION1 (TOC1) contributes to regulation of the circadian clock in Arabidopsis. A 24-h circadian cycle starts with the peak expression of CCA1 at dawn. Although CCA1 is targeted by multiple transcriptional repressors, including PSEUDO-RESPONSE REGULATOR9 (PRR9), PRR7, PRR5 and CCA1 HIKING EXPEDITION (CHE), activators of CCA1 remain elusive. Here we use mathematical modelling to infer a co-activator role for LIGHT-REGULATED WD1 (LWD1) in CCA1 expression. We show that the TEOSINTE BRANCHED 1-CYCLOIDEA-PCF20 (TCP20) and TCP22 proteins act as LWD-interacting transcriptional activators. The concomitant binding of LWD1 and TCP20/TCP22 to the TCP-binding site in the CCA1 promoter activates CCA1. Our study reveals activators of the morning gene CCA1 and provides an action mechanism that ensures elevated expression of CCA1 at dawn to sustain a robust clock.
Molecular simulations in conjunction with experimental studies illustrate internal protein logic controlled by conformation and dynamic structural change
(2015-03-22)
Morrell, Thomas
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Rafalska-Metcalf, Ilona
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Chu, Jhih-Wei
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Yang, Haw
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生物科技學系
Structural basis for overhang excision and terminal unwinding of DNA duplexes by TREX1
(2018-05-01)
Huang, Kuan-Wei
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Liu, Tung-Chang
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Liang, Ruei-Yue
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Chu, Lee-Ya
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Cheng, Hiu-Lo
Three prime repair exonuciease 1 (TREX1) is an essential exonuclease in mammalian cells, and numerous in vivo and in vitro data evidenced its participation in immunity regulation and in genotoxicity remediation. In these very complicated cellular functions, the molecular mechanisms by which duplex DNA substrates are processed are mostly elusive because of the lack of structure information. Here, we report multiple crystal structures of TREX1 complexed with various substrates to provide the structure basis for overhang excision and terminal unwinding of DNA duplexes. The substrates were designed to mimic the intermediate structural DNAs involved in various repair pathways. The results showed that the Leu24Pro25-Ser26 cluster of TREX1 served to cap the nonscissile 5'-end of the DNA for precise removal of the short 3'-overhang in L- and Y-structural DNA or to wedge into the double-stranded region for further digestion along the duplex. Biochemical assays were also conducted to demonstrate that TREX1 can indeed degrade double-stranded DNA (dsDNA) to a full extent. Overall, this study provided unprecedented knowledge at the molecular level on the enzymatic substrate processing involved in prevention of immune activation and in responses to genotoxic stresses. For example, Arg128, whose mutation in TREX1 was linked to a disease state, were shown to exhibit consistent interaction patterns with the nonscissile strand in all of the structures we solved. Such structure basis is expected to play an indispensable role in elucidating the functional activities of TREX1 at the cellular level and in vivo.
Structure-mechanics statistical learning unravels the linkage between local rigidity and global flexibility in nucleic acids
(2020-05-21)
Chen, Yi-Tsao
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Yang, Haw
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Chu, Jhih-Wei
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交大名義發表
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生物資訊及系統生物研究所
The mechanical properties of nucleic acids underlie biological processes ranging from genome packaging to gene expression, but tracing their molecular origin has been difficult due to the structural and chemical complexity. We posit that concepts from machine learning can help to tackle this long-standing challenge. Here, we demonstrate the feasibility and advantage of this strategy through developing a structure-mechanics statistical learning scheme to elucidate how local rigidity in double-stranded (ds)DNA and dsRNA may lead to their global flexibility in bend, stretch, and twist. Specifically, the mechanical parameters in a heavy-atom elastic network model are computed from the trajectory data of all-atom molecular dynamics simulation. The results show that the inter-atomic springs for backbone and ribose puckering in dsRNA are stronger than those in dsDNA, but are similar in strengths for base-stacking and base-pairing. Our analysis shows that the experimental observation of dsDNA being easier to bend but harder to stretch than dsRNA comes mostly from the respective B- and A-form topologies. The computationally resolved composition of local rigidity indicates that the flexibility of both nucleic acids is mostly due to base-stacking. But for properties like twist-stretch coupling, backbone springs are shown to play a major role instead. The quantitative connection between local rigidity and global flexibility sets foundation for understanding how local binding and chemical modification of genetic materials effectuate longer-ranged regulatory signals.
Systems-level Modeling with Molecular Resolution Elucidates the Rate-limiting Mechanisms of Cellulose Decomposition by Cellobiohydrolases
(2013-10-04)
Shang, Barry Z.
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Chang, Rakwoo
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Chu, Jhih-Wei
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生物科技學系
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生物資訊及系統生物研究所
Interprotein and enzyme-substrate couplings in interfacial biocatalysis induce spatial correlations beyond the capabilities of classical mass-action principles in modeling reaction kinetics. To understand the impact of spatial constraints on enzyme kinetics, we developed a computational scheme to simulate the reaction network of enzymes with the structures of individual proteins and substrate molecules explicitly resolved in the three-dimensional space. This methodology was applied to elucidate the rate-limiting mechanisms of crystalline cellulose decomposition by cellobiohydrolases. We illustrate that the primary bottlenecks are slow complexation of glucan chains into the enzyme active site and excessive enzyme jamming along the crowded substrate. Jamming could be alleviated by increasing the decomplexation rate constant but at the expense of reduced processivity. We demonstrate that enhancing the apparent reaction rate required a subtle balance between accelerating the complexation driving force and simultaneously avoiding enzyme jamming. Via a spatiotemporal systems analysis, we developed a unified mechanistic framework that delineates the experimental conditions under which different sets of rate-limiting behaviors emerge. We found that optimization of the complexation-exchange kinetics is critical for overcoming the barriers imposed by interfacial confinement and accelerating the apparent rate of enzymatic cellulose decomposition.
Trajectory Entropy of Continuous Stochastic Processes at Equilibrium
(2014-03-20)
Haas, Kevin R.
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Yang, Haw
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Chu, Jhih-Wei
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生物科技學系
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生物資訊及系統生物研究所
We propose to quantify the trajectory entropy of a dynamic system as the information content in excess of a free-diffusion reference model. The space time trajectory is now the dynamic variable, and its path probability is given by the Onsager-Machlup action. For the time propagation of the overdamped Langevin equation, we solved the action path integral in the continuum limit and arrived at an exact analytical expression that emerged as a simple functional of the deterministic mean force and the stochastic diffusion. This work may have direct implications in chemical and phase equilibria, bond isomerization, and conformational changes in biological macromolecules as well transport problems in general.