Myoglobin's old and new clothes: from molecular structure to function in living cells

With an introduction and perspectives by Jonathan Wittenburg.

Myoglobin, a mobile carrier of oxygen, is developed in red muscle and heart cells in response to increased demand for oxygen during exercise and transports oxygen from the sarcolemma to the mitochondria of vertebrate heart and red muscle cells (Wittenberg and Wittenberg, 2003; Kanatous et al., 2009). Random displacement of oxymyoglobin molecules within a gradient of oxymyoglobin concentration (translational diffusion of oxymyoglobin) provides a flux of oxygen additional to the simple diffusive flux (Wyman, 1966). The flux of oxymyoglobin is, of course, accompanied by an equal and opposite flux of deoxymyoglobin (Hemmingsen, 1965). A myoglobin molecule carrying oxygen does not transverse the entire distance from sarcolemma to mitochondrion. Instead, there is a continuing reaction in which myoglobin combines with and dissociates oxygen, achieving near equilibrium. Direct transfer of oxygen from one myoglobin molecule to another does not occur (Gibson, 1959). The rate of dissociation of ligands from myoglobin (‘off’ constant) plays an important role. If this rate is very small, as is oxygen dissociation from hemoglobin H or carbon monoxide dissociation from myoglobin, carrier-mediated ligand diffusion nearly vanishes (Mochizuki and Forster, 1962; Wittenberg, 1966). Gibson and colleagues and Wittenberg present strong evidence that both the equilibrium and reaction rate constants of leghemoglobin (and similar plant symbiotic hemoglobins) and of vertebrate myoglobins with oxygen are subject to Darwinian natural selection (Gibson et al., 1989; Wittenberg, 2007). Their oxygen binding properties are thereby optimized for transport of intracellular oxygen.

A similar system is found in plants. Here, leghemoglobin, a protein similar to myoglobin but with a tenfold greater affinity for oxygen, transports oxygen from the cell membrane of the central cells of the legume root nodule to the symbiosomes, which are membrane-bound intracellular organelles housing the bacteroids, the intracellular nitrogen-fixing form of the bacterium Rhizobium. Symbiosomes, in the root nodule cell, can be considered analogous to the mitochondria of muscle cells. Leghemoglobin is required to maintain the oxygen flux demanded by bacteroidal nitrogen fixation. The concentration of oxygen dissolved in the cytoplasm of the soybean root nodule is vanishingly small, about 10 nanomolar, near the P₅₀ of leghemoglobin, whereas the concentration of leghemoglobin-bound oxygen can exceed millimolar levels. The ratio of leghemoglobin-bound oxygen to free oxygen exceeds one hundred thousand, and essentially all of the oxygen flux must be leghemoglobin mediated (Appleby, 1984). In support of this assertion, carbon monoxide blockade of leghemoglobin in the living nodule essentially abolishes bacteroidal oxidative phosphorylation (Bergersen et al., 1973).

Myoglobin and leghemoglobin are partially deoxygenated in both the nodule cell and the working myocyte (Wittenberg and Wittenberg, 2003), thereby fulfilling the requirement that myoglobin must be desaturated with oxygen somewhere in the system for facilitated diffusion to operate (Wyman, 1966). Desaturation has two important consequences: (1) partially desaturated myoglobin/leghemoglobin is available to buffer and optimize intracellular oxygen pressure near the P₅₀ of the proteins, and (2) bacteroidal nitrogen fixation, inhibited by mere traces of oxygen, is protected from intracellular oxygen.

The usefulness of myoglobin to the cardiac or red muscle cell is established beyond doubt (Wittenberg and Wittenberg, 2003). What is contested is the fraction of the total oxygen flux carried by diffusing oxymyoglobin, relative to the fraction carried by simple diffusive flux of dissolved oxygen. In the root nodule, as we have seen, virtually all the oxygen is carried by leghemoglobin. By contrast, the fraction of the oxygen flux carried by myoglobin in vertebrate heart and muscle is very difficult to determine experimentally. Calculations presented below suggest that it might not be large in some situations but could be considerable in others and is very dependent on the effective myoglobin diffusion coefficient within the cell and on the sarcoplasmic myoglobin concentration. The calculation of the relative contribution, however, depends upon the cell model, cell architecture and the physiological condition. In agreement with these considerations, calculation from experimental data shows that, in the myoglobin-rich muscles of the apneic juvenile elephant seal, a large fraction of the total oxygen flux is supported by myoglobin (see below) (Ponganis et al., 2008). Thus, in both the root nodule and in juvenile seal muscle, where the leghemoglobin/myoglobin concentration is high relative to intracellular oxygen, under steady-state conditions, myoglobin-mediated oxygen flux dominates.

Mb isoforms and concentration vary across species and tissues. The protein appears in root nodules, marine worms, avian gizzards, smooth muscle and striated muscle and has different structural features, which lead to variations in P₅₀ and potentially other functions (Dayhoff and Eck, 1968; Weber and Pauptit, 1972; Vinogradov and Kapp, 1991; Enoki et al., 1995). In terrestrial mammals, Mb resides primarily in cardiac and skeletal muscle. Rodent heart contains about 0.2 mmol l⁻¹. If the volume calculation for cardiomyocyte were to exclude mitochondria (35% of cell volume) and the sarcoplasmic reticulum (4% of the cell volume), the Mb concentration would rise to 0.33 mmol l⁻¹ (Wittenberg and Wittenberg, 2003). Human skeletal muscle has a slightly higher concentration of Mb, ca. 25.4 mg g⁻¹ dry wt (0.4–0.5 mmol l⁻¹) (Jansson and Sylvén, 1981b; Jansson and Sylvén, 1981a; Möller and Sylvén, 1981) In particular, the oxidative, slow-twitch type I fiber has ca. 1.6 times more Mb than the fast-twitch, glycolytic type II fiber (Jansson and Sylvén, 1983). In marine mammals, muscle Mb concentration rises dramatically to 4.5 g per 100 g tissue or ca. 3.8 mmol l⁻¹ in the cytoplasm (Ponganis et al., 1993; Ponganis et al., 2002).

The correlation of Mb concentration with oxidative enzyme activity, capillary density and mitochondrial density appears consistent with its O₂ storage and transport role. Type I fiber has a 50% higher concentration of succinate dehydrogenase, 20% higher capillary/fiber ratio and a higher mitochondrial volume density than type II fibers (Essen et al., 1975). Unfortunately, this correlation does not always hold. The least oxidative intrafusal fibers have very high myoglobin content (James, 1971). Moreover, the highly oxidative heart muscle has a lower myoglobin concentration than skeletal muscle.

With altered physiological demands, Mb levels can increase. Some studies have detected with training an increase in both Mb and the levels of the oxidative enzyme succinate dehydrogenase (Harms and Hickson, 1983; Beyer and Fattore, 1984). Even though type I fiber usually has a higher concentration of succinate dehydrogenase than type II fiber, training can increase the amount in type II fibers to reach the level in type I (Henriksson and Reitman, 1975). Other studies, however, have not detected any increase in Mb with training (Svedenhag et al., 1983; Masuda et al., 2001). At high altitude, Mb expression also appears to increase (Gimenez et al., 1977; Terrados et al., 1990; Weber, 2007). More recently, increased myoglobin gene expression, controlled by intracellular calcium pools, has been shown to depend on a combination of both hypoxia and increased work output, which might explain the earlier discrepancies (Kanatous et al., 2009).

To understand the function of Mb in the cell requires an accurate determination of its concentration. Generally, researchers have utilized either an optical or an immunohistochemical approach (Schuder et al., 1979; Möller and Sylvén, 1981; Nemeth and Lowry, 1984; Kunishige et al., 1996). Because of its simplicity, most studies have leaned on the optical method, which relies on spectral differences to distinguish the Mb from the Hb contribution (de Duve, 1948; Reynafarje, 1963). Alternatively, quantitative separation of hemoglobin from myoglobin by affinity chromatography has also been used (Schuder et al., 1979).

In the visible spectrum, Mb and Hb exhibit almost indistinguishable spectral features. However, as noted by deDuvet and Reynafarje, HbCO shows an equal absorbance intensity in the β (538 nm) and the α (568 nm) bands (de Duve, 1948; Reynafarje, 1963). By contrast, the MbCO β peak has a 20% lower intensity than the α peak. As a consequence, the signal intensity difference at 568 and 538 nm divided by the Mb extinction coefficient leads to a determination of the Mb concentration in blood-perfused tissue. An alternative approach uses the spectral differences in the Soret band centered at ca. 400 nm (Nakatani, 1988).

Many researchers use the Reynafarje method ‘as is’, whereas others have included a calibration step that recognizes the limited Mb database used in the initial standardization (Kanatous et al., 2002). Nevertheless, such spectral differencing strategy underpins current determination of heme proteins, such as neuroglobin (Williams et al., 2008).

By contrast, ¹H NMR detects distinct MbCO and HbCO signals in a clear spectral window (Jue, 1994). The HbCO γ-CH₃ Val E11 signals of the α and β subunits appear at −1.72 p.p.m. and −1.92 p.p.m. The MbCO γ-CH₃ Val-E11 signal appears distinctly at −2.40 p.p.m. These signals resonate outside the crowded spectral region between 0–10 p.p.m. The corresponding MbO₂ or metMb signals resonate at −2.8 p.p.m. and −3.7 p.p.m. (Chung et al., 1996; Kreutzer and Jue, 2004). The HbO₂ and metHb signals do not co-resonate (Ho and Russu, 1981). In tissue samples, the cytochrome concentration appears too low to produce any significant spectral interference (Feng et al., 1990; Masuda et al., 2008).

The NMR analysis, however, requires a more highly concentrated sample but yields an accurate Mb concentration of 0.28 and 0.26 mmol l⁻¹ from perfused and unperfused rat heart homogenate, respectively. In buffer-perfused heart, no Hb signals can interfere. By comparison, the optical differencing method detects a Mb concentration of 0.26 mmol l⁻¹ in perfused heart but 0.36 mmol l⁻¹ in the homogenate derived from in situ heart, overestimating the Mb concentration by ca. 50% (Masuda et al., 2008).

Using the Reynafarje method without appropriate calibration and baseline correction can lead to an erroneous Mb determination. ¹H NMR analysis overcomes the HbCO interference but requires a higher Mb concentration (Ho and Russu, 1981; Kreutzer et al., 1992). Nevertheless, a correct analysis of Mb function demands an accurate determination of the Mb concentration (Masuda et al., 2008).

A relevant question is: do the kinetics of the reaction between O₂ and Mb, especially MbO₂ dissociation kinetics, limit Mb-facilitated diffusion? That a minimum velocity of the chemical reaction is required to enable a carrier-mediated transport of the ligand has been recognized early on by Wyman and also Wittenberg and more recently was discussed with respect to Mb (Wittenberg, 1966; Wyman, 1966; Wittenberg and Wittenberg, 2003). Wyman showed, for example, that rotational diffusion cannot support a facilitated O₂ diffusion because much faster reaction kinetics would be required in view of the small distance over which the ligand is transported, namely the molecular diameter of the carrier (Gros et al., 1984). Also, Wittenberg has shown that CO, whose dissociation kinetics from Mb is almost 1000 times slower than that of O₂, does not exhibit Mb-facilitated CO diffusion in 150 μm Millipore filters, in which O₂ diffusion was facilitated by Mb by 100% (Wittenberg, 1966). Kutchai and colleagues have analyzed the problem in quantitative terms for hemoglobin-facilitated O₂ diffusion and arrived at the conclusion that the oxygen dissociation rate limitation of hemoglobin-facilitated diffusion is of minor importance when diffusion distances of >10 μm are considered (Kutchai et al., 1970).

The internal kinetic energy fuels the random molecular rotational and translational self-diffusion. These motions underlie the ability of the cell to regulate its host of chemical reactions and therefore its function. An impeded cellular diffusion would modulate the molecular interactions, and the reactions observed in vitro might not even occur in a cellular environment. Consequently an understanding of protein function in the cell requires insight into the dynamic processes, as reflected in rotational and translational diffusion.

Riveros-Moreno and Wittenberg first measured the diffusion coefficient of myoglobin to elucidate the mechanism of hemoglobin- and myoglobin-facilitated O₂ diffusion (Riveros-Moreno and Wittenberg, 1972). Both C¹⁴ labeling and spectrophotometric measurements were used. They found a diffusion coefficient of Mb at infinite dilution (D_Mb,0) of ca. 10×10⁻⁷ cm² s⁻¹ at 20°C at low concentration when myoglobin diffuses freely in a protein-free buffer solution. At a myoglobin concentration of 25 g/100 ml, a concentration corresponding to the total protein concentration found in muscle tissues, they found diffusion slows to D_Mb≈3.2×10⁻⁷ cm² s⁻¹. Extrapolating the diffusion data obtained at the various Mb concentrations to infinite dilution, [Mb]=0, one predicts a slightly higher D_Mb,0 of 12×10⁻⁷ cm² s⁻¹ (Gros, 1978). Using the Q₁₀ of 1.38 given by Keller and colleagues for the diffusion of hemoglobin in solution (Keller et al., 1971), one obtains a D_Mb,0 at 37°C of 17–21×10⁻⁷ cm² s⁻¹. At a [Mb] of 25 g% and 37°C, one then expects D_Mb=5.5×10⁻⁷ cm² s⁻¹, one-quarter to one-third of the D_Mb,0.

Intact heart and skeletal muscle cells contain ca. 25 g% total protein. The presence of the proteins in highly ordered myofilaments, M lines, Z disks or their location in intracellular membrane systems such as the sarcoplasmic reticulum (SR), T tubuli and mitochondria might present more serious obstacles to protein diffusion (Fig. 1). In fact, it has been shown that these structures render very large proteins such as ferritin (450 kDa, molecular diameter 12.2 nm) or even larger ones such as earthworm haemoglobin (mol. mass 3700 kDa, molecular diameter 30 nm) almost completely immobile within muscle cells (Papadopoulos et al., 2000). By contrast, their mobility in highly concentrated solutions of 25 g% always exceeds one-tenth of the value at infinite dilution (Gros, 1978).

The first measurement of Mb diffusion in an intact muscle cell was that of Baylor and Pape, who microinjected MetMb into frog muscle fibers and measured spectrophotometrically the MetMb concentration profile along the longitudinal fiber axis at various times (Baylor and Pape, 1988). From the time-courses, they calculated the intracellular D_Mb. The diffusion distances observed were of the order of 1000 μm, and the observed direction of diffusion was exclusively longitudinal (Table 1). The measurements were performed at 16°C and 22°C, and their result, corrected to 37°C with the temperature dependence Q₁₀=1.5 for D_Mb in muscle cells given by Papadopoulos and colleagues, was 2.3×10⁻⁷ cm² s⁻¹ (Papadopoulos et al., 1995). Subsequently, measurements were reported for mammalian muscle cells. Jürgens and colleagues oxidized the native Mb of a small section of rat diaphragm muscle cells by UV light and observed spectrophotometrically the diffusion of MetMb out of the illuminated area, obtaining a D_Mb of 2.1(±0.1)×10⁻⁷ cm² s⁻¹ (Jürgens et al., 1994). The diffusion distances observed by them were 70–500 μm in the longitudinal direction.

Next, Papadopoulos and colleagues measured D_Mb by MetMb injection and photometrically following the time-course of the longitudinal concentration profile (Papadopoulos et al., 1995). They compared D_Mb in (slow) rat soleus [2.2(±0.1)×10⁻⁷ cm² s⁻¹ at 37°C] and (fast) rat extensor digitorum longus, finding a somewhat higher diffusivity of Mb in the latter. In addition, they reported that contractile activity of the muscle fibers does not enhance the apparent D_Mb, and they determined the temperature dependence of intramuscular D_Mb (see above).

In the last of this series of papers from the same laboratory, Papadopoulos and colleagues added two important insights to the previous findings (Papadopoulos et al., 2001). The technique employed was to inject fluorescently labelled Mb into rat soleus fibers or isolated rat cardiomyocytes and wait until the labelled myoglobin had spread out widely throughout the cells. Then the fluorescent Mb was bleached by illumination within a 10 μm wide area. Depending on the position of this area relative to the fiber longitudinal axis, the diffusion of unbleached Mb into this area either in the longitudinal or in the radial direction of the fiber could be observed (Fig. 1). The result was that D_Mb (a) is identical in soleus fibers [2.2(±0.1)×10⁻⁷ cm² s⁻¹, at 37°C] and in cardiomyocytes [2.0(±0.1)×10⁻⁷ cm² s⁻¹, at 37°C] and (b) is identical in the longitudinal and in the radial direction in both soleus fibers and in cardiomyocytes. Identical diffusion properties in soleus and cardiomyocytes and in the longitudinal versus the radial direction were also observed for lactalbumin (mol. mass 14 kDa) and for ovalbumin (mol. mass 45 kDa). This allowed the authors to conclude that D_Mb at 37°C is ca. 2×10⁻⁷ cm² s⁻¹ in both heart and skeletal muscle independent of the direction of diffusion. This value of D_Mb is about one-half to one-third of the value in a 25 g% Mb solution. The authors suggested that the diffusion obstacles for Mb in muscle cells are greater than is expected from the presence of 25 g% soluble proteins. Among the candidates for structural obstacles listed above, Z disks and M lines per se appeared unlikely because, with diffusion distances ≥10 μm, they would be expected to affect longitudinal diffusion more strongly than radial diffusion (Fig. 1).

An understanding of protein function in the cell requires insight into dynamic processes, as reflected in rotational and translational diffusion. In rotational diffusion, the motion involves only an angular displacement – the molecule does not need to move in space from point A to point B. Molecular re-orientation illustrates the dynamic search for an optimal spatial arrangement that can lead to molecular binding and a subsequent chemical reaction. In translational diffusion, the motion involves a linear displacement from point A to point B – substrate must move from the source at a distant site, for example the cell membrane, to a distant enzyme site, where the chemical reaction takes place.

Using the Stokes–Einstein equation to obtain a rotational τ_c does assume Brownian motion in an ideal, homogeneous, and isotropic solvent. How well this assumption holds in the cell remains a topic of research inquiry as the cell does not represent an ideal solution (Lavalette et al., 2006). Nevertheless, these fundamental concepts underpin the NMR methods to measure Mb diffusion and, therefore, function in the cell (Price, 1997; Jue, 2009).

NMR measures diffusion using either relaxation or a pulsed-field gradient (PFG) technique. These methods observe different time-scales of molecular motions. With the relaxation approach, the NMR experiments measure predominantly rotational correlation time (τ_c), usually in the 10⁻⁸ to 10⁻¹² s time-scale. Water has a τ_c of 10⁻¹² s, whereas a protein the size of Mb should exhibit a τ_c of ca. 10⁻⁹ s.

In particular, the hyperfine-shifted proximal histidyl N_δH signal of paramagnetic deoxy-Mb provides a convenient way to use field-dependent transverse relaxation to determine the rotational correlation time in solution and in the cell (Gueron, 1975; Vega and Fiat, 1976). With increasing magnetic field strengths, the N_δH signal line-width increases quadratically and can actually lead to a counter-intuitive decrease in signal to noise at higher fields. Indeed, the field-dependent relaxation analysis of the solution deoxy-Mb and Hb proximal histidyl N_δH resonances have yielded rotational correlation times of 9.7×10⁻⁹ s and 37.7×10⁻⁹ s at 25°C, respectively (Wang et al., 1997). The values agree with the Stokes–Einstein equation-predicted values of 11×10⁻⁹ s and 44×10⁻⁹ s and with the expected relative increase in the hydrodynamic radius from a monomeric (16 kDa) to a tetrameric (64 kDa) protein. Moreover, the NMR and fluorescence determinations also stand in excellent agreement (Stryer, 1965; Yguerabide et al., 1970).

Researchers have also utilized the proximal histidyl N_δH resonances to determine the Mb rotational τ_c of 14.7×10⁻⁹ s in excised heart tissue (Livingston et al., 1983). In a standard, respiring physiological heart model, the field-dependent relaxation analysis yields a slightly lower Mb rotational τ_c of 13.6×10⁻⁹ s at 25°C (Kreutzer and Jue, 1991; Wang et al., 1997).

The rotational τ_c gives two specific insights into cellular function: the 1.4 τ_c ratio of Mb in solution versus in the cell implies a corresponding ratio in viscosity, which precludes the presence of any gel-matrix (Pollack, 2001). Even though studies have postulated a crowded cellular environment based on the total amount of assayed substances within a cell volume, the propensity for proteins to form multi-subunit aggregates, to become embedded in membranes, and to bind substrates creates an uncertainty in the calculated effective concentration (Goodsell, 1991). Experimental observation, however, does not detect any significant Mb interaction, and Mb appears completely NMR visible (Kreutzer et al., 1993). No significant crowding or gel matrix exists in the cell to impede the rotational diffusion of Mb.

Because Mb has a rotational τ_c of 13.6×10⁻⁹ s, any proposed model of a protein chain that allows O₂ to hop from heme to heme cannot contribute significantly to the transport of O₂. Such a model would produce a dramatic change in the τ_c and the line shape. Moreover, even a transient O₂ hopping from Mb seems improbable given the uncorrelated rotational diffusion of 10⁻⁹ s versus Mb k_off on the order of 15 s⁻¹ (Carver et al., 1992). Mb will have diffused through countless angles and must pair precisely at the ligand pathway leading from the heme to the protein surface when O₂ releases from one heme to hop to another Mb with an unoccupied heme. A ‘bucket brigade’ mechanism of oxygen transport by myoglobin is ruled out (Gibson, 1959; Wyman, 1966).

As described above, diffusion measurements of microinjected metMb or Mb with an attached fluorophore in isolated muscle fibers have yielded a D_Mb of ca. 1.2×10⁻⁷ cm² s⁻¹ at 22°C, well below the value calculated to support a significant Mb role in facilitating oxygen flux in the cell under steady-state conditions (Papadopoulos et al., 2001). By contrast, the NMR measurements have yielded a much greater D_Mb of ca. 8×10⁻⁷ cm² s⁻¹ (see Table 1).

The pulsed-field gradient (PFG) method can also assess biomolecule diffusion and does not rely on molecular relaxation. Instead, it follows the signal intensity in a spin–echo pulse sequence in the presence of an applied magnetic field gradient. Without any applied gradient, the spin-echo signal intensity will decrease with echo time at an intrinsic transverse relaxation rate characterized by the time T₂. With the application of a linear gradient, the diffusion trajectory of the nuclear spins become phase encoded. Consequently, the signal becomes further attenuated. The additional attenuation reflects then self-diffusion and provides the basis for determining the translational diffusion coefficient (Stejskal and Tanner, 1965; Price, 1997; Jue, 2009).

Cardiac physiology experiments often use a Langendorff perfused heart model to study bioenergetics, metabolism and respiratory control. Through the aorta, the isolated heart receives temperature-regulated, oxygenated (95% O₂, 5% CO₂) buffer with the correct balance of ions and nutrients to sustain its cellular bioenergetics and contractile performance over several hours. Buffer can flow at a constant flow rate or at constant pressure, maintained either by a peristaltic pump or a calibrated column of buffer above the heart.

A strain gauge transducer measures the heart rate and perfusion pressure, while an oscillographic unit records heart rate and cycle. At 35°C, a normoxic rat heart beats with a heart rate (HR) of ca. 240 beats min⁻¹ and exhibits a left-ventricular developed pressure (LVDP) of 100 mmHg. The rate pressure product (HR×LVDP), an index of work, approaches 30,000, while the P_O2 difference between the inflow and outflow buffer multiplied by the flow rate yields an oxygen consumption rate of 40 μmol min⁻¹ g⁻¹ dry weight (Chung and Jue, 1999; Kreutzer and Jue, 2004).

In the PFG measurements of Mb, the hearts received 30 mmol l⁻¹ K⁺ to arrest the contraction. Except for a new balance of NaCl and KCl, all components of the Krebs–Hensleit buffer remained unchanged. The KCl stops the heart for 4–6 h. Upon reperfusion with normal Krebs–Hensleit buffer, the heart regains to its normal physiological function.

Instead of following the proximal histidyl N_δH signal of deoxy Mb, PFG monitors ¹H NMR γ-CH₃ Val E11 signal of diamagnetic MbO₂ in myocardium at −2.8 p.p.m. In contrast to the proximal histidyl N_δH signal, the Val E11 signal reaches its maximum signal upon Mb oxygenation and disappears upon deoxygenation (Kreutzer et al., 1992). Because the Val E11 resides above the heme plane, it interacts with the heme ring current and shifts the γ-CH₃ resonance of MbO₂ to −2.8 p.p.m. and the MbCO resonance to −2.4 p.p.m., outside the 0–10 p.p.m. spectral window where other amino acid peaks appear (Kreutzer et al., 1992). Moreover, the diamagnetic γ-CH₃ Val E11 peak of MbO₂ has a much longer relaxation time than the proximal histidyl N_δH of deoxy Mb. The longer relaxation time confers a distinct advantage in implementing the PFG measurement of Mb translational diffusion in vivo. Fortunately, comparative in vitro and in vivo experiments indicate a free, mobile and total NMR-detectable pool of Mb (Kreutzer et al., 1993). No significant compartmentalization exists.

The ¹H NMR γ-CH3 Val E11 MbO₂ signal in the presence of an increasing field gradient strength along a given direction, such as along the X axis, will decrease progressively. The analysis of the natural logarithm of plot of MbO₂ signal intensity versus the square of the gradient strength yields the 1.8 mmol l⁻¹ solution Mb translational diffusion coefficient of 17×10⁻⁷ cm² s⁻¹ at 35°C. Applying the gradient along the Y or Z direction produces the same result, consistent with isotropic diffusion. In the myocyte, the cellular MbO₂ translational diffusion drops to 7.9×10⁻⁷ cm² s⁻¹. Within the diffusion boundaries measured in the PFG experiments, cellular Mb diffuses isotropically and about two times slower than in solution. Indeed, the observed translational diffusion coefficient in the cell corresponds to the value predicted by NMR rotational diffusion analysis (Lin et al., 2007b; Lin et al., 2007a).

Given the literature-reported values for K₀ and the experimentally determined value of D_Mb, an estimate emerges about the relative contribution of Mb to O₂ diffusion. The literature contains values for the Krogh's diffusion constant (K₀) that range from 2.52–4.28×10⁻⁵ ml O₂ cm⁻¹ min⁻¹ atm⁻¹ (Bentley et al., 1993). The contribution from free-O₂ flux increases linearly with P_O2. By contrast, the O₂ flux from Mb rises nonlinearly with P_O2 and depends upon Mb concentration, P₅₀ and D_Mb. Experiments have determined a P₅₀ of 1.5 mmHg at 25°C but have extrapolated empirically a P₅₀ of 2.0 mmHg at 35°C. Using the D_Mb value of 1.7–7.9×10⁻⁷cm² s⁻¹ at 35°C, a P₅₀ of 2.0 mmHg and Mb concentration of 0.19 mmol l⁻¹, the equipoise diffusion P_O2 is ~0–1.7 mmHg. Only when the cell P_O2 falls below 0–1.7 mmHg will the Mb role in O₂ transport dominate (Table 2) (Papadopoulos et al., 1995; Papadopoulos et al., 2001; Lin et al., 2007b; Lin et al., 2007a).

Endeward and colleagues have recently presented calculations on the basis of a Krogh cylinder model, as it has been described (Groebe, 1995; Endeward et al., 2010). First, O₂ supply in the maximally working human heart is analysed in the phase of the diastole, in which the heart is continuously perfused by means of the coronary arteries. Second, it is studied in the phase of the systole, in which, owing to the elevated intraventricular pressure, the left coronary artery is compressed and perfusion of the left ventricle wall nearly comes to a standstill. Thus, one can consider two fundamentally different situations: a steady-state situation with constant O₂ supply from the vascular system during diastole, and a non-steady-state situation during systole, when, after the cessation of coronary perfusion, the O₂ consumed by the cardiac muscle comes only from the O₂ stores present in the ventricle wall at the end of the diastole – i.e. O₂ bound to haemoglobin in red cells that have remained in the capillaries, O₂ bound to muscular Mb and the O₂ physically dissolved in the cardiac tissue. When these stores are exhausted, the O₂ supply of the left ventricular wall is at an end and the tissue becomes anoxic.

The basic approach employed in the model calculation is: (1) to consider diastolic O₂ supply on the basis of values of specific O₂ consumption and coronary perfusion as they are expected for conditions of the maximally working heart and (2) to model O₂ supply during systole for a systolic period of 150 ms as it prevails at a heart beat frequency of 200 min⁻¹. The diastolic perfusion rate is treated as a parameter that is variable within reasonable limits and chosen so that anoxia, when the 150 ms have passed, just begins in the most unfavourably located part of the tissue, thus just avoiding tissue anoxia. The time until anoxia begins in the heart tissue during systole is defined as the ‘time to anoxia’ (t_a). Under all the various conditions studied with this model, the necessary coronary perfusion rates remain within the range of values expected from experimental data (4300–4900 ml min⁻¹ kg⁻¹).

The model calculations were performed with a cardiac myoglobin concentration of 0.19 mmol l⁻¹ (Swaanenburg et al., 2001) and an intracellular Mb diffusion coefficient of 2×10⁻⁷ cm² s⁻¹ (see Table 1). They show that, while, at the end of the systole, the P_O2 at the venous end of the outer circumference of the Krogh cylinder just reaches 0 mmHg, the end capillary P_O2 falls from a diastolic value of 16.9 mmHg to an endsystolic value of 10.5 mmHg, and mean tissue P_O2 falls from the diastolic value of 24.6 to an endsystolic value of 15.7 mmHg. Mean Mb oxygen saturation falls from a diastolic value of 85% to the endsystolic value of 77.5%.

The general range of these Mb saturations agrees approximately with the values reported in the literature from NMR measurements, which range between 76 and >90% (Jelicks and Wittenberg, 1995; Bache et al., 1999; Zhang et al., 1999). In vivo¹H NMR experiments do not detect any deoxy-Mb-proximal histidyl NδH protons in the in situ myocardium up to 2–3 times above the basal workstate (Kreutzer et al., 1998; Zhang et al., 1999; Kreutzer et al., 2001). Because the NMR sensitivity can detect a 10% deoxygenated Mb signal, the observation implies that cellular P_O2 saturates over 90% of MbO₂ and exceeds 10 mmHg even at elevated workstates. Blood-perfused heart does not set the cellular P_O2 to a level that only partially saturates Mb. Even though, in whole animals, the myocardial work can increase about five times above the basal level, the P_O2 would have to drop from approximately 10 mmHg to well below 2 mmHg, before Mb can compete significantly with the free O₂ flux. This agrees with the conclusion of Endeward and colleagues that the contribution of facilitated O₂ diffusion by Mb diffusion to total O₂ diffusion during the diastolic steady-state situation is only 1.3% and thus insignificant (Endeward et al., 2010).

However, Mb is of much greater importance during the systolic non-steady-state period, when greater regions of Mb desaturation are predicted to occur in the tissue (Endeward et al., 2010). It should be noted here that the fall in saturation during systole has not been observed experimentally by NMR (Chung and Jue, 1999). NMR measures, however, the transient fluctuation averaged over the entire heart, whereas the Endeward model predicts a fluctuation in localized regions and sets up a provocative idea to investigate with future experiments.

The role of Mb during the systolic phase of interrupted coronary perfusion is borne out more clearly by another parameter estimated by Endeward and colleagues, the ‘time to anoxia’, t_a, during the phase of the systole. They chose the model parameters such that the heart just survives the systolic period of 150 ms without tissue anoxia – that is, t_a=150 ms. When myoglobin was immobilized by setting D_Mb=0, t_a fell to 128 ms, indicating that 22 ms of the systole, or 15%, are supported by myoglobin-facilitated O₂ diffusion. Elimination of all myoglobin by setting C_Mb=0 caused an additional reduction of t_a to 116 ms. This indicates that 12 ms, or 8%, of the systole are supported by immobile myoglobin – that is, by the storage function of myoglobin. Altogether, Mb was responsible for O₂ supply to the left ventricular wall for 23% of the systole. The remainder was due to the O₂ bound to the hemoglobin present in the unperfused capillaries (53%) and to the dissolved O₂ present in the tissue at the end of the diastole (24%). These numbers are shifted moderately in favour of both functions of Mb, when an intracellular Mb concentration of 0.3 rather than 0.19 mmol l⁻¹ is assumed (Endeward et al., 2010). The role of Mb during systole might be further enhanced when it is considered that, during systole, the intracapillary volume, and thus the number of red cells present in the tissue, appears to be reduced by 40% owing to the compression of the capillaries (Toyota et al., 2002).

Endeward and colleagues have used this model to estimate quantitatively the adaptations that are necessary to compensate for the absence of Mb in the heart: a diastolic perfusion rate increased by 5.2%, or a hematocrit increased by 4.2% or a capillary density increased by 16% can functionally offset the loss of Mb (Endeward et al., 2010). These numbers are of the order of those observed as adaptations in the hearts of mice in which the gene encoding Mb has been knocked out (Gödecke et al., 1999).

The main conclusions from these theoretical studies are: (1) Mb plays hardly any role during the steady-state situation that prevails during diastole, (2) its main role is during the systolic interruption of the coronary perfusion, when Mb accounts for ca. one-quarter of the O₂ supply to the cardiac tissue, and (3) the contributions of Mb to systolic O₂ supply are by facilitated O₂ diffusion (15%) and by Mb as an O₂ store (8%).

In resting muscle, blood supplies sufficient P_O2 to fully saturate Mb. ¹H NMR does not detect any proximal histidyl NδH signal. Skeletal muscle, however, contains a range of Mb concentrations. Type II glycolytic fibers have much less Mb than type I oxidative fibers, which can exceed the Mb concentration in heart.

At the onset of contraction, Mb desaturates immediately (Chung et al., 2005). Within ~30 s, Mb desaturates to a steady-state level. The rapid desaturation of Mb points to a transient mismatch between O₂ supply and demand, in contrast to the hypothesized perfect match of O₂ delivery and utilization at all times (Behnke et al., 2000; Kindig et al., 2003). Mb provides an immediate intracellular source of O₂, until the blood flow can adjust to meet the increased energy demand. The Mb desaturation kinetics imply also a sudden drop in intracellular P_O2. If the intracellular P_O2 falls below the equipoise P_O2 of ca. 4 mmHg, slightly above the Mb P₅₀ of 3.2 mmHg at 40°C, then Mb-facilitated O₂ transport will begin to dominate. Indeed, at the start of muscle contraction and at high-intensity exercise, the P_O2 might well fall below 4 mmHg (Molé et al., 1999; Chung et al., 2005).

The Mb desaturation kinetics yield insight into the cellular $V.O2$⁠, which whole-body oxygen $V.O2$ must extrapolate based on a number of assumptions. Given the cellular Mb concentration of ca. 0.4 mmol l⁻¹ in human gastrocnemius muscle, the dMbO₂/dt at the start of muscle contraction implies an intracellular $V.O2$ of ca. 9 μmol l⁻¹ O₂ s⁻¹ (Chung et al., 2005). Based on the estimated ATP cost per contraction, oxidative phosphorylation can actually support over one-third of the energy cost at the start of contraction, a much higher fraction than previously expected (Blei et al., 1993). Nevertheless, the derived value for the intracellular $V.O2$ falls well within the range of the $V.O2$ determined by near-infra-red spectroscopy (NIRS) of 3.0 to 31.2 O₂ μmol l⁻¹ s⁻¹ in forearm muscle during isometric handgrip exercise (Tran et al., 1999; Sako et al., 2000). Indeed, the dynamic relationship between O₂, bioenergetics and Mb requires further clarification and will yield invaluable insights into Mb function.

The analysis of the contribution of Mb to intracellular O₂ supply often relies on a simplified cell model and the resultant equipoise P_O2. However, the equipoise P_O2 depends upon accurate values for D_Mb, K₀, C_Mb and P₅₀. In particular, the temperature-dependent values for K₀ and P₅₀ still remain uncertain (Wittenberg, 1970; Schenkman et al., 1997). The experimentally determined free-O₂ diffusion in the cell varies widely (Bentley et al., 1993). Moreover, experimentally measuring the Mb P₅₀ at 37°C poses many challenges. Nevertheless, given the observed Mb diffusion in cells and the available data, Mb appears to play a limited role in facilitating O₂ diffusion in the steady state but might have a significant role in the transient state, as reflected in the Mb desaturation dynamics at the start of skeletal muscle contraction as well as during systolic cessation of coronary blood flow.

The prolonged breath holds (apneas) of seals during sleep presents a unique model to gather insight on Mb function. These spontaneous apneas last about 10 min in northern elephant seals (Mirougna angustirostris) and approximate the physiological state in a dive (Castellini et al., 1994; Le Boeuf et al., 2000). Unlike terrestrial mammals, seal muscle has 3.8 mmol l⁻¹ Mb, which can serve as an O₂ store during a dive (Scholander, 1940; Scholander et al., 1942). Indeed during apnea, MbO₂ desaturates about 20% from its control level, as muscle blood flow (MBF) decreases. During eupnea, Mb resaturates rapidly (Ponganis et al., 2002; Ponganis et al., 2008).

Throughout eupnea and apnea, the phosphocreatine (PCr) and ATP levels remain unchanged, even though the intracellular P_O2, $V.O2$ and MBF have fallen. Without any change in the PCr/ATP ratio, the tissue shows no sign of hypoxemia or ischemia, even though MBF has decreased to 30% of the eupneic level and vascular P_O2, has dropped from 60 to 21 mmHg. Even at a decreased O₂ level, oxidative phosphorylation supplies adequate ATP for the cell. An insufficient oxidative ATP production would trigger a breakdown of PCr and enhanced glycolysis to buffer the energy loss. This non-limiting oxidative phosphorylation in the face of declining intracellular O₂ clarifies a long-standing supposition that PCr breakdown supplements glycolysis during a dive to compensate for the hypoxemia (Hochachka and McClelland, 1997). Neither PCr breakdown nor enhanced glycolysis occurs. During sleep apnea, no hypoxemia appears, consistent with the absence of any lactate washout (Castellini et al., 1994).

During sleep apnea, the decreased blood flow and intracellular P_O2 can still maintain a sufficient O₂ gradient to ensure O₂ delivery that meets a declining energy demand. In essence, the O₂ delivery matches dynamically the energy demand to avoid any hypoxemia. Oxidative metabolism during apnea, however, poses a challenge for CO₂ disposal. Excess CO₂ can alter respiratory drive as well as sleep structure (Milsom et al., 1996; Skinner and Milsom, 2004; Stephenson, 2005). Even though marine and terrestrial mammals have a similar hypercarbic chemosensitivity, the higher blood buffering capacity allows marine mammals to accommodate the CO₂ build-up

In diving mammals then, Mb provides a temporary but immediate source of O₂. It plays a predominant role in transporting and/or supplying O₂ as the equipoise P_O2 stands around 14–71 mmHg (Ponganis et al., 2008).

Without Mb, mice appear to show no superficial physiological deficits (Garry et al., 1998; Flögel et al., 2005). However, homeostatic mechanisms, including increased capillary density that tends to steepen the oxygen pressure gradient to the mitochondria, effectively shorten the diffusion path for oxygen (Gödecke et al., 1999; Grange et al., 2001; Meeson et al., 2001). In addition, myocardial metabolism of Mb-deficient mice switches away from fatty acid towards glucose metabolism to compensate, presumably, for the absence of Mb (Flögel et al., 2005). A new explanation envisions Mb playing a role in binding and facilitating fatty acid diffusion.

The idea, however, remains controversial as the literature contains inconsistent evidence. ¹⁴C oleic acid does bind to a rat heart cytosolic fraction of 16 kDa. Because Mb has a molecular mass of 16 kDa, the observation implicates Mb (Gloster and Harris, 1977). Unfortunately, fatty acid binding protein (FABP) has a molecular mass of 15 kDa. FABP studies have determined that Mb binds to fatty acids with a lower affinity than albumin on a per-mole basis, with a Kd of 12.2 to 48 μmol l⁻¹ (Gloster, 1977; Gloster and Harris, 1977; Götz et al., 1994). However, other studies have refuted such findings (Glatz and Veerkamp, 1983; Said and Schulz, 1984).

Recently, ¹H NMR experiments have measured the interaction of palmitate (PA) with metmyoglobin cyanide (MbCN), a surrogate model of MbO₂, and have found a selective change in the 8-heme methyl signal intensity, consistent with a specific interaction in a localized region of Mb. Moreover, palmitate appears much more soluble in Mb solution than in buffer (Sriram et al., 2008).

Such observations do not agree with a monolithic viewpoint that only FABP traffics fatty acid in the cell (Glatz and van der Vusse, 1989). The low fatty acid binding affinity for Mb seems to militate against any significant role. However, dissociating fatty acid from a high-affinity FABP must involve a complex release mechanism (Corsico et al., 2004). Moreover the higher concentration of Mb, its capacity to increase fatty acid solubility and rapid diffusivity in the cell might well confer a distinct advantage under certain physiological conditions. Future experiments must determine the relative contribution of fatty acid transport by myoglobin and by FABP present simultaneously in the tissue (Moore et al., 1993).

However, the attractive hypothesis that myoglobin serves to transport fatty acids through the sarcoplasm must await the determination of the dissociation rate, which will quantitate the capacity of Mb to support a transport function. Even though a ligand interacts with a protein, it does not necessarily indicate that the protein transports the ligand. For example, even though Hb binds to bromothymol blue and chloride anions and produces a consequent allosteric rearrangement, it is still not usually considered as a mobile transporter of either ligand (Antonini et al., 1963; Antonini et al., 1965; Chiancone et al., 1976; Norne et al., 1978).

In a first attempt to approximate the situation of Mb diffusion in a cell, Riveros-Moreno and Wittenberg used free and ¹⁴C-iodoacetamide-attached sperm whale Mb to determine the protein diffusion at high protein concentrations across 150–300 μm Millipore filters with a pore size of 0.45 μm (Riveros-Moreno and Wittenberg, 1972). With 1 mmol l⁻¹ Mb at 20°C, they determined a diffusion coefficient at infinite dilution (D_Mb,0) of ca. 10×10⁻⁷cm² s⁻¹. The calculated D_Mb,0 would be ca. 17×10⁻⁷ cm² s⁻¹ at 37°C. Above 6 mmol l⁻¹ Mb, D_Mb decreased monotonically with protein concentration, reaching ca. 6×10⁻⁷cm² s⁻¹ (10 mmol l⁻¹ or 17 g dl⁻¹, 20°C) and 3×10⁻⁷ cm² s⁻¹ (15 mmol l⁻¹ or 25 g dl⁻¹, 20°C). This type of dependence of protein diffusion on protein concentration has been shown to reflect the macroscopic viscosity η of the protein solution. For proteins of different molecular masses, diffusivity is proportional to 1/η, where η is defined as the solution viscosity determined in an Ostwald-type viscometer (Gros, 1978). What can be learned about the apparent viscosity inside a muscle cell from measurements of intracellular diffusion of Mb?

Clearly, in the cellular environment, D_Mb should deviate significantly from the D_Mb,0. The cellular D_Mb based on fluorescence recovery after photobleaching (FRAP) and NMR, however, differs significantly. FRAP yields consistently a D_Mb of ≈2×10⁻⁷ cm² s⁻¹ at 37°C (Papadopoulos et al., 2001). By contrast, the PFG-NMR method gives ~8×10⁻⁷ cm² s⁻¹ at 37°C (Lin et al., 2007a). These values differ by a factor of four and alter substantially the predicted contribution of facilitated O₂ diffusion inside muscle cells. However, these two biophysical approaches do not measure diffusion over the same distance.

With NMR, based on the Einstein–Smoluchowski equation of <r²>=6Dt, the reported experiments detect only a mean-squared displacement ranging from 2.5 to 3.5 μm. Improved NMR spectrometer design can extend the range, as observed for small cellular metabolites (de Graaf et al., 2000). By contrast, the FRAP measurements were performed over a diffusion distance of 10 μm. The NMR and fluorescence then give snapshots of the local and extended cellular environment.

Indeed, the close relationship between NMR-determined translational and rotational diffusion coefficients indicates that NMR senses a local environment. Cellular Mb exhibits a rotational diffusion ca. 1.4 times slower than solution Mb (Wang et al., 1997). Translational diffusion is ca. 2.7 times slower. The observation agrees with experiments showing green fluorescent protein in the cytoplasm exhibits a rotational (1.5 times) and translational diffusion (3.2 times) times slower than in saline solution (Swaminathan et al., 1997). The presence of local domains agrees with current cell models, which envision the cytoplasm as a concentrated macromolecular solution, a rigid gel network or an entangled filament network. All cytoplasm models envision proteins diffusing in the local domain about at least three times slower in the cell than in solution (Luby-Phelps et al., 1988; Kataoka et al., 1995).

When interpreting measurements of intracellular translational or rotational protein diffusion in terms of an apparent ‘microviscosity’ in the cell, it should be kept in mind that many techniques exhibit their own specific dependence on the intracellular environment and usually do not reflect the macroscopic viscosity as defined above. This phenomenon can be observed in concentrated protein solutions, as has been pointed out for protein diffusivities obtained from light-scattering spectroscopy (Alpert and Banks, 1976; Veldkamp and Votano, 1976; Gros, 1978). Lavalette and colleagues have made a similar observation when they measured translational protein diffusion in concentrated polymer solutions using fluorescence correlation spectroscopy (FCS) (Lavalette et al., 2006). These authors found also that the translational diffusion coefficient decreases to a lesser extent with increasing polymer concentration than expected from the macroscopic viscosity. It appears that proteins in these spectroscopic methods experience less interaction with neighboring macromolecules, or less ‘microviscosity’, than when observed by tracking protein diffusion over longer distances, as for example in the measurements of Riveros-Moreno and Wittenberg (Riveros-Moreno and Wittenberg, 1972). ‘Microviscosity’ here is defined as the apparent η one obtains when one applies Eqns 3 and 4 to protein diffusion coefficients that are reduced in comparison with their value in pure solvents.

An even lesser local viscosity, or ‘microviscosity’, is experienced by proteins rotating in a protein-crowded environment. Intuitively, this is expected because rotation of a protein does not require more space than is taken up by the molecule itself. It has been confirmed experimentally by Lavalette and colleagues, who observed a considerably lesser decrease of rotational diffusivity of serum albumin with increasing 17.5 kDa dextran concentration than for translational diffusion of this protein (Lavalette et al., 2006). Similarly, Zorrilla and colleagues reported that the rotational diffusivity of equine apo-myoglobin depends much less on increasing concentrations of RNAse A or human serum albumin than does its translational diffusivity (Zorrilla et al., 2007). It can be concluded that the microviscosity sensed by rotational diffusion at higher protein concentrations is considerably lower than that sensed by translational diffusion. In both these studies, FCS was used to determine translational diffusivities, and the apparent ‘microviscosity’ sensed by translational protein diffusion when using this and other spectroscopic techniques appears to be somewhat lower than the macroscopic viscosity, which governs classical ‘long-range’ diffusion measurements.

What is the ‘microviscosity’ inside a muscle cell? This environment has a high total protein concentration, an extended network of myofilaments and a substantial volume fraction (up to 40% in cardiomyocytes) occupied by mitochondria, sarcoplasmic reticulum and T tubules (Eisenberg, 1983). All of these might present major obstacles to protein diffusion For small molecules such as O₂, CO₂, ATP and PCr, diffusion in the sarcoplasm is reduced by only 20–35% compared with diffusion in water (Kawashiro et al., 1975; Hubley et al., 1995). So the ‘microviscosity’ experienced by small molecules inside muscle cells is rather close to the viscosity of water. Protein diffusivity in the muscle cell as measured by classical techniques has been reported to be quite low and range between one-tenth and one-fiftieth of their diffusivity in water, or even zero, depending on the size of the protein (Papadopoulos et al., 2000). This indicates an extremely broad range of apparent ‘microviscosities’. By contrast, rotational mobility of Mb in heart tissue and of hemoglobin in red cells is only moderately reduced by 30% and 50%, respectively (Wang et al., 1997), indicating fundamentally lower ‘microviscosities’ than seen with translational diffusion. BCECF [2′,7′-Bis-(carboxyethyl)-5(6)-carboxyfluorescein], with a molecular mass of 520 Da and with only modestly reduced mobility in Swiss 3T3 fibroblasts, is three times more reduced in its translational than in its rotational mobility in cytoplasm compared with the values in water (Kao et al., 1993).

What then is the nature of the intracellular obstacles that appear to become effective at longer diffusion distances of 10 μm but not at the shorter distance of ~3 μm observed by NMR? In the A-band, the zone in which the thick and thin filaments overlap, the surface-to-surface distance of the filaments is between 8 and 15 nm (Elliott et al., 1963). In view of the molecular diameter of Mb of 3.5 nm, these distances should not constitute serious obstacles to Mb (although they might be responsible for the complete immobility of earthworm hemoglobin, given its diameter of 30 nm), but they will cause some tortuosity of the diffusion path in the radial direction and some restriction of mobility in the longitudinal direction. However, the distance between filaments clearly does not have the postulated spacing of between 3 and 10 μm. The only structures that fulfil this latter criterion are the intracellular membrane systems sarcoplasmic reticulum, mitochondria and the T system (Fig. 1). These occupy between 39% (in mouse heart), 15% (in rat soleus) and 8% (in human slow skeletal muscle fibers) of the intracellular space (Davey and Wong, 1980; Eisenberg, 1983). Fig. 1 shows that core mitochondria (that make up 95% of total mitochondria in human skeletal muscle), terminal cisternae (i.e. junctional sarcoplasmic reticulum, making up one-third to one-half of total sarcoplasmic reticulum) and T tubules are all assembled in the region around the Z disks. Between Z disks, somewhat lesser volume fractions are occupied by peripheral mitochondria and by the ‘light’ sarcoplasmic reticulum, which is associated with the A and the I band (Eisenberg, 1983). The sarcomere length – that is, the distance between Z disks – is between 2.5 and 3.2 μm. Therefore, there appears to be a space whose length is of that order of magnitude and that presents fewer diffusion obstacles due to mitochondria and sarcoplasmic reticulum than the region close to the Z disks. Thus, if observation of protein diffusion is restricted to <<3.2 μm, a significantly greater diffusivity might be found than when diffusion is observed over several sarcomeres and Z disks. The view that these intracellular organelles, especially core mitochondria, junctional sarcoplasmic reticulum and T tubules, are major diffusion obstacles in muscle cells would be compatible with a recent modelling study of intracellular diffusion of macromolecules by Novak and colleagues, which concludes that intracellular membranes represent the major intracellular diffusion impediments for proteins rather than filamentous structures (Novak et al., 2009).

If this concept of the interaction of Mb diffusion and cytoarchitecture is correct, then the diffusion coefficient of 2×10⁻⁷ cm² s⁻¹ will be applicable when longer diffusion distances of intracellular O₂ transport are considered. These include skeletal muscle, with its fiber radius between 25 and 50 μm. Some compromise value, between 2 and 8×10⁻⁷ cm² s⁻¹ might be relevant in cardiomyocytes of human hearts with a radius of 7 μm (Armstrong et al., 1998). In these cells, diffusion distances consequently vary between 0 and 7 μm (Endeward et al., 2010). Future NMR and fluorescence diffusion experiments will give insights into the diffusion of Mb, the function of Mb and the nature of cytoarchitecture.

Myoglobin facilitation of oxygen diffusion in aqueous solutions is well understood as a physico-chemical process. As we turn our attention to myoglobin-mediated oxygen transport within the living cell, and, in so doing, we join the myriad ranks of biologists who have studied the functions of proteins within cytoplasm (Wang, 1954). For this purpose, myoglobin has the advantage that, by virtue of its color and of certain residues that give NMR signals outside the envelope of other protein resonances, both its reactions with ligands and its mobility can be monitored non-invasively using spectrophotometry or NMR.

Myoglobin is found to enjoy surprising mobility within the sarcoplasm of both cardiac myocytes and skeletal muscle cells. Injected and native myoglobin have identical mobilities. The diffusion coefficient of myoglobin is found to be the same, whether measured parallel or perpendicular to the long axis of the cell. NMR detects no significant impediment to rotational diffusion of myoglobin, and sarcoplasmic myoglobin is completely NMR visible. We recall that myoglobin molecules in very concentrated solutions apparently move past each other with minimal frictional interaction (Riveros-Moreno and Wittenberg, 1972). In a sense, the molecules must be ‘slippery’. All of these observations are consonant with the idea that sarcoplasmic myoglobin is in free solution in an aqueous phase. Why then is the diffusivity of myoglobin in sarcoplasm less than that in dilute solution? Part of the explanation might be found in the ‘obstruction effect’ (Wang et al., 1954). Large protein molecules obstruct the paths of other protein molecules, making their diffusion paths tortuous. In cytoplasm, protein molecules must follow longer paths to traverse the same distance as they wend their way around other molecules. Accordingly, the macroscopic diffusion coefficient is decreased. Recent NMR experiments revise our understanding of myoglobin diffusivity in sarcoplasm (Lin et al., 2007b; Lin et al., 2007a). Previous determinations of the diffusion coefficient of myoglobin in muscle used observation windows, across which diffusion occurred 10 μm or more gave concordant values of D_Mb,37°C=2.0–2.2×10⁻⁷ cm² s⁻¹ (Table 1). This is about one-tenth the value in a dilute solution. Lin and colleagues, using an NMR technique with a much smaller observation window of ca. 3 μm, find D_Mb,37°C=8×10⁻⁷ cm² s⁻¹, or approximately four times faster than previous values. Thus, D_Mb within small domains in the sarcoplasm is fully eight-tenths as fast as that in dilute solution. The authors suggest that the intrinsic viscosity of sarcoplasm is small but that, over any long distance, membranous elements of the cytoarchitecture, including sarcoplasmic reticulum, mitochondria and T-tubules, might obstruct movement of myoglobin molecules. The diameter of the observing window, ca. 3 μm (Lin et al., 2007b; Lin et al., 2007a), within which the NMR-determined myoglobin diffusion coefficient is fourfold greater than that found using larger observation distances coincides with the distance between Z disks, ca. 2.5 to 3.2 μm. This suggests that myoglobin might diffuse relatively freely within each sarcomere but might not readily move between sarcomeres.

With the intention of constructing a model for oxygen flow to the mitochondria, it might be helpful to assemble the known parameters bearing on that flow, assuming always that the muscle is operating in the steady-state characteristic of sustained work and the heart is operating in the steady-state characteristic of normal beating (Wittenberg, 1970). A more detailed treatment of some parameters is given elsewhere (Wittenberg and Wittenberg, 2003). The oxygen pressure at the mitochondrial surface will fall somewhere between the P₅₀ for half-maximal mitochondrial oxygen uptake, 0.04 mmHg, and sarcoplasmic P_O2 (Wittenberg and Wittenberg, 2007). The oxygen pressure at the sarcolemma has not been measured but can be estimated from the mean capillary P_O2 of ca. 50 mmHg less the oxygen pressure drop across the capillary wall, approximately 20–25 mmHg, to give an estimated mean P_O2 at the sarcolemma of the order of 30 mmHg. This value is of course a maximum, as myoglobin-facilitated oxygen diffusion will dissipate the local concentration of oxygen. The volume-average P_O2 in the sarcoplasm and in the beating heart is well known from optical and NMR measurements. As myoglobin in working heart and muscle is always partially desaturated with oxygen, the P_O2 cannot be far from the P₅₀ of Mb, 2.5 mmHg at 37°C. The oxygen partial pressure difference from the external medium to the mitochondria of isolated cardiac myocytes respiring in steady states at low P_O2 is very small, of the order of 2–3 mmHg (Katz et al., 1984; Wittenberg and Wittenberg, 1985). We ourselves picture the heart cell and contracting red skeletal muscle fibers as nearly, but crucially not quite, equipotential in oxygen.

Geometric factors will enter the analysis. The diffusion path for oxygen through the sarcoplasm is tortuous. The capillary endothelial surface offers by far the smallest cross-sectional area in the diffusion path for oxygen from blood to cytochrome oxidase, and the largest oxygen pressure difference is expected at this point. The surface area of the mitochondria is 30–150-fold greater than the area of the capillary lumen serving each cell. Accordingly, the oxygen pressure drop across the mitochondrial surface need only be small, and the oxygen pressure experienced by cytochrome oxidase will closely approach sarcoplasmic oxygen pressure.

The rate and equilibrium constants for reaction of myoglobin with oxygen obviously play a governing role in myoglobin-mediated oxygen diffusion. The oxygen equilibrium binding constant and the oxygen combination rate constant are probably subject to control by Darwinian natural selection. As the oxygen dissociation rates of different myoglobins are tightly clustered, these constants too must be subject to selection pressure and thereby optimized (Wittenberg, 2007). The root-mean-square displacement of myoglobin molecules during the time that oxygen is resident (which depends on the off-rate) is calculated to be 3.8–12 μm. These displacements are within an order of magnitude of the dimensions of a cardiac myocyte, say 20×120 μm. It is difficult to fit this awkward fact into any present description of oxygen flow within the myocyte, since our present descriptions imply that oxygen must combine and dissociate from myoglobin many times as it makes its way towards the mitochondrion. We have a square peg, not fitting any round hole of theory.

In this concluding article, we have attempted to gather the facts that must be compatible with any theoretical description of oxygen flow with the muscle cell. They are at variance with the modified Krogh–Erlang formulation because the latter does not take into account the steep oxygen pressure drop across the capillary wall and, accordingly, invokes a steep intracellular oxygen pressure gradient encompassing the full pressure difference between capillary blood and cytochrome oxidase. The enumerated facts suggest a much shallower intracellular gradient. Clearly, we need more facts. Ideally, we would like to measure the oxygen flow in the muscle of a marathon runner during a race.

We gratefully acknowledge Jonathan Wittenberg's guidance in writing this review. Without his astute intellect, his open attitude towards scientific inquiry, his outstanding collegiality and enduring friendship, this article would never have materialized. Michael Berenbrink also provided much leadership and encouragement in helping us realize this article. T.J. acknowledges funding support from NIH GM 58688 and Philip Morris 005510 and also the scientific discussions with and the assistance in manuscript preparation of Ulrike Kreutzer. G.G. acknowledges funding support from the Deutsche Forschungsgemeinschaft (EN 908/1-1; GR 489/20-1), the invaluable discussion with Volker Endeward and the help of Samer Al-Samir in producing Fig. 1 in this article. Deposited in PMC for release after 12 months.