On the dispersion index of a Markovian molecular clock

The number of nucleotide substitutions accumulated in a gene or in a lineage is an important random variable in the study of molecular evolution. Of particular interest is the ratio of the variance to the mean of that random variable, often known as the dispersion index. Because nucleotide substitution is most commonly modeled by a continuous-time four-state Markov chain, this paper provides a systematic method of computing the dispersion indices exhibited by a continuous-time four-state Markov chain. Using this method along with computer algebra and Monte Carlo simulation, this paper offers partially proven conjectures that were supported by thorough computer experiments. It is believed that the Tamura model, the equal-input model and the Takahata-Kimura model always exhibit dispersion indices less than 2. It is also believed that a general four-state model can be chosen to exhibit a dispersion index of any desired magnitude, although the chance of a randomly chosen such model exhibiting a dispersion index greater than 2 is as small as about 2%. Relevance of these findings to the neutral theory is discussed.     

Ultrasonic study of the kinetics of counterion site binding in aqueous solutions of polyelectrolytes

The excess ultrasonic absorption due to counterion binding has been studied as a function of frequency for a series of polysalts in the range 1–150 MHz. All the relaxation spectra can be represented by a relaxation equation with two relaxation terms. The relaxation frequencies appear concentration independent and the relaxation amplitudes seem proportional to concentration. The low frequency relaxation process appears to depend mainly on the nature of the counterion while the high frequency relaxation process seems to be mostly dependent on the nature of the polyion. These results are quite similar to those obtained in ultrasonic studies of ion-pairing in solutions of divalent sulfates. The kinetic model used for the quantitative analysis of these results has been modified for polysalts through introducing the concept of“counterion condensation”. In this modified model the excess absorption is assigned to the perturbation by the ultrasonic waves of the equilibria between the three states of hydration of ths complex formed by a counterion and that part of the polyion where it is bound. Analytical expressions of the relaxation amplitudes have been derived using classical procedures for this modified kinetic model. In the case of cobalt-polyphosphate (Co-PP), the ultrasonic data together with the results of NMR measurements on either Co²⁺ or Co-PP have been used for the evaluation of the volume changes, the rate constants and the fractions of counterions in the three states of hydration involved in the binding equilibria. The volume changes obtained in this manner depend only slightly on the method of calculation and appear to be consistent with volume changes for outer-sphere and inner-sphere complex formation. These results are discussed.     

Effect of Spatial Dispersion on Evolutionary Stability: A Two-Phenotype and Two-Patch Model

In this paper, we investigate a simple two-phenotype and two-patch model that incorporates both spatial dispersion and density effects in the evolutionary game dynamics. The migration rates from one patch to another are considered to be patch-dependent but independent of individual’s phenotype. Our main goal is to reveal the dynamical properties of the evolutionary game in a heterogeneous patchy environment. By analyzing the equilibria and their stabilities, we find that the dynamical behavior of the evolutionary game dynamics could be very complicated. Numerical analysis shows that the simple model can have twelve equilibria where four of them are stable. This implies that spatial dispersion can significantly complicate the evolutionary game, and the evolutionary outcome in a patchy environment should depend sensitively on the initial state of the patches.     

Four-state models and regulation of contraction of smooth muscle. II. Properties of the solutions and identification.

We analyze the behavior and the identification problem of cyclic four-state models. We find that for any state, or a weighted combination of two states, there can be at most one maximum, or one minimum, and two inflection points. We obtain necessary conditions for overshoot and undershoot and give examples. We describe procedures to estimate all the rate constants and discuss certain experimental aspects of the identification problem. Finally, we give an example of identification by obtaining the 10 model parameters from experimental data on skinned fibers from smooth muscle. These results, in conjunction with the results of the previous paper, can help in testing four-state models of regulation of contraction of smooth muscle and of a variety of other physiological and biochemical phenomena.     

Resonance raman investigations of site-directed mutants of myoglobin: effects of distal histidine replacement

The resonance Raman spectra of met-, deoxy-, and (carbonmonoxy)myoglobin (MbCO) are studied as a function of amino acid replacement at the distal histidine-E7 position. The synthetic wild type is found to be spectroscopically identical with the native material. The methionine and glycine replacements do not affect the met or deoxy spectra but do lead to distinct changes in the nu Fe-CO region of the MbCO spectrum. The native MbCO displays a pH-dependent population redistribution of the nu Fe-CO modes, while the analogous population in the mutant systems is found to be pH independent. This indicates that histidine-E7 is the titratable group in native MbCO. Moreover, the pH dependence of the population dynamics is found to be inconsistent with a simple two-state Henderson-Hasselbalch analysis. Instead, we suggest a four-state model involving the coupling of histidine protonation and conformational change. Within this model, the pK of the distal histidine is found to be 6.0 in the "open" configuration and 3.8 in the "closed" conformation. This corresponds to a 3 kcal/mol destabilization of the positively charged distal histidine within the hydrophobic pocket and suggests how protonation can lead to a larger population of the "open" conformation. At pH 7, the pocket is found to be "open" approximately 3% of the time. Further work, involving both IR and Raman measurements, allows the electron-nuclear coupling strengths of the various nu Fe-CO and nu C-O Raman modes to be determined. The slowly rebinding conformational state, corresponding to nu Fe-CO = 518 cm-1 (nu C-O = 1932 cm-1), displays unusually weak coupling of the Fe-CO mode to the Soret transition. Studies of the nu Fe-CO region as a function of temperature reveal that the equilibria between the conformational states are quenched in both the native and glycine mutant below the freezing point of the solvent. Unusual line narrowing of the nu Fe-CO modes at the phase transition is also observed in all samples studied. This line narrowing stands in marked contrast to the other heme Raman modes and suggests that Fe-CO librational motion and/or distal pocket vibrational (or conformational) excitations are involved in the line broadening at room temperature.     

Characterizing multistationarity regimes in biochemical reaction networks

Switch like responses appear as common strategies in the regulation of cellular systems. Here we present a method to characterize bistable regimes in biochemical reaction networks that can be of use to both direct and reverse engineering of biological switches. In the design of a synthetic biological switch, it is important to study the capability for bistability of the underlying biochemical network structure. Chemical Reaction Network Theory (CRNT) may help at this level to decide whether a given network has the capacity for multiple positive equilibria, based on their structural properties. However, in order to build a working switch, we also need to ensure that the bistability property is robust, by studying the conditions leading to the existence of two different steady states. In the reverse engineering of biological switches, knowledge collected about the bistable regimes of the underlying potential model structures can contribute at the model identification stage to a drastic reduction of the feasible region in the parameter space of search. In this work, we make use and extend previous results of the CRNT, aiming not only to discriminate whether a biochemical reaction network can exhibit multiple steady states, but also to determine the regions within the whole space of parameters capable of producing multistationarity. To that purpose we present and justify a condition on the parameters of biochemical networks for the appearance of multistationarity, and propose an efficient and reliable computational method to check its satisfaction through the parameter space.     

Measurement of three-dimensional radiation dose distributions using MRI

Recent investigations have shown that nuclear magnetic resonance (NMR) can be used in conjunction with a suitable chemical dosimeter to estimate the dose from ionizing radiation (Gore et al., Phys Med. Biol. 29, 1189-1197, 1984). Based on this fact it was proposed that spatial dose distributions can be measured in gels infused with the chemical dosimeter using NMR imaging. There have been few such attempts and they provided only qualitative results. In this paper, we report results demonstrating the feasibility of obtaining quantitative dose distribution measurements by this technique. It is shown that quantitative dose distribution measurements necessitate the calculation of relaxation rate maps. We have determined that the spin-spin relaxation rate is a more sensitive parameter than the spin-lattice relaxation rate. It is also demonstrated that the addition of chemical sensitizers could improve the dose sensitivity of the measured NMR parameters. The two features characterizing a photon beam, depth-dose relationship, and beam profile as measured by this technique are in good agreement with the measurements using conventional methods, ionization chambers, and film dosimetry.     

Dielectric increment and dielectric dispersion of solutions containing simple charged linear macromolecules. II. Experimental results with synthetic polyelectrolytes

The electric permittivity of aqueous solutions of different synthetic polyelectrolytes have been measured as a function of frequency in the range 5 kHz up to 100 MHz in the absence of added salt. Solutions of polymethacrylic acid and polyacrylic acid of different degrees of polymerization, both partially neutralized with NaOH, were investigated as well as solutions of Na-polystyrenesulphonate at different concentrations.For all systems a dispersion profile with two separated dispersion regions was obtained with a molecular weight dependent value of the static electric permittivity. The low frequency dispersion region was found to be characterized by a molecular weight dependent mean relaxation time while for the high frequency dispersion region both the mean relaxation time and the dielectric increment are molecular weight independent. It is shown that the reciprocal values of the specific increments and of the relaxation times depend linearly on the macromolecular concentration. Extrapolation of the corresponding quantities to infinite dilution was found to be possible. A comparison of these extrapolated values with calculated ones according to the previously derived theory also applicable to flexible macromolecules establishes that this theory describes satisfactorily the dielectric behaviour of the systems investigated.The conclusion is reached that the high frequency dispersion and relaxation can be attributed to fluctuations in the distribution of bound counterions along limited parts of the macromolecule. The relaxation time of the low frequency dispersion region seems to be essentially determined by the rotation of the complete molecule and the static electric permittivity can he explained in terms of fluctuations in the counterion density extending over the whole macromolecule.     

Equilibrium analysis and phase synchronization of two coupled HR neurons with gap junction

The properties of equilibria and phase synchronization involving burst synchronization and spike synchronization of two electrically coupled HR neurons are studied in this paper. The findings reveal that in the non-delayed system the existence of equilibria can be turned into intersection of two odd functions, and two types of equilibria with symmetry and non-symmetry can be found. With the stability and bifurcation analysis, the bifurcations of equilibria are investigated. For the delayed system, the equilibria remain unchanged. However, the Hopf bifurcation point is drastically affected by time delay. For the phase synchronization, we focus on the synchronization transition from burst synchronization to spike synchronization in the non-delayed system and the effect of coupling strength and time delay on spike synchronization in delayed system. In addition, corresponding firing rhythms and spike synchronized regions are obtained in the two parameters plane. The results allow us to better understand the properties of equilibria, multi-time-scale properties of synchronization and temporal encoding scheme in neuronal systems.     

RankAggreg, an R package for weighted rank aggregation

Background

Determining the parameters of a mathematical model from quantitative measurements is the main bottleneck of modelling biological systems. Parameter values can be estimated from steady-state data or from dynamic data. The nature of suitable data for these two types of estimation is rather different. For instance, estimations of parameter values in pathway models, such as kinetic orders, rate constants, flux control coefficients or elasticities, from steady-state data are generally based on experiments that measure how a biochemical system responds to small perturbations around the steady state. In contrast, parameter estimation from dynamic data requires time series measurements for all dependent variables. Almost no literature has so far discussed the combined use of both steady-state and transient data for estimating parameter values of biochemical systems.

Results

In this study we introduce a constrained optimization method for estimating parameter values of biochemical pathway models using steady-state information and transient measurements. The constraints are derived from the flux connectivity relationships of the system at the steady state. Two case studies demonstrate the estimation results with and without flux connectivity constraints. The unconstrained optimal estimates from dynamic data may fit the experiments well, but they do not necessarily maintain the connectivity relationships. As a consequence, individual fluxes may be misrepresented, which may cause problems in later extrapolations. By contrast, the constrained estimation accounting for flux connectivity information reduces this misrepresentation and thereby yields improved model parameters.

Conclusion

The method combines transient metabolic profiles and steady-state information and leads to the formulation of an inverse parameter estimation task as a constrained optimization problem. Parameter estimation and model selection are simultaneously carried out on the constrained optimization problem and yield realistic model parameters that are more likely to hold up in extrapolations with the model.     

Fast and accurate fitting of relaxation dispersion data using the flexible software package GLOVE

Relaxation dispersion spectroscopy is one of the most widely used techniques for the analysis of protein dynamics. To obtain a detailed understanding of the protein function from the view point of dynamics, it is essential to fit relaxation dispersion data accurately. The grid search method is commonly used for relaxation dispersion curve fits, but it does not always find the global minimum that provides the best-fit parameter set. Also, the fitting quality does not always improve with increase of the grid size although the computational time becomes longer. This is because relaxation dispersion curve fitting suffers from a local minimum problem, which is a general problem in non-linear least squares curve fitting. Therefore, in order to fit relaxation dispersion data rapidly and accurately, we developed a new fitting program called GLOVE that minimizes global and local parameters alternately, and incorporates a Monte-Carlo minimization method that enables fitting parameters to pass through local minima with low computational cost. GLOVE also implements a random search method, which sets up initial parameter values randomly within user-defined ranges. We demonstrate here that the combined use of the three methods can find the global minimum more rapidly and more accurately than grid search alone.     

Extending the range of amide proton relaxation dispersion experiments in proteins using a constant-time relaxation-compensated CPMG approach

Relaxation compensated constant-time Carr–Purcell–Meiboom–Gill relaxation dispersion experiments for amide protons are presented that detect s-ms time-scale dynamics of protein backbone amide sites. Because of their ten-fold larger magnetogyric ratio, much shorter 180° pulses can be applied to ¹H than to ¹⁵N spins; therefore, off-resonance effects are reduced and a wider range of effective rf fields can often be used in the case of ¹H experiments. Applications to [¹H-¹⁵N]-ubiquitin and [¹H-¹⁵N]-perdeuterated HIV-1 protease are discussed. In the case of ubiquitin, we present a pulse sequence that reduces artifacts that arise from homonuclear ³J(H_N-H) coupling. In the case of the protease, we show that relaxation dispersion of both ¹H and ¹⁵N spins provides a more comprehensive picture of slow backbone dynamics than does the relaxation dispersion of either spin alone. We also compare the relative merits of ¹H versus ¹⁵N transverse relaxation measurements and note the benefits of using a perdeuterated protein to measure the relaxation dispersion of both spin types.     

Determination of a unique solution to parallel proton transfer reactions using the genetic algorithm

Kinetic analysis of the dynamics as measured in multiequilibria systems is readily attained by curve-fitting methodologies, a treatment that can accurately retrace the shape of the measured signal. Still, these reconstructions are not related to the detailed mechanism of the process. In this study we subjected multiple proton transfer reactions to rigorous kinetic analysis, which consists of solving a set of coupled-nonlinear differential rate equations. The manual analysis of such systems can be biased by the operator; thus the analysis calls for impartial corroboration. What is more, there is no assurance that such a complex system has a unique solution. In this study, we used the Genetic Algorithm to investigate whether the solution of the system will converge into a single global minimum in the multidimensional parameter space. The experimental system consisted of proton transfer between four proton-binding sites with seven independent adjustable parameters. The results of the search indicate that the solution is unique and all adjustable parameters converge into a single minimum in the multidimensional parameter space, thus corroborating the accuracy of the manual analysis.     

Multiphysics simulation of left ventricular filling dynamics using fluid-structure interaction finite element method

To relate the subcellular molecular events to organ level physiology in heart, we have developed a three-dimensional finite-element-based simulation program incorporating the cellular mechanisms of excitation-contraction coupling and its propagation, and simulated the fluid-structure interaction involved in the contraction and relaxation of the human left ventricle. The FitzHugh-Nagumo model and four-state model representing the cross-bridge kinetics were adopted for cellular model. Both ventricular wall and blood in the cavity were modeled by finite element mesh. An arbitrary Lagrangian Eulerian finite element method with automatic mesh updating has been formulated for large domain changes, and a strong coupling strategy has been taken. Using electrical analog of pulmonary circulation and left atrium as a preload and the windkessel model as an afterload, dynamics of ventricular filling as well as ejection was simulated. We successfully reproduced the biphasic filling flow consisting of early rapid filling and atrial contraction similar to that reported in clinical observation. Furthermore, fluid-structure analysis enabled us to analyze the wave propagation velocity of filling flow. This simulator can be a powerful tool for establishing a link between molecular abnormality and the clinical disorder at the macroscopic level.     

A density dependent model of pheromone trapping

A discrete, deterministic model is presented to represent a use of pheromones in insect pest control. Males are assumed to be attracted to traps containing virgin females releasing male-attracting pheromone (or equivalent pheromone preparation); males entering the traps are killed. Females are assumed monogamous, and males can mate only once per day; both sexes have the same density dependent mortality. If the amount of pheromone is above a certain value, so that males become limiting, there are two positive equilibria; the larger is locally asymptotically stable, thus control becomes possible. As the amount of pheromone is further increased, these two equilibria merge at a critical release rate, above this value the pest is eradicated. Density dependence makes eradication by this means much easier, for parameter values considered this critical release rate is about two orders of magnitude less than that in a density independent case.     

A new view of water dynamics in immobilized proteins.

The inflection frequency of the deuteron magnetic relaxation dispersion from water in rotationally immobilized protein samples has recently been found to be essentially independent of temperature and protein structure. This remarkable invariance has been interpreted in terms of a universal residence time of 1 microseconds for protein-associated water molecules. We demonstrate here that this interpretation is an artifact of the conventional perturbation theory of spin relaxation, which is not valid for rotationally immobile proteins. Using a newly developed non-perturbative, stochastic theory of spin relaxation, we identify the apparent correlation time of 1 microseconds with the inverse of the nuclear quadrupole frequency, thus explaining its invariance. The observed dispersion profiles are consistent with a broad distribution of residence times, spanning the microseconds range. Furthermore, we argue that the deuteron dispersion is due to buried water molecules rather than to the traditional surface hydration previously invoked, and that the contribution from rapidly exchanging protein hydrogens cannot be neglected. The conclusions of the present work are also relevant to proton relaxation in immobilized protein samples and to magnetic resonance imaging of soft tissue.     

Mapping the dynamics of ligand reorganization via 13CH3 and 13CH2 relaxation dispersion at natural abundance

Flexible ligands pose challenges to standard structure-activity studies since they frequently reorganize their conformations upon protein binding and catalysis. Here, we demonstrate the utility of side chain ¹³C relaxation dispersion measurements to identify and quantify the conformational dynamics that drive this reorganization. The dispersion measurements probe methylene ¹³CH₂ and methyl ¹³CH₃ groups; the latter are highly prevalent side chain moieties in known drugs. Combining these side chain studies with existing backbone dispersion studies enables a comprehensive investigation of μs–ms conformational dynamics related to binding and catalysis. We perform these measurements at natural ¹³C abundance, in congruence with common pharmaceutical research settings. We illustrate these methods through a study of the interaction of a phosphopeptide ligand with the peptidyl-prolyl isomerase, Pin1. The results illuminate the side-chain moieties that undergo conformational readjustments upon complex formation. In particular, we find evidence that multiple exchange processes influence the side chain dispersion profiles. Collectively, our studies illustrate how side-chain relaxation dispersion can shed light on ligand conformational transitions required for activity, and thereby suggest strategies for its optimization.     

pH-Dependent Conformational Changes in Proteins and Their Effect on Experimental pKas: The Case of Nitrophorin 4

The acid-base behavior of amino acids is an important subject of study due to their prominent role in enzyme catalysis, substrate binding and protein structure. Due to interactions with the protein environment, their pK_as can be shifted from their solution values and, if a protein has two stable conformations, it is possible for a residue to have different “microscopic”, conformation-dependent pK_a values. In those cases, interpretation of experimental measurements of the pK_a is complicated by the coupling between pH, protonation state and protein conformation. We explored these issues using Nitrophorin 4 (NP4), a protein that releases NO in a pH sensitive manner. At pH 5.5 NP4 is in a closed conformation where NO is tightly bound, while at pH 7.5 Asp30 becomes deprotonated, causing the conformation to change to an open state from which NO can easily escape. Using constant pH molecular dynamics we found two distinct microscopic Asp30 pK_as: 8.5 in the closed structure and 4.3 in the open structure. Using a four-state model, we then related the obtained microscopic values to the experimentally observed “apparent” pK_a, obtaining a value of 6.5, in excellent agreement with experimental data. This value must be interpreted as the pH at which the closed to open population transition takes place. More generally, our results show that it is possible to relate microscopic structure dependent pKa values to experimentally observed ensemble dependent apparent pK_as and that the insight gained in the relatively simple case of NP4 can be useful in several more complex cases involving a pH dependent transition, of great biochemical interest.     

Fluctuation-driven directional flow in biochemical cycle: further study of electric activation of Na,K pumps

Directional flow of information and energies is characteristic of many types of biochemical reactions, for instance, ion transport, energy coupling during ATP synthesis, and muscle contraction. Can a fluctuating force field, or a noise, induce such a directional flux? Previous work has shown that Na,K-ATPase of human erythrocyte can absorb free energy from an externally applied random-telegraph-noise (RTN) electric field to pump Rb+ up its concentration gradient. However, the RTN field used in these experiments was constant in amplitude and would not mimic fluctuating electric fields of a cell membrane. Here we show that electric fields which fluctuate both in life time and in amplitude, and thus, better mimicking the transmembrane electric fields of a cell, can also induce Rb+ pumping by Na,K-ATPase. A Gaussian-RTN-electric field, or a field with amplitude fluctuating according to the Gaussian distribution, with varied standard deviation (sigma), induced active pumping of Rb+ in human erythrocyte, which was completely inhibited by ouabain. Increased values for sigma led to a nonmonotonic reduction in pumping efficiency. A general formula for calculating the ion transport in a biochemical cycle induced by fluctuating electric field has been derived and applied to a simple four-state electroconformational coupling (ECC) model. It was found that the calculated efficiency in the energy coupling decreased with increasing sigma value, and this effect was relatively small and monotonic, whereas experimental data were more complex: monotonic under certain sets of conditions but nonmonotonic under different sets. The agreement in general features but disagreement in some fine features suggest that there are other properties of the electric activation process for Na,K-ATPase that cannot be adequately described by the simple ECC model, and further refinement of the ECC model is required.     

Protein self-association induced by macromolecular crowding: a quantitative analysis by magnetic relaxation dispersion

In the presence of high concentrations of inert macromolecules, the self-association of proteins is strongly enhanced through an entropic, excluded-volume effect variously called macromolecular crowding or depletion attraction. Despite the predicted large magnitude of this universal effect and its far-reaching biological implications, few experimental studies of macromolecular crowding have been reported. Here, we introduce a powerful new technique, fast field-cycling magnetic relaxation dispersion, for investigating crowding effects on protein self-association equilibria. By recording the solvent proton spin relaxation rate over a wide range of magnetic field strengths, we determine the populations of coexisting monomers and decamers of bovine pancreatic trypsin inhibitor in the presence of dextran up to a macromolecular volume fraction of 27%. Already at a dextran volume fraction of 14%, we find a 30-fold increase of the decamer population and 510(5)-fold increase of the association constant. The analysis of these results, in terms of a statistical-mechanical model that incorporates polymer flexibility as well as the excluded volume of the protein, shows that the dramatic enhancement of bovine pancreatic trypsin inhibitor self-association can be quantitatively rationalized in terms of hard repulsive interactions.