Orthogonal Invariant Sets of the Diffusion Tensor and the Development of a Curvilinear Set Suitable for Low-Anisotropy Tissues

We develop a curvilinear invariant set of the diffusion tensor which may be applied to Diffusion Tensor Imaging measurements on tissues and porous media. This new set is an alternative to the more common invariants such as fractional anisotropy and the diffusion mode. The alternative invariant set possesses a different structure to the other known invariant sets; the second and third members of the curvilinear set measure the degree of orthotropy and oblateness/prolateness, respectively. The proposed advantage of these invariants is that they may work well in situations of low diffusion anisotropy and isotropy, as is often observed in tissues such as cartilage. We also explore the other orthogonal invariant sets in terms of their geometry in relation to eigenvalue space; a cylindrical set, a spherical set (including fractional anisotropy and the mode), and a log-Euclidean set. These three sets have a common structure. The first invariant measures the magnitude of the diffusion, the second and third invariants capture aspects of the anisotropy; the magnitude of the anisotropy and the shape of the diffusion ellipsoid (the manner in which the anisotropy is realised). We also show a simple method to prove the orthogonality of the invariants within a set.     

Spike-frequency adaptation generates intensity invariance in a primary auditory interneuron

Adaptation of the spike-frequency response to constant stimulation, as observed on various timescales in many neurons, reflects high-pass filter properties of a neuron's transfer function. Adaptation in general, however, is not sufficient to make a neuron's response independent of the mean intensity of a sensory stimulus, since low frequency components of the stimulus are still transmitted, although with reduced gain. We here show, based on an analytically tractable model, that the response of a neuron is intensity invariant, if the fully adapted steady-state spike-frequency response to constant stimuli is independent of stimulus intensity. Electrophysiological recordings from the AN1, a primary auditory interneuron of crickets, show that for intensities above 60 dB SPL (sound pressure level) the AN1 adapted with a time-constant of approximately 40 ms to a steady-state firing rate of approximately 100 Hz. Using identical random amplitude-modulation stimuli we verified that the AN1's spike-frequency response is indeed invariant to the stimulus' mean intensity above 60 dB SPL. The transfer function of the AN1 is a band pass, resulting from a high-pass filter (cutoff frequency at 4 Hz) due to adaptation and a low-pass filter (100 Hz) determined by the steady-state spike frequency. Thus, fast spike-frequency adaptation can generate intensity invariance already at the first level of neural processing.     

Mathematical modeling of living cell metabolism using the method of steady-state stoichiometric flux balance

This approach uses a set of algebraic linear equations for reaction rates (the method of steady-state stoichiometric flux balance) to model the purposeful metabolism of the living self-reproducing biochemical system (i.e. cell), which persists in steady-state growth. Linear programming (SIMPLEX method) is used to derive the solution for the model equations set (determining reaction rates which provide flux balance at given conditions). Here, we demonstrate the approach through the mathematical modeling of steady-state metabolism in Saccharomyces cerevisiae mitochondria.     

A comparison of the intrinsic fluorescence of red kangaroo, horse and sperm whale metmyoglobins

Several metmyoglobins (red kangaroo, horse and sperm whale), containing different numbers of tyrosines, but with invariant tryptophan residues (Trp-7, Trp-14), exhibit intrinsic fluorescence when studied by steady-state front-face fluorometry. The increasing tyrosine content of these myoglobins correlates with a shift in emission maximum to shorter wavelengths with excitation at 280 nm: red kangaroo (Tyr-146) emission maximum 335 nm; horse (Tyr-103, -146) emission maximum 333 nm; sperm whale (Tyr-103, -146, -151) emission maximum 331 nm. Since 280 nm excites both tyrosine and tryptophan, this strongly suggests that tyrosine emission is not completely quenched but also contributes to this fluorescence emission. Upon titration to pH 12.5, there is a reversible shift of the emission maximum to longer wavelengths with an increase greater than 2-fold in fluorescence intensity. With excitation at 305 nm, a tyrosinate-like emission is detected at a pH greater than 12. These studies show that: (1) metmyoglobins, Class B proteins containing both tyrosine and tryptophan residues, exhibit intrinsic fluorescence; (2) tyrosine residues also contribute to the observed steady-state fluorescence emission when excited by light at 280 nm; (3) the ionization of Tyr-146 is likely coupled to protein unfolding.     

Estimation of phylogeny and invariant sites under the general Markov model of nucleotide sequence evolution

The models of nucleotide substitution used by most maximum likelihood-based methods assume that the evolutionary process is stationary, reversible, and homogeneous. We present an extension of the Barry and Hartigan model, which can be used to estimate parameters by maximum likelihood (ML) when the data contain invariant sites and there are violations of the assumptions of stationarity, reversibility, and homogeneity. Unlike most ML methods for estimating invariant sites, we estimate the nucleotide composition of invariant sites separately from that of variable sites. We analyze a bacterial data set where problems due to lack of stationarity and homogeneity have been previously well noted and use the parametric bootstrap to show that the data are consistent with our general Markov model. We also show that estimates of invariant sites obtained using our method are fairly accurate when applied to data simulated under the general Markov model.     

Plasmid repopulation kinetics in Staphylococcus aureus

We have analyzed the kinetic route by which the indirectly controlled Staphylococcus aureus plasmid, pT181, responds to and corrects fluctuations in copy number. The kinetics of copy number correction from low to steady-state levels (termed repopulation) were determined using two different methods of copy number reduction. Thermosensitive replication (Tsr) mutants of pT181 were grown at nonpermissive temperatures to lower copy number and then shifted to a permissive temperature to allow repopulation. After the downshift, both wild-type and copy mutant plasmids, with active inhibitors, exhibited a burst of exponential replication that resulted in a two- to threefold overshoot of normal steady-state copy numbers. This was followed by inhibition of replication and eventual reestablishment of the steady-state replication rate. Similar replication kinetics were observed when these plasmids were introduced into naive cells by high-frequency transduction. By contrast, a pT181 copy mutant with a nonfunctional inhibitor-target regulation did not overshoot its steady-state copy number, but instead repopulated asymptotically. These results suggest that at low copy numbers, pT181 and its derivatives replicate at near-maximal rates and overshoot prior to the establishment of an inhibitory concentration of repressor. The maximal replication rate is independent of the plasmid's cop genotype. As the copy number increases, inhibitor accumulates and eventually reduces the replication rate. In the absence of an active inhibitor, the steady-state copy number is established at a level that must be limited by some other invariant factor.     

Learning invariant object recognition in the visual system with continuous transformations

The cerebral cortex utilizes spatiotemporal continuity in the world to help build invariant representations. In vision, these might be representations of objects. The temporal continuity typical of objects has been used in an associative learning rule with a short-term memory trace to help build invariant object representations. In this paper, we show that spatial continuity can also provide a basis for helping a system to self-organize invariant representations. We introduce a new learning paradigm “continuous transformation learning” which operates by mapping spatially similar input patterns to the same postsynaptic neurons in a competitive learning system. As the inputs move through the space of possible continuous transforms (e.g. translation, rotation, etc.), the active synapses are modified onto the set of postsynaptic neurons. Because other transforms of the same stimulus overlap with previously learned exemplars, a common set of postsynaptic neurons is activated by the new transforms, and learning of the new active inputs onto the same postsynaptic neurons is facilitated. We demonstrate that a hierarchical model of cortical processing in the ventral visual system can be trained with continuous transform learning, and highlight differences in the learning of invariant representations to those achieved by trace learning.     

t-mixing: from fluctuation relations to response and irreversibility in MD

This paper provides an introduction to the theory of steady-state fluctuation relations for molecular dynamics systems, that led to a general theory of response. The main ingredient of this theory is a new dynamical condition now known as t-mixing. We use such a condition to identify necessary and sufficient conditions for the relaxation to a steady state (whether equilibrium or not) of an ensemble of identical systems, as well as of a single system. This allows us to address the problem of the irreversibility of time reversal invariant (conservative as well as dissipative) particle systems.     

A framework for whole-cell mathematical modeling

The default framework for modeling biochemical processes is that of a constant-volume reactor operating under steady-state conditions. This is satisfactory for many applications, but not for modeling growth and division of cells. In this study, a whole-cell modeling framework is developed that assumes expanding volumes and a cell-division cycle. A spherical newborn cell is designed to grow in volume during the growth phase of the cycle. After 80% of the cycle period, the cell begins to divide by constricting about its equator, ultimately affording two spherical cells with total volume equal to twice that of the original. The cell is partitioned into two regions or volumes, namely the cytoplasm (Vcyt) and membrane (Vmem), with molecular components present in each. Both volumes change during the cell cycle; Vcyt changes in response to osmotic pressure changes as nutrients enter the cell from the environment, while Vmem changes in response to this osmotic pressure effect such that membrane thickness remains invariant. The two volumes change at different rates; in most cases, this imposes periodic or oscillatory behavior on all components within the cell. Since the framework itself rather than a particular set of reactions and components is responsible for this behavior, it should be possible to model various biochemical processes within it, affording stable periodic solutions without requiring that the biochemical process itself generates oscillations as an inherent feature. Given that these processes naturally occur in growing and dividing cells, it is reasonable to conclude that the dynamics of component concentrations will be more realistic than when modeled within constant-volume and/or steady-state frameworks. This approach is illustrated using a symbolic whole cell model.     

Stability of steady states in nucleic acid poly (A-T) synthesis and the stirred flow reactor.

The template directed synthesis of poly[d(A-T)] from the nucleoside triphosphates in the presence of DNA polymerase I is carried out continuously in a stirred flow reactor for the first time. The initial objective is to test the kinetic stability of the established steady states at various flow rates. Graphical analysis predicts instable steady states for certain high flow rates. As a consequence of instabilities multiple steady states and steady-state hysteresis may occur. Steady-state hysteresis has now been found experimentally. For a different enzyme fraction of low exonuclease activity we found the steady-state absorbance at 260 nm to be almost invariant with flow rate at high enzyme concentrations even if the flow rate was increased by a large factor. We call this phenomenon kinetic buffering. Relaxation of a large flow perturbation approaches the steady state in a sigmoidal fashion. Concentration oscillations at 260 nm occurred in one experiment using an enzyme fraction of low exonuclease activity after perturbing the steady state by monomer (dATP). Advantages of the stirred flow reactor method over serial transfer are discussed.     

Global analysis of fluorescence lifetime imaging microscopy data

Global analysis techniques are described for frequency domain fluorescence lifetime imaging microscopy (FLIM) data. These algorithms exploit the prior knowledge that only a limited number of fluorescent molecule species whose lifetimes do not vary spatially are present in the sample. Two approaches to implementing the lifetime invariance constraint are described. In the lifetime invariant fit method, each image in the lifetime image sequence is spatially averaged to obtain an improved signal-to-noise ratio. The lifetime estimations from these averaged data are used to recover the fractional contribution to the steady-state fluorescence on a pixel-by-pixel basis for each species. The second, superior, approach uses a global analysis technique that simultaneously fits the fractional contributions in all pixels and the spatially invariant lifetimes. In frequency domain FLIM the maximum number of lifetimes that can be fit with the global analysis method is twice the number of lifetimes that can be fit with conventional approaches. As a result, it is possible to discern two lifetimes with a single-frequency FLIM setup. The algorithms were tested on simulated data and then applied to separate the cellular distributions of coexpressed green fluorescent proteins in living cells.     

Insight of Endo-1,4-Xylanase II from Trichoderma reesei: Conserved Water-Mediated H-Bond and Ion Pairs Interactions

Endo-1,4-Xylanase II is an enzyme which degrades the linear polysaccharide beta-1,4-xylan into xylose. This enzyme shows highest enzyme activity around 55 °C, even without being stabilized by the disulphide bridges. A set of nine high resolution crystal structures of Xylanase II (1.11–1.80 Å) from Trichoderma reesei were selected and analyzed in order to identify the invariant water molecules, ion pairs and water-mediated ionic interactions. The crystal structure (PDB-id: 2DFB) solved at highest resolution (1.11 Å) was chosen as the reference and the remaining structures were treated as mobile molecules. These structures were then superimposed with the reference molecule to observe the invariant water molecules using 3-dimensional structural superposition server. A total of 37 water molecules were identified to be invariant molecules in all the crystal structures, of which 26 invariant molecules have hydrogen bond interactions with the back bone of residues and 21 invariant water molecules have interactions with side chain residues. The structural and functional roles of these water molecules and ion pairs have been discussed. The results show that the invariant water molecules and ion pairs may be involved in maintaining the structural architecture, dynamics and function of the Endo-1,4-Xylanase II.     

The catalytic mechanism of gastric H+/K+-ATPase: simulations of pre-steady-state and steady-state kinetic results

A reaction cycle for the gastric H+/K+-ATPase is proposed. This has been used to simulate the results from four types of pre-steady-state and steady-state kinetic experiments: (1) the K+ dependence of the dephosphorylation of the phosphoenzyme; (2) the rate of phosphorylation of the enzyme by ATP at different concentrations; (3) the effect of ATP concentration on the steady-state rate of ATP hydrolysis; (4) the phosphoenzyme levels in the steady state at various ATP concentrations. A single set of equilibrium and rate constants can be used to reproduce the results from all four sets of experiments quite well. It is suggested that the steady-state rate equation is nonhyperbolic because ATP can react with the enzyme in both the E1 and the E2 state, but with a lower affinity in E2. No single step is by itself limiting the maximum turnover rate.     

Dimerization and Bifunctionality Confer Robustness to the Isocitrate Dehydrogenase Regulatory System in Escherichia coli

An important goal of systems biology is to develop quantitative models that explain how specific molecular features give rise to systems-level properties. Metabolic and regulatory pathways that contain multifunctional proteins are especially interesting to study from this perspective because they have frequently been observed to exhibit robustness: the ability for a system to perform its proper function even as levels of its components change. In this study, we use extensive biochemical data and algebraic modeling to develop and analyze a model that shows how robust behavior arises in the isocitrate dehydrogenase (IDH) regulatory system of Escherichia coli, which was shown in 1985 to experimentally exhibit robustness. E. coli IDH is regulated by reversible phosphorylation catalyzed by the bifunctional isocitrate dehydrogenase kinase/phosphatase (IDHKP), and the level of IDH activity determines whether carbon flux is directed through the glyoxylate bypass (for growth on two-carbon substrates) or the full tricarboxylic acid cycle. Our model, which incorporates recent structural data on IDHKP, identifies several specific biochemical features of the system (including homodimerization of IDH and bifunctionality of IDHKP) that provide a potential explanation for robustness. Using algebraic techniques, we derive an invariant that summarizes the steady-state relationship between the phospho-forms of IDH. We use the invariant in combination with kinetic data on IDHKP to calculate IDH activity at a range of total IDH levels and find that our model predicts robustness. Our work unifies much of the known biochemistry of the IDH regulatory system into a single quantitative framework and highlights the importance of constructing biochemically realistic models in systems biology.     

Mechanism of interaction between leucine-based sorting signals from the invariant chain and clathrin-associated adaptor protein complexes AP1 and AP2

The cytoplasmic tail of the invariant chain contains two leucine-based sorting signals, and each of those seems sufficient to route the invariant chain to its intracellular destination in either normal or polarized cells. It is believed that the intracellular routing of the invariant chain is mediated by its interactions with the clathrin-associated adaptor protein complexes AP1 and AP2. We () have previously demonstrated the in vitro interactions between the cytoplasmic tail of the invariant chain and AP1/AP2 complexes. These interactions were specific and depended on the critical leucine residues in the invariant chain's sorting signals. In the present study, we decided to investigate the molecular mechanism of these interactions. To this end, we constructed a set of glutathione S-transferase fusion proteins that contained the intact cytoplasmic tail of the invariant chain and its various mutants to define residues important for its interactions with AP1 and AP-2. Our results demonstrated the importance of several residues other than the critical leucine residues for such interactions. A strong correlation between in vitro binding of AP2 to the invariant chain and in vivo internalization of the invariant chain was observed, confirming the primary role of AP2 in recognition of endocytic signals. In addition, we demonstrated different requirements for AP1 and AP2 binding to cytoplasmic tail of the invariant chain, which may reflect that the different sorting pathways mediated by AP1 and AP2 involve their recognition of the primary structure of the sorting signal.     

Isotopic probes yield microscopic constants: separation of binding energy from catalytic efficiency in the bovine plasma amine oxidase reaction

Bovine plasma amine oxidase catalyzes the oxidative deamination of primary amines. The reaction can be viewed as two half-reactions: enzyme reduction by substrate followed by enzyme reoxidation by dioxygen. Anaerobic stopped-flow kinetic measurements of the first half-reaction indicate very large deuterium isotope effects for benzylamine, m-tyramine, and dopamine, Dk = 13.5 +/- 1.3, which are ascribed to an intrinsic isotope effect. From the insensitivity of these isotope effects to amine concentration, stopped-flow data provide substrate dissociation constants, K1, and rate constants for the C-H bond cleavage step, k3, directly. Steady-state isotope effects have also been measured for benzylamine and six ring-substituted phenethylamines. Whereas a small range of values for kcat, 0.38-1.2 s-1, and Dkcat, 5.4-8.8, is observed, kcat/Km = 1.3 X 10(2) to 3.8 X 10(4) M-1 S-1 and D(kcat/Km) = 5.6-16.1 indicate a marked effect of ring substituent. As described earlier [Miller, S., & Klinman, J.P. (1982) Methods Enzymol. 87, 711], the availability of an intrinsic isotope effect for an enzymatic reaction permits calculation of microscopic constants from steady-state data. By employment of a minimal mechanism for bovine plasma amine oxidase involving a single precatalytic and multiple postcatalytic enzyme-substrate complexes, equations have been derived that allow calculation of k3 and K1 when DKeq congruent to 1 less than Dk. Unexpectedly, in the case of K1, we have shown that this parameter can be calculated from steady-state parameters without the requirement for an intrinsic isotope effect. This result should have general application to both ping-pong and sequential ternary-complex enzyme mechanisms. Of significance for future applications of steady-state isotope effects to the calculation of microscopic constants, values for K1 and k3 derived from steady-state parameters and single turnover measurements indicate excellent agreement. Compilation of parameters among six ring-substituted phenethylamines reveals alteration in delta G for enzyme-substrate complex formation by 2.8 kcal/mol, together with an essentially invariant rate constant for C-H bond activation. A detailed discussion of the relevance of these findings to the interrelationship of binding energy and catalytic efficiency in enzyme reactions is presented.     

Quantification of the effects of vasomotion on mass transport to tissue from axisymmetric blood vessels

The process known as vasomotion, rhythmic oscillations in vessel diameter, has been proposed to act as a protective mechanism for tissue under conditions of reduced perfusion, since it is frequently only observed experimentally when perfusion levels are reduced. This could be due to a resultant increase in oxygen transport from the vasculature to the surrounding tissue, either directly or indirectly. It is thus potentially of significant clinical interest as a warning signal for ischemia. However, there has been little analysis performed to quantify the effects of vessel wall movement on time-averaged mass transport. We thus present a detailed analysis of such mass transport for an axisymmetric vessel with a periodically oscillating wall, by solving the non-linear mass transport equation, and quantify the differences between the time-averaged mass transport under conditions of no oscillation (i.e. the steady-state) and varying wall oscillation amplitude. The results show that if the vessel wall alone is oscillated, with an invariant wall concentration, the time-averaged mass transport is reduced relative to the steady-state, but if the vessel wall concentration is also oscillated, then mass transport is increased, although this is generally only true when these oscillate in phase with each other. The influence of Péclet number and the non-dimensional rate of consumption of oxygen in tissue, as well as the amplitude of oscillations, are fully characterised. We conclude by considering the likely implications of these results in the context of oxygen transport to tissue.     

Noninferior periodic operation of the biological reactor

The optimal periodic operation of the biological reactor was studied from the standpoint of the two-objective programming problem. The noninferior set with respect to the cell productivity and the conversion of the substrate into the biomass was determined by use of the optimization technique due to Miele. It was shown that the noninferior set was composed in general of the repeated batch branch and the repeated fed-batch branch, which occupy the high-productivity portion and the high-conversion portion of the noninferior set, respectively. However, the latter branch disappears in the case of growth kinetics with no substrate inhibition. In addition, the extreme points of the noninferior set yielding the maximal productivity and the maximal conversion represent such operations that are equivalent to the steady-state operation (chemostat culture) and the batch operation, respectively.     

Chemical transformation is not rate-limiting in the reaction catalyzed by Escherichia coli 6-hydroxymethyl-7,8-dihydropterin pyrophosphokinase

6-Hydroxymethyl-7,8-dihydropterin pyrophosphokinase (HPPK) catalyzes the transfer of pyrophosphate from ATP to 6-hydroxymethyl-7,8-dihydropterin (HMDP). Because HPPK is essential for microorganisms but is absent from human and animals, the enzyme is an excellent target for developing antimicrobial agent. Thermodynamic analysis shows that Mg(2+) is important not only for the binding of nucleotides but also for the binding of HMDP. Transient kinetic analysis shows that a step or steps after the chemical transformation are rate-limiting in the reaction catalyzed by HPPK. The pre-steady-state kinetics is composed of a burst phase and a steady-state phase. The rate constant for the burst phase is approximately 50 times larger than that for the steady-state phase. The latter is very similar to the k(cat) value measured by steady-state kinetics. A set of rate constants for the individual steps of the HPPK-catalyzed reaction has been determined by a combination of stopped-flow and quench-flow analyses. These results form a thermodynamic and kinetic framework for dissecting the roles of active site residues in the substrate binding and catalysis by HPPK.