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PubMed2004

Optimal context quantization in lossless compression of image data sequences.

In image compression context-based entropy coding is commonly used. A critical issue to the performance of context-based image coding is how to resolve the conflict of a desire for large templates to model high-order statistic dependency of the pixels and the problem of context dilution due to insufficient sample statistics of a given input image. We consider the problem of finding the optimal quantizer Q that quantizes the K-dimensional causal context Ct = (Xt-t1,Xt-t2,...,X t-tK) of a source symbol Xt into one of a set of conditioning states. The optimality of context quantization is defined to be the minimum static or minimum adaptive code length of given a data set. For a binary source alphabet an optimal context quantizer can be computed exactly by a fast dynamic programming algorithm. Faster approximation solutions are also proposed. In case of m-ary source alphabet a random variable can be decomposed into a sequence of binary decisions, each of which is coded using optimal context quantization designed for the corresponding binary random variable. This optimized coding scheme is applied to digital maps and alpha-plane sequences. The proposed optimal context quantization technique can also be used to establish a lower bound on the achievable code length, and hence is a useful tool to evaluate the performance of existing heuristic context quantizers.

Algorithms↗

PubMed1999

A sexually dimorphic pattern of growth hormone secretion in the elderly.

In rodents, the sexually dimorphic pattern of pulsatile GH secretion is an important determinant of growth, liver enzyme function and insulin-like growth factor I (IGF-I) expression. Whether this difference is present in humans at different ages is unclear. We studied GH secretory patterns in the elderly by constructing 24-h serum GH profiles in 45 male and 38 female (age, 59.4-73.0 yr) volunteers and related patterns to IGF-I, IGF-binding protein-3 (IGFBP-3), and GH-binding protein levels; body mass index; and waist/hip ratio. Serum GH concentrations were measured in samples drawn at 20-min intervals and analyzed using a sensitive chemiluminescent assay (Nichols Institute Diagnostics: sensitivity, 0.036 mU/L). The 24-h serum GH profiles were analyzed using a concentration distribution method to determine GH peak and trough levels, spectral analysis, and assessment of serial irregularity by approximate entropy (ApEn). There was a highly significant difference in mean 24-h serum GH concentrations in females compared to males (males, 0.88 mU/L; females, 1.31 mU/L; P = 0.009) as a result of significantly higher trough GH levels (males, 0.04 mU/L; females, 0.16 mU/L; P < 0.001). Peak values were not significantly different. Serum IGF-I levels were significantly higher in males (males, 162.4 ng/mL; females, 87.8 ng/ mL; P < 0.001). Peak GH values were related to serum IGF-I levels (males: r = 0.39; P = 0.009; females: r = 0.5; P = 0.002), whereas trough GH levels were not. IGFBP-3 levels were similar and related to GH peaks only in males (r = 0.32; P = 0.03). GH was secreted with a dominant periodicity of 200 min in males and 280 min in females (P < 0.025). The proportion of time taken up by regular oscillatory activity was less in females (females, 11.1%; males, 14.7%; P = 0.01). GH secretion assessed by ApEn was more disordered in females (males, 0.60; females, 0.81; P < 0.001), and increasing disorder was associated with lower IGF-I levels. Body mass index was negatively related to GH in both sexes. In males, trough values were the major determinant (r = -0.31; P = 0.04), whereas in females, the peak value was the major determinant (r = 0.35; P = 0.04). Trough GH levels were inversely related in both sexes to waist/hip ratio (males: r = -0.40; P = 0.006; females: r = -0.44; P = 0.006) and to increasing secretory disorder (ApEn; r = -0.46; P < 0.001). These data demonstrate a sexually dimorphic pattern of GH secretion in the elderly.

Aged↗

PubMed1995

DNA cytometric abnormalities in human esophageal squamous intraepithelial and invasive carcinomas.

DNA ploidy abnormalities of 21 archival human esophageal intraepithelial neoplasia samples were assessed, using image cytometry of deparaffinized samples, with reference to invasive squamous cell carcinoma and corresponding uninvolved squamous epithelium. Cytometric parameters investigated were proportion of G0G1 aneuploid cell population, histogram typing, proportion of G0G1 diploid nuclei, coefficient of variation, mean DNA content, crude 5c exceeding proportion, 2c deviation index, malignancy index and grade, and entropy. The distributions of the above parameters were compared using the paired t test and Fisher's exact test. Among 10 parameters used, Auer typing of DNA histograms, crude 5c exceeding rate, 2c deviation index and malignancy grade according to Böcking allowed discrimination between uninvolved epithelium and invasive squamous cell carcinoma as well as intraepithelial neoplasia. In particular, the distribution of 2c deviation index in the uninvolved epithelium did not overlap that of intraepithelial and invasive carcinomas. The above four parameters, however, were unable to discriminate intraepithelial neoplasia from invasive carcinoma.

Aneuploidy↗

PubMed2000

Omega-space adaptive acquisition technique for magnetic resonance imaging from projections.

An omega-space adaptive acquisition technique for MRI from projections is presented. It is based on the evaluation of the information content of a set composed of four initial projections, measured at angles 0 degrees, 45 degrees, 90 degrees, and 135 degrees, followed by the selection of new angles where the information content is maximum. An entropy function is defined on the power spectrum of the projections that is useful for evaluating the information content of each projection. The method makes it possible to reduce the total acquisition time with little degradation of the reconstructed image and it adapts to the arbitrary shape of the sample. For this reason, it can be particularly useful in those applications where acquisition from projections is strongly recommended to save acquisition time, such as functional MRI, imaging of species having very short T(2), or angiography. The method has been tested both on simulated data and on experimental data collected by a commercial MRI apparatus. The method has also been compared to the regular acquisition method, that is, the standard acquisition method in MRI from projections.

Image Processing, Computer-Assisted↗

PubMed1989

A real-time autoregressive spectrum analyzer for Doppler ultrasound signals.

A system based on a digital signal processor and a microcomputer has been programmed to estimate the maximum entropy autoregressive (AR) power spectrum of ultrasonic Doppler shift signals and display the results in the form of a sonogram in real-time on a computer screen. The system, which is based on a TMS 320C25 digital signal processor chip, calculates spectra with 128 frequency components from 64 samples of the Doppler signal. The samples are collected at a programmable rate of up to 40.96 kHz, and the computation of each spectrum takes typically 3.2 ms. The feasibility of on-line AR spectral estimation makes this type of analysis an attractive alternative to the more conventional fast Fourier transform approach to the analysis of Doppler ultrasound signals.

Analog-Digital Conversion↗

PubMed2006

Improved lossless intra coding for H.264/MPEG-4 AVC.

A new lossless intra coding method based on sample-by-sample differential pulse code modulation (DPCM) is presented as an enhancement of the H.264/MPEG-4 AVC standard. The H.264/AVC design includes a multidirectional spatial prediction method to reduce spatial redundancy by using neighboring samples as a prediction for the samples in a block of data to be encoded. In the new lossless intra coding method, the spatial prediction is performed based on samplewise DPCM instead of in the block-based manner used in the current H.264/AVC standard, while the block structure is retained for the residual difference entropy coding process. We show that the new method, based on samplewise DPCM, does not have a major complexity penalty, despite its apparent pipeline dependencies. Experiments show that the new lossless intra coding method reduces the bit rate by approximately 12% in comparison with the lossless intra coding method previously included in the H.264/AVC standard. As a result, the new method is currently being adopted into the H.264/AVC standard in a new enhancement project.

Algorithms↗

PubMed2004

How many clusters? An information-theoretic perspective.

Clustering provides a common means of identifying structure in complex data, and there is renewed interest in clustering as a tool for the analysis of large data sets in many fields. A natural question is how many clusters are appropriate for the description of a given system. Traditional approaches to this problem are based on either a framework in which clusters of a particular shape are assumed as a model of the system or on a two-step procedure in which a clustering criterion determines the optimal assignments for a given number of clusters and a separate criterion measures the goodness of the classification to determine the number of clusters. In a statistical mechanics approach, clustering can be seen as a trade-off between energy- and entropy-like terms, with lower temperature driving the proliferation of clusters to provide a more detailed description of the data. For finite data sets, we expect that there is a limit to the meaningful structure that can be resolved and therefore a minimum temperature beyond which we will capture sampling noise. This suggests that correcting the clustering criterion for the bias that arises due to sampling errors will allow us to find a clustering solution at a temperature that is optimal in the sense that we capture maximal meaningful structure--without having to define an external criterion for the goodness or stability of the clustering. We show that in a general information-theoretic framework, the finite size of a data set determines an optimal temperature, and we introduce a method for finding the maximal number of clusters that can be resolved from the data in the hard clustering limit.

Journal Article↗

PubMed1997

Description of the ventriculoarterial interaction dynamics using recurrence plot strategies.

OBJECTIVE: The classical description of ventriculoarterial coupling by calculating the ratio between the effective arterial elastance Ea to the end-systolic elastance Ees does not give insight into the underlying dynamics of the interaction between left-ventricular pressure (LVP) and aortic pressure (AOP) and flow (AOF). The aim of this study was to introduce a state space representation for the ventriculoarterial coupling and to quantify changes of the coupling state. METHODS: A ventriculoarterial state space orbit VAO was defined to be dependent on three variables: VAO = [LVP(t), AOP(t + delta t), AOF(t + delta t)]. Changes in the coupling effect directly or indirectly on the time series of these parameters. They reflect the actual state of the cardiovascular system. The time delay delta t between the LVP and the aortic signals takes respect to the short delay between the heart action and the resulting waves in the arterial tree. The recurrence map of the VAO(i) (i = 1 .. N, N = number of points) is constructed by plotting the index i of every single point on the orbit (x-axis) against the indices of his 10 nearest neighbors (y-axis) in distance. The data were recorded in 9 anaesthetized pigs with a sample frequency of 512 Hz over a period of 6 seconds using piezoelectric pressure sensors and a Doppler flowmeter. A control condition was compared to a total occlusion of the descending aorta as a strong artificial disturbance of ventriculoarterial interaction. The nonlinear parameters percent recurrence, percent determinism and the entropy were calculated from the plot. RESULTS: Periodic crossing points and forbidden zones in all plots identify the nonlinear character of the chosen variables. The recurrent patterns are less rigid for control conditions than for total occlusion. Entropy (2.3% rise) and determinism (24% rise) are significantly (p < 0.003) increased. Total aortic occlusion leads to more complex time correlation patterns. CONCLUSIONS: These results may reflect the loss of an ideal coupling state leading to a more complex deterministic behavior of the overall regulatory system. Because recurrence plots do not impose rigid constraints on data set size, stationarity, or statistical distribution, we hypothesize that this technique might be useful to describe the nonlinear dynamics between left ventricle and arterial system.

Animals↗

PubMed1997

Thermodynamic analysis of antigen-antibody binding using biosensor measurements at different temperatures.

The thermodynamic parameters of the interaction between hen egg white lysozyme and Fab D1.3 were determined by measuring the temperature dependence of the ratio of its kinetic association and dissociation rate constants. Biosensor technology (BIAcore 2000) was used to measure the rate constants at temperatures ranging from 5 to 40 degrees C. The value of DeltaG degrees at 25 degrees C (-49 kJ M-1) calculated by this method was very close to that obtained previously from fluorescence quenching measurements (-48.5 kJ M-1). However, the value of DeltaH degrees measured at 25 degrees C by biosensor technology (-35 kJ M-1) was smaller than that determined previously by microcalorimetry (-90 kJ M-1). Another difference was the limited variation of ln K and DeltaG with temperature observed with BIAcore compared to the steady decrease of ln K with temperature found by calorimetry. Our data showed that the binding reaction was driven only by enthalpy below 23 degrees C, by enthalpy and entropy between 23 and 35 degrees C, and only by entropy above 35 degrees C. This suggests, inter alia, that the contribution from the enthalpy of hydration due to the water molecules present at the interface in the lysozyme-antibody complex is progressively eliminated as the temperature increases. Whereas calorimetric data pertain to all the components present in the sample, including solvent molecules, BIAcore measurements monitor only the physical association and dissociation of the two macromolecular species. The difference between the two sets of data may also reflect the complexity of the binding mechanism between lysozyme and Fab D1.3.

Animals↗

PubMed2004

Entropy effects on protein hinges: the reaction catalyzed by triosephosphate isomerase.

Many proteins utilize segmental motions to catalyze a specific reaction. The Omega loop of triosephosphate isomerase (TIM) is important for preventing the loss of the reactive enediol(ate) intermediate. The loop opens and closes even in the absence of the ligand, and the loop itself does not change conformation during movement. The conformational changes are localized to two hinges at the loop termini. Glycine is never observed in native TIM hinge sequences. In this paper, the hypothesis that limited access to conformational space is a requirement for protein hinges involved in catalysis was tested. The N-terminal hinge was mutated to P166/V167G/W168G (PGG), and the C-terminal hinge was mutated to K174G/T175G/A176G (GGG) in chicken TIM. The single-hinge mutants PGG and GGG had k(cat) values 200-fold lower than that of the wild type and K(m) values 10-fold higher. The k(cat) of double-hinge mutant P166/V167G/W168G/K174G/T175G/A176G was reduced 2500-fold; the K(m) was 10-fold higher. A combination of primary kinetic isotope effect measurements, isothermal calorimetric measurements, and (31)P NMR spectroscopic titration with the inhibitor 2-phosphoglycolate revealed that the mutants have a different ligand-binding mode than that of the wild-type enzyme. The predominant conformations of the mutants even in the presence of the inhibitor are loop-open conformations. In conclusion, mutation of the hinge residues to glycine resulted in the sampling of many more hinge conformations with the consequence that the population of the active-closed conformation is reduced. This reduced population results in a reduced catalytic activity.

Animals↗

PubMed1998

Reaction path and free energy calculations of the transition between alternate conformations of HIV-1 protease.

Two different structures of ligand-free HIV protease have been determined by X-ray crystallography. These structures differ in the position of two 12 residue, beta-hairpin regions (or "flaps") which cap the active site. The movements of the flaps must be involved in the binding of substrates since, in either conformation, the flaps block the binding site. One of these structures is similar to structures of the ligand-bound enzyme; however, the importance of both structures to enzyme function is unclear. This transformation takes place on a time scale too long for conventional molecular dynamics simulations, so the process was studied by first identifying a reaction path between the two structures and then calculating the free energy along this path using umbrella sampling. For the ligand-free enzyme, it is found that the two structures are nearly equally stable, with the ligand-bound-type structure being less stable, consistent with X-ray crystallography data. The more stable open structure does not have a lower potential energy, but is stabilized by entropy. The transition occurs through a collapse and reformation of the beta-sheet structure of the conformationally flexible, glycine-rich flap ends. Additionally, some problems in studying conformational changes in proteins through the use of a single reaction path are addressed.

Crystallography, X-Ray↗

PubMed1995

Deconvolution of STM images using entropy as a regularization functional.

The Maximum Entropy approach is applied to restore and sharpen scanning tunneling microscopy (STM) images with atomic resolution. Based on the STM theory of Tersoff and Hamann the process of data acquisition can be approximated by the convolution of a localised atomic density of states (i.e. narrow spots in the reconstruction) of the sample and a Gaussian resolution function which limits the resolution. In STM practice a good and robust estimation of the atomic core positions is necessary for different reasons, such as to be able to calculate the characteristics of the atomic lattice or to study non-periodicities.

Journal Article↗

PubMed2005

Peptide conformational equilibria computed via a single-stage shifting protocol.

We study the conformational equilibria of two peptides using a novel statistical mechanics approach designed for calculating free energy differences between highly dissimilar conformational states. Our results elucidate the contrasting roles of entropy in implicitly solvated leucine dipeptide and decaglycine. The method extends earlier work by Voter and overcomes the notorious "overlap" problem in free energy computations by constructing a mathematically equivalent calculation with high conformational similarity. The approach requires only equilibrium simulations of the two states of interest, without the need for sampling transition states. We discuss possible extensions and optimizations of the approach.

Peptides↗

PubMed2005

Decreased physiologic variability as a generalized response to human endotoxemia.

OBJECTIVE: To test the effect in normal human volunteers of transient systemic inflammation on the variability in time-series behaviors of widely divergent physiologic measures of the human inflammatory response. DESIGN: Prospective study of human volunteers who were tested on 2 consecutive days, a control day and a treatment day. Each participant served as his or her own control. SETTING: Critical care facility of a university medical center. SUBJECTS: Subjects were eight healthy human volunteers. INTERVENTIONS: Participant subjects were tested on both a baseline day with no intervention and on a treatment day when they received 4 ng/kg intravenous Escherichia coli endotoxin. MEASUREMENTS AND MAIN RESULTS: Continuous electrocardiographic recordings and serial blood sampling (performed every 5 mins) were used to create time-series of heart rate (R-R intervals), neutrophil function (phagocytosis), and plasma cortisol concentrations. For each primary measure, we recorded a significant increase in the regularity (decreased variability) of the functional measurement as assessed by the statistical entity, approximate entropy. CONCLUSIONS: Increased regularity, or decreased variability, of organ functions is a generalized response to systemic inflammation that occurs in widely divergent systems during endotoxemia.

Adult↗

PubMed1999

Short-term fasting suppresses leptin and (conversely) activates disorderly growth hormone secretion in midluteal phase women--a clinical research center study.

Short term fasting activates the corticotropic and somatotropic, and suppresses the reproductive, axis in men. Analogous neuroendocrine responses are less well characterized in women. Recently, we identified a negative association between the adipocyte-derived nutritional signaling peptide, leptin, and pulsatile GH secretion in older fed women. In the present study, we investigated the impact of acute nutrient deprivation on pulsatile GH and LH secretion and mean leptin concentrations in eight healthy young women in the sex-steroid replete milieu of the midluteal phase of the normal menstrual cycle. Volunteers underwent 24-h blood sampling during randomly ordered, short term (2.5-day), fasting vs. fed sessions in separate menstrual cycles. Pulsatile GH and LH secretion over 24 h was quantified by deconvolution analysis, nyctohemeral rhythmicity was quantified by cosinor analysis, and the orderliness of the GH or LH release process was quantified by the approximate entropy statistic. By paired statistical analysis, a 2.5-day fast failed to alter mean (pooled) 24-h serum concentrations of LH, progesterone, estradiol, or PRL, but increased cortisol levels more than 1.5-fold (P = 0.0003). Concurrently, mean (pooled) serum leptin concentrations fell by 75% (P = 0.0003), and insulin-like growth factor I (IGF-I; P < 0.05) and insulin decreased significantly (P = 0.0018). In contrast, the daily pulsatile GH secretion rate rose 3-fold (P < 0.001). Amplified daily GH secretion was attributable mechanistically to a 2.3-fold rise in GH secretory burst mass, reflecting an increased GH secretory burst amplitude (P < 0.01). The GH half-life, duration of GH secretory bursts, and GH pulse frequency did not vary during short term fasting. The disorderliness of GH release increased significantly with nutrient restriction (P = 0.005). The mesor and amplitude of the nyctohemeral serum GH concentration rhythm also rose with fasting (P < 0.01), but the timing of maximal serum GH concentrations did not change. Thus, short-term (2.5-day) fasting during the sex steroid-replete midluteal phase of the menstrual cycle in healthy young women profoundly suppresses 24-h serum leptin and insulin (and to a lesser degree, IGF-I) concentrations, augments cortisol release, but fails to alter daily LH, estradiol, or progesterone concentrations. In contrast, the GH axis exhibits strikingly amplified pulsatile secretion, increased nyctohemeral rhythmicity, and marked disorderliness of the release process. We conclude that the somatotropic axis is more evidently vulnerable to short-term nutrient restriction than the reproductive axis in steroidogenically sufficient midluteal phase women. This study invites the question of whether normal (nutritionally replete) GH secretory dynamics can be restored in fasting women by human leptin, insulin, or IGF-I infusions.

Adult↗

PubMed2001

Developmentally delimited emergence of more orderly luteinizing hormone and testosterone secretion during late prepuberty in boys.

To quantitate changing feedback control in the GnRH-LH/FSH-testosterone axis in male puberty, we here quantitate the orderliness of hormone release patterns using the regularity (pattern-sensitive) statistic, approximate entropy (ApEn), in 46 eugonadal boys representing 6 genitally defined stages of normal puberty. ApEn is a single variable, model-free, and scale-independent barometer of coordinate signaling or integrative regulation within a coupled neuroendocrine axis. Accordingly, we quantitated ApEn of LH profiles obtained by immunofluorometric assay of sera sampled every 20 min for 24 h. LH ApEn declined remarkably between early prepuberty (genital stage I-A: mean bone age, 4.6 +/- 1.6 yr; testis volume, <3 mL for at least 3 succeeding yr) and late prepuberty (genital stage I-C: bone age, 8.7 +/- 1.8 yr; testis volume, <3 mL for up to 1 yr thereafter; P: = 0.00019), which indicates the acquisition of more regular LH release patterns in late prepuberty. Maximal LH orderliness occurred in puberty stage II (bone age, 10.7 +/- 1.0 yr; testis volume, 2.8 +/- 0.4 mL). The LH secretory process was more disorderly in mid- and later puberty (Tanner stages III and IV). Transpubertal variations in testosterone ApEn manifested a similar tempo, i.e. the greatest regularity of testosterone secretion (lowest ApEn) emerged in Tanner genital stage II (P: < 10(-)(7)), with less orderly patterns evident both earlier and later in sexual development. In contrast, FSH ApEn values remained invariant of pubertal status. Analysis of bihormonal coupling using the theoretically related bivariate cross-ApEn statistic disclosed maximal 2-hormone synchrony for LH and testosterone secretion in genital stage II (P: = 0.031), with relative deterioration of coordinate LH and testosterone release patterns both before and after. LH and FSH release became maximally synchronous at the end of prepuberty (genital stage I-C; P: = 0.029), and FSH and testosterone synchrony peaked in pubertal stage III (P: = 0.037). As mean 24-h serum concentrations of LH, FSH, and testosterone rose transpubertally by 35-fold (LH), 68-fold (FSH), and 70-fold (testosterone), respectively, we infer that pubertal developmental stage per se rather than level of hormone output dictates coordinate GnRH-LH/FSH-testosterone secretion. In summary, in eugonadal boys, the regularity of 24-h LH and testosterone secretory patterns undergoes well defined pubertal stage-specific control. No sexually developmentally delimited regulation is inferable for FSH. The concept of temporally biphasic puberty-dependent variations in neurohormone secretory regularity contrasts with the unidirectional rise in daily hormone output. Accordingly, we infer that late prepuberty and early puberty (Tanner genital stages IC and II) embody a physiologically unique sexual developmental window, marked by transiently enhanced LH and testosterone feedback stability in boys. Whether analogous plasticity of hypothalamo-pituitary-gonadal interactions unfolds during female adolescence is not known.

Child↗

PubMed2000

Analysis of vocal disorders in a feature space.

This paper provides a way to classify vocal disorders for clinical applications. This goal is achieved by means of geometric signal separation in a feature space. Typical quantities from chaos theory (like entropy, correlation dimension and first lyapunov exponent) and some conventional ones (like autocorrelation and spectral factor) are analysed and evaluated, in order to provide entries for the feature vectors. A way of quantifying the amount of disorder is proposed by means of a healthy index that measures the distance of a voice sample from the centre of mass of both healthy and sick clusters in the feature space. A successful application of the geometrical signal separation is reported, concerning distinction between normal and disordered phonation.

Algorithms↗

PubMed1975

Dynamics of ligand binding to myoglobin.

Myoglobin rebinding of carbon monoxide and dioxygen after photodissociation has been observed in the temperature range between 40 and 350 K. A system was constructed that records the change in optical absorption at 436 nm smoothly and without break between 2 musec and 1 ksec. Four different rebinding processes have been found. Between 40 and 160 K, a single process is observed. It is not exponential in time, but approximately given by N(t) = (1 + t/to)-n, where to and n are temperature-dependent, ligand-concentration independent, parameters. At about 170 K, a second and at 200 K, a third concentration-independent process emerge. At 210 K, a concentration-dependent process sets in. If myoglobin is embedded in a solid, only the first three can be seen, and they are all nonexponential. In a liquid glycerol-water solvent, rebinding is exponential. To interpret the data, a model is proposed in which the ligand molecule, on its way from the solvent to the binding site at the ferrous heme iron, encounters four barriers in succession. The barriers are tentatively identified with known features of myoglobin. By computer-solving the differential equation for the motion of a ligand molecule over four barriers, the rates for all important steps are obtained. The temperature dependences of the rates yield enthalpy, entropy, and free-energy changes at all barriers. The free-energy barriers at 310 K indicate how myoglobin achieves specificity and order. For carbon monoxide, the heights of these barriers increase toward the inside; carbon monoxide consequently is partially rejected at each of the four barriers. Dioxygen, in contrast, sees barriers of about equal height and moves smoothly toward the binding site. The entropy increases over the first two barriers, indicating a rupturing of bonds or displacement of residues, and then smoothly decreases, reaching a minimum at the binding site. The magnitude of the decrease over the innermost barrier implies participation of heme and/or protein. The nonexponential rebinding observed at low temperatures and in solid samples implies that the innermost barrier has a spectrum of activation energies. The shape of the spectrum has been determined; its existence can be explained by assuming the presence of many conformational states for myoglobin. In a liquid at temperatures above about 230 K, relaxation among conformational states occurs and rebinding becomes exponential.

Binding Sites↗