Computational Assessment of Transport Distances in Living Skeletal Muscle Fibers Studied In Situ

Abstract

Transport distances in skeletal muscle fibers are mitigated by these cells having multiple nuclei. We have studied mouse living slow (soleus) and fast (extensor digitorum longus) muscle fibers in situ and determined cellular dimensions and the positions of all the nuclei within fiber segments. We modeled the effect of placing nuclei optimally and randomly using the nuclei as the origin of a transportation network. It appeared that an equidistant positioning of nuclei minimizes transport distances along the surface for both muscles. In the soleus muscle, however, which were richer in nuclei, positioning of nuclei to reduce transport distances to the cytoplasm were of less importance, and these fibers exhibit a pattern not statistically different from a random positioning of nuclei. We also simulated transport times for myoglobin and found that they were remarkably similar between the two muscles despite differences in nuclear patterning and distances. Together, these results highlight the importance of spatially distributed nuclei to minimize transport distances to the surface when nuclear density is low, whereas it appears that the distribution are of less importance at higher nuclear densities.

Significance

We show that in vivo labeling of nuclei within muscle fibers in combination with confocal microscopy enables us to computationally three-dimensionally model the effect of placing nuclei randomly and optimally to predict physical limitations to cellular transportation and diffusion. Our results emphasize the importance of keeping close to a uniform nuclear distribution when nuclear densities are low, whereas at higher densities, nuclear positioning is predicted to be of less importance for sustaining nuclear transcription and translation. Although the two muscles showed different nuclear number and distribution, they displayed similar diffusion transport times. We think our study would be of interest to the muscle field in general but also highlight the potential of uniting modeling, physics, and biology together.

Introduction

The largest cells in the vertebrate body are the muscle fibers, and their vast size introduces logistical problems with respect to synthetic capacity and transport of macromolecules. For example, a human sartorius muscle has an average fiber length of 42 cm (1) and a fiber cross-sectional area of ∼2500 μm² (2), which lead to a volume of 1050 nL. Most other mononucleated cells range 5–20 μm in diameter and, assuming a spherical shape, have volumes less than 0.004 nL. When cells become larger, the average transport distances increase and, thus, influence the overall transport times. Hence, small and large cells operate under different timescales (3). In larger cells, molecules would acquire longer transport times to reach their target. Because diffusion times, t, scale to the distance traveled, L as t ∝ L², diffusion is more efficient at shorter distances. In contrast, active transport times scale linearly to the distance traveled, although the speed relies heavily upon the molecular kinetics of the motor proteins. Typically, a kinesin motor moves at a speed of 1 μm/s along the microtubules (4), whereas the myosin V moves at 3 μm/s on the actin filaments (5).

Mammalian skeletal muscle cells are syncytia, and human muscle cells might have several thousand nuclei (6). The high number density of nuclei is believed to be required because of the large fiber volume and long transport distances, and both the positioning and the nuclear number seem to be regulated to overcome these challenges based on restricted synthetic capacity and the physical limitations to intracellular transport (7, 8, 9, 10). We have previously suggested that the myonuclei seem to be distributed as if to repel each other to minimize cytoplasmic transport distances in mammalian fibers (7,9). This optimization seems to be important in the fruit fly as well (11) because perturbation of the nuclear positioning has been reported to impair muscle function (8,12).

The fact that these cells are multinucleated has led to the proposal of a so-called cytoplasmic-to-nucleus domain that signifies a theoretical cytoplasmic domain governed by a single nucleus (13). Each nucleus is surrounded by a synthetic machinery that seems to remain localized (14,15), and it has been shown that some proteins are localized in proximity to site of expression both in vitro (14,16,17) and in vivo (18,19).

In hybrid myotubes in which one or a few nuclei were derived from myoblasts expressing nonmuscle proteins, it was found that messenger RNAs for nuclear, cytoplasmic, and endoplasmic-reticulum-targeted proteins had similar distributions and were, in most cases, confined to distances 25–100 μm from their nucleus of origin (17). Similar observations were made with virus-infected, isolated muscle fibers in culture, with the additional observation that virus coding messenger RNA did not venture into the fiber interior but remained around the nucleus at the fiber surface at least for the first 40 h after infection (20). These and other observations support the notion that myonuclei support the expression of different genes within a compartmentalized region of a fiber (14,16,21). The most prominent example is the acetylcholine receptor that is normally expressed only in the nuclei near the endplate, and expression outside this area (acetylcholine supersensitivity) seems to require that the receptor is transcribed in nuclei along the entire length of the fiber (22,23).

On the other hand, experiments in which aggregated blastomeres from two different mouse strains expressing distinct forms of cytosolic metabolic enzymes display heterodimers of the enzyme in the syncytial muscle tissue (24), and histological examination displayed no mosaicism in the cellular distribution of the enzyme variants in muscle fibers (25). This means that some smaller proteins that are not trapped in structures such as synapses and sarcomeres can be distributed over longer distances and that the nucleus of origin is of less importance.

To get a better understanding of how nuclear positioning may affect transport distances in muscle fibers, we employed precise confocal imaging and three-dimensional (3D) reconstruction of fibers labeled in vivo in the soleus and extensor digitorum longus (EDL) muscle. These muscles differ in their myonuclear density and their nuclear distribution (7) and, thus, potentially highlight variations in nuclear pattern because of the differences in functional needs and composition of the interior. We therefore modeled each nucleus as an origin of a transportation network and derived the area and volume of responsibility by Voronoi segmentation (26) as well as transport distances along the surface and within the cytoplasm. Additionally, we used simulations of transport distances to analyze the energy trade-off by letting a protein either diffuse locally from a nucleus or being actively transported.

Materials and Methods

Animals

A total of 23 3-month-old female Naval Medical Research Institute (NMRI) mice (20–30 g) were used. The animal experiments were approved by the Norwegian Animal Research Authority and were conducted in accordance with the Norwegian Animal Welfare Act of December 20, 1974. The Norwegian Animal Research Authority provided governance to ensure that facilities and experiments were in accordance with the National Regulations Act of January 15, 1996, and the European Convention for the Protection of Vertebrate Animals Used for Experimental and Other Scientific Purposes of March 18, 1986.

Preparing the mice for imaging

Before surgery, animals were anesthetized by a single intraperitoneal injection of a zrf cocktail (18.7 mg zolazepam, 18.7 mg tiletamine, 0.45 mg xylazine, and 2.6 mg fentanyl/mL) that were administered at a dose of 0.08 mL/20 g body weight.

The skin over the tibialis anterior and gastrocnemius was shaved, and a small incision was made to expose the overlaying muscles that were subsequently retracted laterally to expose the EDL and soleus muscle, respectively. The epimysium was gently removed, and care was taken not to damage the muscle. The exposed muscle was covered with a mouse Ringer’s solution: NaCl 154 mM, KCl 5.6 mM, MgCl₂ 2.2 mM, and NaHCO₃ 2.4 mM, and held in place with a coverslip mounted ∼2 mm above the muscle. In some muscles, the neuromuscular end plate was visualized by applying Alexa-488-conjugated α-bungarotoxin (Molecular Probes, Eugene, OR) to the surface of the muscle for 2–3 min.

In vivo intracellular injections of intravital dyes were essentially done as described previously (27). Animals were placed under a fixed-stage fluorescence microscope (Olympus BX50WI; Tokyo, Japan) with a 20×, NA 0.3, long working distance water immersion objective. For in vivo labeling of nuclei and cytoplasm, single fibers in the EDL and soleus were injected with a solution containing 5′-TRITC or FITC-labeled random 17-mer oligonucleotide with a phosphorothioated backbone (Yorkshire Biosciences, Heslington, UK) and 2 mg/mL Cascade blue dextran (10 kDa; Molecular Probes) dissolved in an injection buffer (10 mm NaCl, 10 mm Tris (pH 7.5), 0.1 mM ethylenediaminetetraacetic acid, and 100 mM potassium gluconate) at a final concentration of 0.5 and 1 mM, respectively. The oligonucleotides are taken up into the nuclei inside the injected fibers probably by an active transport mechanism (28), whereas dextran remain in the cytoplasm.

Confocal imaging

Muscle fibers and nuclei in the EDL and soleus were imaged with a confocal microscope (Olympus FluoView 1000, BX61W1) in optical sections separated by z axis steps of 1 μm to have the full 3D data set of nuclei. Sequential scanned fields of 0.09 mm² were captured with a pixel dwell time of 2 μs. Movements induced by the mouse heartbeat or ventilation sometimes caused displacements of individual nuclei within stacks. Therefore, mice were given an overdose of the anesthetics, and image acquisition continued for 20 min after euthanasia. No difference in myonuclear positioning and fiber morphology was observed before and after euthanasia.

Image analysis

In vivo images (320 × 640 pixels × 2-μm voxel depth) from optical sections of muscle fibers were imported and analyzed for myonuclear number, 3D myonuclear positioning, fiber volume, and surface area using the Imaris Bitplane 8.3.1 software (Zürich, Switzerland). Using the spot function in Imaris, a spot was automatically assigned to each nucleus based on the fluorescence intensity from the injected oligonucleotides, and if misaligned, spots were manually re-positioned to the center of each myonucleus. Highlighted spots/nuclei within fibers were then given a 3D coordinate in a relative Euclidean space using the Imaris Vantage extension. Volume and surface rendering were performed using the fluorescence from Cascade-Blue-conjugated dextran (Thermo Fisher Scientific, Waltham, MA) in the cytoplasm. Rendering of fiber volume was performed using the fluorescence perimeter border of the fiber in the cross-sectional direction as an outer limit and thereby preserving fiber morphology during quantification.

Computer modeling of nuclear distribution

Once the Euclidian coordinates of the myonuclei were obtained in Imaris, fiber surfaces were approximated by fitting the points to the surface of an idealized elliptical cylinder surface.

An elliptical cylinder oriented along the z axis, with its major and minor axes oriented along the x and y axes, respectively, can be parameterized as follows:

$$\frac{{\left(x-x_0\right)}^2}{a^2}+\frac{{\left(y-y_0\right)}^2}{b^2}=1$$

where a and b are the major and minor axes, respectively, and is the point where the center of the cylinder intercepts the x-y-plane.

The cylinder was parameterized by the Euler angles pointing along the z axis of the cylinder, the major and minor axes of the ellipse, the angle of the major axis relative to the x axis, and the x- and y-coordinates of the center of the ellipse for a total of eight free variables. The optimization was carried out in two rounds, in which the Euler angles were chosen first, after which the remaining ellipse fitting was completed for the given angles using a direct least-squares fitting. The angles were then updated using MATLAB’s interior point optimization solver (The MathWorks, Natick, MA) until convergence was achieved.

Subsequently, the 3D coordinates of the nuclei retrieved from Imaris were mapped to the two-dimensional coordinates consisting of the position along the z axis and distance along the ellipse circumference relative to the major axis. Next, we calculated area and volumes defined by Voronoi geometries of each nucleus and distance maps for all points on the cylinder surface to the nearest nuclei based on the distance along the cylinder surface, noting that the surface is periodic along the circumference.

Additionally, we ran Monte Carlo simulations, and nuclei were randomly placed on the parameterized surface to compare the distance map with that of the observed fibers. By Lloyd’s algorithm, we also placed the nuclei optimally on the surface such that the Voronoi domains for all nuclei were equal and compared it with an observed distribution of nuclear patterns.

All codes used in the computational modeling can be found at https://github.com/CINPLA/cylinderTools.

Statistics

Data were derived from 53 fibers injected on the lateral surface of EDL (N = 9 animals) and 51 fibers injected on the dorsal surface of the soleus muscle (N = 12 animals). Descriptive statistics are shown as the pooled sum of fibers for a given muscle. If not stated otherwise, numbers are presented as mean ± standard deviation (SD) (mean ± SD). Frequently, point pattern analysis, including Voronoi segmentation, subdivide the area or volume of interest into a subset of polygons (two-dimensional) or polyhedrons (3D). In Voronoi segmentation, the number of Voronoi polygons is proportional to number of points in the plane or room. In our case, the number of points is equal to the nuclear number within each muscle fiber. Thus, if an area (or volume) is divided into the same number of domain objects, the average size of objects remains the same despite differences in the nuclear patterning (i.e., because the total area, volume, and number of points are unchanged). Instead, we used the variation (SD) as a measure of similarity between the observed mean size of polygons to those when nuclei were placed randomly or optimally. We statistically compared SDs of domain areas and domain volumes by differences in nuclear patterning by a one-way analysis of variance followed by a Tukey’s correction for multiple comparisons.

In comparison, we are not faced with the same problem as outlined for the Voronoi segmentation when analyzing transport distances between different nuclear distributions. They were therefore analyzed statistically by a Brown-Forsythe analysis followed by a Games-Howell’s multiple comparisons test at a 5% significance level.

All graphs and statistical analysis were plotted and performed in Prism 8 (GraphPad).

Results

We have analyzed fiber segments of live surface fibers of EDL and soleus muscle in situ. Nuclei were generally confined to the fiber surface; hence, we analyzed nuclear domains both in two-dimensional (as surface domains along the sarcolemma) and in 3D (as volume domains of the fiber cytoplasm).

Cross-sectional area and nuclear number in individual muscle fibers from the EDL and soleus muscle

Quantification of nuclear number after injecting fluorescently labeled oligonucleotides and fluorescent dextran into the fibers of the EDL (Fig. 1 A) and soleus muscle (Fig. 1 B) revealed that the EDL fibers had approximately half as many nuclei (47 ± 10 nuclei/mm) compared with the soleus fibers (88 ± 29 nuclei/mm) (Fig. 1 E). Reconstructions of fiber segments from confocal stacks (Fig. 1, C and D) showed that the EDL fibers were ∼31% larger than those in the soleus (Fig. 1 F). On average, the EDL muscle had a cross-sectional area of 1091 ± 278.4 μm², and the soleus was 831.8 ± 241.3 μm². To normalize for size differences between the two muscles, we expressed the area or volume per nucleus, the so-called nuclear domain. EDL fibers had volume domains 130% larger than soleus fibers (Fig. 1 G) and 93% larger surface domains than in the soleus (Fig. 1 H). The domain volumes were on average 23,468 ± 5930 and 10,187 ± 3432 μm³ in the EDL and soleus muscle, respectively. In the EDL, surface domains were on average 2907 ± 582 μm², whereas in the soleus, they were on average 1508 ± 582 μm².

Figure 1.

3D modeling of muscle fibers after in vivo injections and confocal imaging. (A and B) Shown are representative collapsed z-stacks of fibers from the EDL (A) and soleus (B) muscles injected with dextran (blue) and labeled oligonucleotides (red). (C and D) Shown are 3D modeled fibers based on the fluorescence from the cytosolic dextran, whereas (D) shows an example of automatic assignment of spots to each myonucleus to define their 3D coordinates for subsequent analysis. (E) Given is the frequency distribution of the nuclear number and (F) cross-sectional area, domain volume (G), and surface domains (H) in the EDL (blue) and soleus muscle (red).

EDL and soleus display pattern differences of their domains

Although the average nuclear domains are a simple function of the density of nuclei and independent of how the nuclei are distributed, our data allowed us to study the distribution of the individual myonuclear domains. In a nuclear distribution in which the distance between the nuclei is optimized, all domains would in principle be of equal size. Any deviation from this would imply a nonoptimized positioning of nuclei, which may indicate that some domains would be unnecessarily large and represent “lacuna” where transport distances might be rather long.

To analyze how individual surface domains is influenced by the distribution of nuclei, we performed a detailed quantification of individual nuclear domains for each nucleus within fibers of the EDL and soleus muscle (Fig. 2 A). Voronoi segmentation is a common way of formally quantifying areas or volumes of “influence” for distributed entities of the same kind. We modeled individual Voronoi domains by partitioning the fiber into subcompartmentalized surface areas and volumes based on the position of each individual nucleus. To compare differences between nuclear distributions in the two muscles, we used the SD of the domain sizes. In EDL, the observed cumulative frequency for the SD as measured for the Voronoi areas were more similar to those obtained when the nuclei were placed optimally, compared with a random placement of nuclei (Fig. 2 B). However, some fibers showed much larger variability (Fig. 2 D), indicating a less optimal distribution with some large lacunar domains. In the soleus, however, the observed pattern resembled more the values obtained when nuclei were placed randomly on the fiber surface (Fig. 2 C). There was no obvious difference in the variation of nuclear position with fiber size in either muscles (Fig. 2, F and G).

Figure 2.

EDL and soleus display differences in patterning of Voronoi areas. (A) Shown are representative Voronoi diagrams of individual surface domain areas in fibers from the EDL and soleus in the observed distribution and where nuclei were placed randomly or optimally on the fiber surface. (B and C) Given is a cumulative comparison of observed and simulated nuclear patterns in the EDL and soleus muscle, respectively, plotted as the variation (SD) of individual domains within a single fiber. (D) and (E) show the interspecific variation between nuclear patterns in the EDL and soleus muscle, respectively, and lines signify the interspecific connection between standardized variation of random observed and observed optimal. (F and G) Shown is the relationship between the SD of surface domains plotted against the cross-sectional area in the EDL and soleus, respectively. The relationship between variation and cross-sectional area was nonsignificant: random (EDL, p = 0.1625; SOL, p = 0.5160), observed (EDL, p = 0.8844; SOL, p = 0.5858), and optimal (EDL, p = 0.6892; SOL, p = 0.9167).

Statistically, domain areas of the observed distribution were different from a random positioning of nuclei (mean difference in the SD of 641 μm²; p < 0.0001 in the EDL and 194 μm²; p < 0.0001 in the soleus) as well as optimal positioning (mean difference in the SD of 405 μm²; p < 0.0001 for the EDL and 432 μm²; p < 0.0001 for the soleus).

Next, we analyzed the Voronoi volumes (Fig. 3 A), and the difference between the two muscles resembled the values obtained when comparing Voronoi areas. Thus, in the EDL, the variability resembled an optimal distribution (Fig. 3, B and D), whereas in the soleus, the variability was closer to placing the nuclei randomly (Fig. 3, C and E). There was no clear effect of fiber size on the variation in Voronoi volumes (Fig. 3, F and G).

Figure 3.

EDL and soleus display differences in the patterning of Voronoi volumes. (A) Shown are representative Voronoi diagrams of individual domain volumes in fibers from the EDL and soleus in the observed distribution and where nuclei were placed randomly or optimally on the fiber surface. (B and C) Given is a cumulative comparison of observed and simulated nuclear patterns in the EDL and soleus muscle, respectively, plotted as the variation (SD) of individual domains within a single fiber. (D) and (E) show the interspecific variation between various nuclear patterns in the EDL and soleus muscle, respectively, and lines signify the interspecific connection between standardized variation of random observed and observed optimal. (F and G) Shown is the relationship between the SD of volume domains plotted against the cross-sectional area in the EDL and soleus, respectively. The relationship between variation and cross-sectional area was nonsignificant: random (EDL, p = 0.9651; SOL, p = 0.9965), observed (EDL, p = 0.9843; SOL, p = 0.8789), and optimal (EDL, p = 0.0627; SOL, p = 0.4078).

Statistically, domain volumes of the observed distribution were different from a random positioning of nuclei (mean difference, 6521 μm³; p < 0.0001 in the EDL and mean difference, 224 μm³; p < 0.0001 in the soleus) as well as optimal positioning (mean difference, 6306 μm³; p < 0.0001 for the EDL and 4084 μm³; p < 0.0001 for the soleus).

Transport distances vary between the EDL and soleus muscle

The perhaps most relevant measure for judging the importance of nuclear density and positioning is the actual transport distances along the surface or through the cytoplasm. To model this, a grid of points corresponding to the pixel size was placed on the surface or in the volume of each fiber. The closest distance from these points to the nearest nuclei was calculated both on the surface (Fig. 4, A and B) and in the fiber volume (Fig. 5 A).

Figure 4.

Quantitated measurements of transport distances along the fiber surface. (A and B) Shown are fiber surface distance maps of fibers in the observed distribution and where nuclei were placed randomly or optimally on the fiber surface in the EDL and in the soleus muscle, respectively. (C–H) The average surface distance to the nearest nucleus was calculated and compared with optimal and random transport distances. Transport distances in the EDL (blue, C) and soleus muscle (red, D) compared by their mean value per fiber lines signifies the interspecific connection between random observed and observed optimal. (E) and (F) compare transport distances on a per-nucleus basis after a Gaussian or lognormal fit on data from the EDL (E) and soleus (F). In (E), all nuclear distributions were fitted with a Gaussian distribution (the probability of a Gaussian distribution >99.99% for all three distributions), whereas in (F), only the optimal distribution were fitted with a Gaussian distribution (probability >99.99%), and random (Gaussian probability <0.01%) and observed (Gaussian probability <0.01%) distributions were based on a lognormal distribution. (G) and (H) show the cumulative frequency curve of (E) and (F) for both the EDL (G) and soleus (H). The red-green-blue color gradient in (A) and (B) signifies the distance at 0, 30, and 60 μm from its nuclear origin.

Figure 5.

Measurements of transport distances to the cytoplasm. (A) Shown are fiber volume distance maps of fibers where nuclei are placed optimally, randomly, and compared with the observed positioning. (B–G) The average volume distance to the nearest nucleus were calculated and compared with optimal and random transport distances. Shown are transport distances in the EDL (blue, B) and soleus muscle (red, C) compared by their mean value per fiber, whereas (D) and (E) compare transport distances on a per-nucleus basis after a Gaussian or lognormal fit on data from the EDL (D) and soleus (E). (F) and (G) show the integrated curve of (D) and (E) for both the EDL (F) and soleus (G). Lines in (B) and (C) signify the interspecific connection between random observed and observed optimal. The Gaussian or lognormal fit in (D) and (E) were selected based on Akaike’s information criterion. In (D), all nuclear distributions were fitted with an Gaussian (the probability of a Gaussian distribution >99.99% for all three distributions), whereas in (E), only the optimal distribution were fitted with an Gaussian (probability >99.99%), and random (Gaussian probability <0.01%) and observed (Gaussian probability <0.01%) distributions were based on a lognormal distribution. The red-green-blue color gradient in (A) signifies the distance at 0–30 and 30–60 μm from its nuclear origin.

Comparing transport distances along the surface between different nuclear distributions showed that on average, EDL fibers had 10% longer transport distances (23 ± 4.7 μm) compared with the value obtained by placing nuclei optimally (21 ± 2.5 μm). When nuclei were placed randomly (27 ± 3.2 μm), the average distance was increased by 29% relative to optimal distances, and the observed distances were closer to an optimal distribution (Fig. 4, C, E, and G). In the soleus muscle, fibers had transport distances (20 ± 9.2 μm) 25% longer compared with the average transport distance when nuclei were placed optimally (16 ± 4.2 μm), whereas placing nuclei randomly (20 ± 5.5 μm) was essentially identical to the observed distances (Fig. 4, D, F, and H).

For average transport distances in the cytoplasm (Fig. 5 A), the observed distances in the EDL (22 ± 3.6 μm) were closer to those seen when nuclei were placed randomly (23 ± 2.7 μm) compared with nuclei placed optimally (21 ± 2.6 μm) (Fig. 5 B). Interestingly, distances to the cytoplasm retrieved from placing nuclei randomly (20 ± 5.5 μm) or optimally (17 ± 5.7 μm) in the soleus (Fig. 5 C) were not significantly different from observed transport distances (20 ± 7.5 μm).

In the EDL, there was a statistical improvement in placing nuclei optimally compared with the observed and random transport distances to the surface and in the cytoplasm (Fig. S1, A and B). However, there was no statistical difference between the observed placement of nuclei and their transport distances to the cytoplasm when compared with optimally and randomly positioned nuclei (Fig. S1 B). In fibers of the soleus muscle, there was a statistical improvement when placing nuclei optimally for transport distances along the surface (Fig. S1 C) but no improvement within the cytoplasm (Document S1. Fig. S1, Document S2. Article plus Supporting Material).

The average transport distance might not reflect whether there are some areas or volumes that are far away from any given nucleus, giving rise to lacunas that might have an insufficient supply of macromolecules. Therefore, we analyzed the distances on a per-nucleus basis. Distribution of individual nuclear distances were fitted with either a Gaussian or lognormal function (Figs. 4, E and F and 5, D and E) and also integrated to obtain the cumulative frequency of distances (Figs. 4, G and H and 5, F and G). It was evident from the curve-linear fit and the cumulative distribution that the actual transport distances along the surface in the EDL were much improved compared with a random distribution and was approximating an optimal distribution (Fig. 4, E and G). Similar conclusions were obtained analyzing optimal transport distances in the cytoplasm (Fig. 5, D and F). Notably, the longer transport distances were less prevalent with the observed placement of nuclei compared with a random distribution. For the soleus, the observed nuclear placement was close to the random distribution along the surface (Fig. 4, F and H) and within the cytoplasm (Fig. 5, E and G).

In the EDL, we found that 23% of the surface was longer than 30 μm from the nearest nucleus, compared with 10% when nuclei were placed optimally and 35% randomly (Fig. 6 A). In the soleus 14% of the fiber surface were longer than 30 μm from the nearest nucleus in the observed distribution, compared with 3 and 17% in the optimal and random distribution, respectively. Interestingly, and despite differences in fiber size and nuclear number, both muscles displayed ∼97% of their surface to be within 50 μm of its nearest nucleus (Table 1).

Figure 6.

The percentage distribution of the cytosol and surface to its nearest nucleus at different distances. (A) and (B) show the percentage of the fiber to its nearest nucleus along the surface (A) and within the cytosol (B) when nuclei were placed randomly and optimally, compared with the observed placement of nuclei along the surface in the EDL (blue) and soleus muscle (red). Transport distances were binned at the following six intervals: 0–10, 11–20, 21–30, 31–40, 41–50, and >50 μm.

Table 1.

Percent of the Cytosol or Surface to its Nearest Nucleus at Different Distances

	Distance (μm)	Observed (%)		Optimal (%)		Random (%)
		EDL	Soleus	EDL	Soleus	EDL	Soleus
Surface	0–10	13	24	13	25	8	22
	11–20	32	40	36	53	27	37
	21–30	34	22	41	18	27	25
	31–40	16	8	9	2	19	11
	41–50	4	3	1	1	10	4
	>50	3	3	<1	<1	6	2
Cytosol	0–10	8	17	8	17	8	15
	11–20	38	48	43	58	35	45
	21–30	37	22	39	18	35	27
	31–40	12	8	7	5	16	9
	41–50	3	3	2	2	5	3
	>50	1	2	<1	<1	1	1

In terms of volumes, 16% of the cytoplasm in the EDL were longer than 30 μm from its nearest nucleus during observed distributions, increasing to 22% when nuclei were placed randomly and 9% with optimal distribution of the nuclei. The soleus muscle displayed overall shorter transport distances because only 13% of the cytoplasm was more than 30 μm from its nearest nucleus, identical to the volume seen when nuclei were randomly distributed. When nuclei were distributed optimally, 8% of the cytoplasm was more than 30 μm from its nearest nucleus (Fig. 6 B). Despite differences between the two muscles with respect to distances within 30 μm, 98% of the cytoplasm resides within a 50-μm proximity to the nearest nucleus (Table 1).

Consequences for diffusion times

We describe here the differences in dimensions, distribution, and density of myonuclei in fibers from soleus and EDL. It has, however, previously been reported that also the effective diffusion coefficient for different proteins differs in fibers from these two muscles (29). When combined with our data, this information allows us to calculate diffusion times (t) from the nearest nuclei to any point in the cell according to t = L²/2σD, where L is the transport distance in σ dimensions, and D is the effective diffusion coefficient. The different diffusion profiles along the surface (Fig. 7 A) and within the cytosol (Fig. 7 B) were used to derive the transport times for the two muscles (Fig. 7, C and D) and compared with transport times that would be obtained by active transport (see Introduction).

Figure 7.

EDL and soleus muscle display roughly equal diffusion times. In combination with the different diffusion coefficients in the soleus and EDL and the root mean-square distance, we were able to analyze the diffusion profiles with respect to time in the two muscles by plotting transport times versus transport distances along the surface (A) and within the fiber volume (B). Because longer intracellular distances would linearly elevate the demand for chemical energy to power active transport, whereas diffusion times, which only rely on thermal noise, would increase with the square of a length characterizing the cell, the relative difference in nuclear patterning and transportation path between the two muscles could be due to the trade-off made by the energy consumption of active transport versus diffusive transport. Therefore, we also compared the diffusion profile in (A) and (B) with active transport at speeds of 1 μm/s (orange dashed line) and 3 μm/s (black dashed line). The corresponding diffusion times and active transport times are highlighted as histograms with a lognormal fit for the surface (C) and fiber volume (D). The vertical dashed line in (C) and (D) corresponds to the diffusion time where the black lines intersect the diffusion profile in (A) and (B). The gray shaded area relates to active transport times at a speed of 3 μm/s, which would be theoretically be faster than diffusion. The numerical inset in (C) and (D) is the geometrical mean, and its SD is based on the lognormal distribution, whereas p signifies the probability of equal transport times between the soleus and EDL given the degrees of freedom.

We used myoglobin as an example and calculated the transportation times to each point on the surface. Using the data from Papadopoulos et al. in which diffusion coefficients for myoglobin were measured to be 12.5 × 10⁻⁸ and 18.7 × 10⁻⁸ cm²s⁻¹ in soleus and EDL, respectively, we calculated transport times based on the lognormal fit to be on average 7.5 ± 1.2 and 7.3 ± 1.6 s for EDL and soleus, respectively (Fig. 7 C). For the diffusion times to each point in the entire cytoplasmic volume, the mean was 4.5 ± 1.2 and 4.7 ± 1.3 s for EDL and soleus, respectively (Fig. 7 D). Thus, despite the large differences in dimensions, nuclear density, and nuclear distribution, the diffusion time was remarkably similar for the two muscles.

It is likely that intracellular transport in muscle relies both on active and passive transport. Active transport along microtubules has been described in skeletal muscle (30,31). Although we are not aware of data on myosin-mediated transport along actin in skeletal muscle, myosin IV has been observed localized to the fiber periphery, nuclei, and sarcoplasmic reticulum and at the neuromuscular junction (32), and the thick-filament-associated myosin molecule seems to be replaced in cultured myotubes in the absence of the microtubule system (33), whereas the actin network per se seems to be imperative for transport of membrane-located molecules, e.g., the insulin sensitive glucose translocator GLUT4 (34).

For the transport along the surface (Fig. 7 A), diffusion would be faster than actin-mediated transport (3 μm/s) for distances of less than 15 μm in the soleus and 25 μm in the EDL. When compared with transport along microtubules (1 μm/s), diffusion would be faster for distances up to 50 μm in the soleus and 75 μm in the EDL, which were far outside the range of observed values for transport times.

Using the data related to Fig. 4, actin-mediated transport would on average be faster in 15% of the fibers based on the observed placement in the EDL fibers, whereas the corresponding relative values for random and optimal placement were 62 and 8%, respectively. On a per-nucleus basis, the percentage of nuclei that display transport distances that would yield active transport faster than diffusion were 37% of the nuclei for the observed placement, whereas others were 50 and 33% when nuclei were placed randomly and optimally, respectively. In the soleus, the average transport distances per fiber related to Fig. 4 F would yield active transport faster than diffusion in 78% of the fibers based on the observed placement of nuclei and 88 and 49% of the fibers when placed randomly or optimally, respectively.

When allowed to diffuse in three dimensions (Fig. 7 B), diffusion would be faster than actin-mediated transport for distances of less than 24 μm in the soleus and 36 μm in the EDL. When compared with transport along microtubule, diffusion would be faster for distances up to 75 μm in the soleus and 112 μm in the EDL, corresponding to transport distances and times outside the range of observed values.

The relative amount of fibers in which active transport through the cytosol would be faster than diffusion based on data related to Fig. 5 applies to less than 8% of all fibers independent of nuclear placement in the soleus. For the EDL muscle diffusion throughout, the fiber volume would thus be faster than active transport speeds up to 3 μm/s for all fibers analyzed.

On a per-nucleus basis, there was not much to gain in diffusion times based on nuclear placement in the EDL because only 9% of the nuclei had distances longer than 36 μm, whereas the corresponding values for random and optimal placement were 11 and 4%, respectively. In the soleus, however, 23% of the nuclei had distances longer than 24 μm, whereas 14 and 26% of the nuclei had distances longer than 24 μm when placed optimally and randomly, respectively.

Discussion

We present here precise data for cell dimensions and myonuclear density and positioning in fiber segments of living muscle fibers from EDL and soleus in situ. Fibers from these muscles varied with respect to all these variables, thus the radial size of the EDL fibers was 31% larger than those in the soleus, whereas the surface or volume per nucleus was 93 and 130% larger than in the soleus, respectively. Despite these differences, the calculated transport times to points on the surface or in the cytosol were remarkably similar in soleus and EDL, and we hypothesize that fibers are adapting the number and positioning of nuclei to achieve certain transport times for relevant macromolecules to all parts of the cell.

Given the differences in the variables related to cell dimensions and myonuclei, it might seem like a paradox that the diffusion times are so similar, but it is explained firstly by the diffusion coefficient being 50% higher in the EDL than in the soleus; secondly, we show that the nuclei are placed more optimally in the EDL. If the nuclei in the EDL were placed randomly, the average surface transport distance would yield ∼37% longer diffusion times. This would translate into a 30% lower concentration of a macromolecule at any given point, and because cytoplasmic dilution has been associated with impaired cell function in proliferative cells (35,36), the nonrandom positioning of nuclei thus seem to be important in fibers of the EDL muscle.

In contrast, in the soleus, the placement of nuclei appeared close to random. Moreover, the transport distances observed within here showed that the soleus displayed ∼13% shorter surface transport distances compared with the EDL, which translates into a 32% more concentrated milieu in the soleus relative to the EDL. It is possible that optimalization of positioning is less critical because the density of nuclei is higher. Another explanation is that other needs override those related to internal cell transport in oxidative muscles, thus in the rat soleus, 81% of the nuclei appear next to blood vessels (37).

The interior of any cell is greatly crowded by macromolecules such as ribosomes, RNA, and proteins (38,39), which effectively reduce the motion of molecules and thereby impede diffusion. Additionally, muscle fibers have a dense mesh of myofibrils that occupy 80% of their total volume, which seems to act as exclusion zones for larger macromolecules (40). The EDL and soleus have different properties with regard to cytoarchitecture, with the soleus containing more mitochondria and thicker Z-lines, which may explain the slower diffusion times observed in the soleus than in the EDL (29). Taking into account the effects of macromolecular crowding and scaling of transport times, it has been argued that mononuclear cells have small sizes to optimize diffusion and signaling (41). They found the optimal size of a eukaryotic cell to have a radius of ∼10 μm, about half of the transport distance found for muscle fibers in our calculations. This problem might be aggravated because all nuclei do not express all genes at a given time point (42,43). Thus, the transport distances reported here might represent a bottleneck, limiting cell function and size of muscle fibers.

Hence, because protein concentration within cells are optimized for biochemical reactions (44), whereas the oxidative fibers exhibit higher rates of protein synthesis and degradation compared with glycolytic fibers (45), the apparent random positioning of nuclei in the soleus is optimized to accommodate for the higher biological rates rather than improving its distribution statistically. In fact, for the cytoplasmic distances in the soleus when nuclei were placed randomly were not significantly different from nuclei placed optimally. For example, in a recent study using a mosaic transfection model to create myotubes that contained exactly one nucleus expressing a fluorescent nuclear reporter, its transport was highly influenced by the width of the myotubes and nuclear import rates (46). More specifically, they found that neighboring nuclei were able to import nuclear proteins (e.g., transcription factors) up to a distance of 50 μm from the transfected nucleus, and their mathematical model showed that increasing nuclear density could offset the dilution effects of muscle hypertrophy and restore the original propagation profile of nuclear proteins.

Cytoplasmic dilution has been associated with impaired cell function in proliferative cells (35,36). Thus, our findings of differences between the two muscles in transport distances along the surface and within the cytoplasm may be a mechanism to maintain fibers of different phenotypes with respect to the DNA content to cytoplasmic volume that is required to warrant an appropriate RNA and protein synthesis within transcriptional domains (47).

Disturbance of myonuclear placement impairs muscle function, e.g., after mutations in the Drosophila larvae (8,12). In mammals, denervation, which leads to grave atrophy and is accompanied by myonuclear clustering, eventually gives rise to lacunas with long transport distances (see Results) (48,49). Similarly, nuclear distribution is disturbed in several myopathies and might play a causative role in impairing function (50).

Author Contributions

K.-A.H. performed the animal experiments. A.V.S. performed the computations and modeling. K.-A.H. prepared the figures. J.C.B., K.-A.H., A.V.S., and K.G. designed the experiments. With input from A.V.S., all the remaining authors wrote the article.

Editor: Jason Swedlow.