Modeling of Supramolecular Centrosymmetry Effect on Sarcomeric SHG Intensity Pattern of Skeletal Muscles

Abstract

A theoretical far-field second harmonic generation (SHG) imaging radiation pattern is calculated for muscular myosin taking into account both Gouy effect and light diffraction under high focusing excitation. Theoretical analysis, in agreement with experimental results obtained on healthy Xenopus muscles, shows that the increase on intensity at the middle of the sarcomeric SHG intensity pattern is generated by an off-axis constructive interference related to the specific antipolar distribution of myosin molecules within the sarcomere. The best fit of the experimental sarcomeric SHG intensity pattern was obtained with an estimated size of antiparallel, intrathick filaments' packing-width of 115 ± 25 nm localized at the M-band. During proteolysis, experimental sarcomeric SHG intensity pattern exhibits decrease on intensity at the center of the sarcomere. An effective intra- and interthick filaments centrosymmetry of 320 ± 25 nm, in agreement with ultrastructural disorganization observed at the electron microscopy level, was necessary to fit the experimental sarcomeric SHG intensity pattern. Our results show that sarcomeric SHG intensity pattern is very sensitive to misalignment of thick filaments and highlights the potential usefulness of SHG microscopy to diagnose proteolysis-induced muscular disorders.

Introduction

Second harmonic generation (SHG) imaging microscopy is a powerful tool to image intrinsic subcellular signals from endogenous proteins such as tubulin, myosin, and collagen, which are molecules possessing a permanent dipole and which are well ordered in biological tissue (1–7). Concerning myosin, it has been shown that the elementary dipole responsible for the periodical sarcomeric SHG signal of striated muscles is located along the axis of the α-helix of the myosin molecules (8–10) within the amide groups of the polypeptide chains (11). Sarcomeric SHG intensity pattern is either single- or double-band (SB or DB) when the intensity profile of the SHG signal is considered along the main axis of the myofibril (10,12–19). Different explanations of the DB sarcomeric SHG intensity pattern have been proposed such as nonuniform distribution of myosin heavy chain isoforms along thick filaments, intrathick filaments centrosymmetry resulting from tail-tail interactions of myosin molecules, and lack of optical axial resolution due to thick tissue observation (2,10,13,14,18).

In a previous work, we have shown that the sarcomeric SHG intensity pattern was highly dependent on the freshness of the muscle tissue and was usually SB (∼95%) in fresh tissue and DB (∼90%) in 24 h proteolyzed tissue (20). More recently, we have shown that mechanical stress, chemical stress, and photodamage are able to induce conversion from SB to DB (21,22). We have suggested that appearance of the DB pattern could be due to interthick filaments centrosymmetry resulting from thick filaments misalignment. The SHG intensity pattern of skeletal muscle is undoubtedly related to the particular antipolar organization of myosin molecules within the sarcomere. The presence at its center of a region of antiparallel tail-tail interactions of myosin molecules (see Materials and Methods) is expected to generate intrathick filaments centrosymmetry with no SHG emission. Therefore, the predominant SB sarcomeric SHG intensity pattern with a central increase on intensity observed from fresh muscle tissues, awaits a rational explanation (21).

To this end, based on the summation of the second harmonic fields radiated by the nonlinear polarization induced by the incident laser field, we calculate the far-field SHG radiation pattern. For fresh healthy muscles we demonstrate that, despite the presence of intrathick filaments centrosymmetry centered at the M-band, the antipolar arrangement of myosin molecules leads to constructive interference of the SHG signal resulting in an increase on intensity at the center of the sarcomere and therefore to a SB sarcomeric SHG intensity pattern. Comparison of the theoretical analysis with experimental results enables the estimation of the width M′ = 115 ± 25 nm of the intrathick filaments centrosymmetry. For proteolyzed muscles, electron microscopy (EM) images reveal interthick filaments disorder of width M″ = 200 ± 100 nm. Assuming that this disorder produces an extension M′_a = M′ + M″ of the region of centrosymmetry at the center of the sarcomere, the model predicts a decrease of the SHG intensity for M′_a > 200 nm resulting in a DB sarcomeric SHG intensity pattern. Comparison of the theoretical analysis with experimental results yields an estimate of the width M′_a = 320 ± 25 nm of the intra- and interthick filaments centrosymmetry. This study highlights the sensitivity of SHG microscopy to reveal supramolecular antipolar organization at a nanometric scale and the potential usefulness of SHG microscopy to diagnose proteolysis-induced muscular disorders.

Materials and Methods

Structure of the sarcomere

Striated muscles are built from very thin (∼1 μm) myofibrils formed by a regular and periodical assembly along their main axis of elementary contractile units called sarcomeres. A sarcomere of width ∼L = 2 μm (23,24) consists of a double overlapping of a large number of bipolar thick (myosin II) and thin (actin) filaments (see Fig. 1 A). The myosin II molecule is a hexameric protein (see Fig. 1 B) with two heavy chains and four light chains. The C-terminal end of each heavy chain is α-helical, whereas its N-terminal end is globular. The two heavy chains assemble to form a coiled-coil dimer. Myosin molecules self-associate to form an antiparallel tail-tail interaction of width M′ at the middle of the thick filament. M′ is therefore part of the central bare zone of each thick filament that is characterized by the absence of myosin heads. We consider the M-band as the central sarcomere region encompassing all the bare zones of thick filaments (see Fig. 1 C). Finally, thick filament elongates by staggered association of myosin tails and has a constant length of ∼A = 1.6 μm in relaxed and contracted vertebrate muscles (23,24).

Figure 1.

Schematic diagram illustrating the major sarcomeric components of striated muscles. (A) View of the sarcomere of width L showing overlapping arrays of bipolar myosin thick filaments and actin thin filaments. Thick and thin filaments are transversally interconnected at the M- and Z-bands, respectively, by myomesin or M-protein and α-actinin (not represented). Thick filaments of width A are longitudinally connected at the Z- and M-bands by titin. (B) The myosin molecule is a hexamer of two heavy chains and two pairs of light chains (not presented) and has a length of ∼160 nm. The heavy chain α-helix tails of each myosin molecule form a coiled-coil super-helix dimer. (C) Schematic representation of the antiparallel assembly of myosin molecules at the M-band free of heads. Myosin heads point away from the filament center and myosin tails have antiparallel overlapping of width M′.

Tissue preparation

Stage 50 tadpoles were euthanized in MS222 (0.5 mg.mL⁻¹; Sigma-Aldrich, St. Louis, MO). One group of tadpoles, was immediately fixed overnight in 4% paraformaldehyde containing Mark's Modified Ringer (MMR): NaCl 100 mM, KCl 2 mM, CaCl₂ 2 mM, MgCl₂ 1 mM, HEPES 5 mM, pH 7.8, 4°C). A second group was left in breeding water at room temperature (18–22°C, for 24 h) to obtain a homogenous proteolysis of the tissues, and fixed one day later. The first and second group yields, respectively, the mentioned “fresh” and “proteolyzed” muscle tissue in the text. Blocks of ∼1 mm³ were cut on both tadpoles' tails after being washed three times in MMR. They are then postfixed in glutaraldehyde (2.5%, 4°C, in 0.2 M cacodylate buffer, pH 7.2) for 1 h. For SHG microscopy, the adjacent remaining parts of the tails were mounted in MMR in the imaging chamber (POC system, PeCon, Erbach, Germany). For EM, blocks were washed in cacodylate buffer. The next day, they were postfixed in osmium tetroxide (2%) for 1 h and washed with cacodylate buffer. On the following day, they were dehydrated in a graded acetone series. The day after, they were impregnated firstly into a mix of Epon, Araldite, and acetone during 1.5 h, and secondly into a mix of Epon, Araldite, and DMP30.

Samples were then laid down into molds and polymerization occurred overnight at 60°C. Blocks were desiccated during 24 h and mounted on the cutting stage. Ultrafine slices were made with an ultra-microtome (Ultracut E; Reichert, Depew, NY) and mounted on a copper grid before being stained with uranyl acetate (2.5%) during 1 h. They were then dipped in lead citrate and finally rinsed. Imaging was performed using a 100CXII microscope (JEOL, Tokyo, Japan) at 80 kV accelerating voltage, at the EM mRIC platform (http://microscopie.univ-rennes1.fr/). For thin cryostat-sliced sections, muscle tissues were immerged in an increasing concentration of sucrose solutions (7.5% for 1 h, 15% for 1 h, and 30% for 1 day) after fixation (in paraformaldehyde, 4%, 4°C, overnight). Then, they were dipped in a sucrose (4.5%) plus gelatin (10%) solution at 37°C and stored at 4°C. After solidification, blocks were cut to fit closely the sample, and embedded in cryoresin (Nek50). Polymerization of cryoresin was achieved by dipping blocks in −55°C frozen isopentan and stored at −20°C. Blocks were sliced with a cryostat (Microm, Walldorf, Germany) in 10-μm-thick sections and laid on gelatin-coated slides. Vectashield (Vector Laboratories, Burlingame, CA) mounting medium was added and the coverslip was sealed with nail polish.

SHG imaging system

SHG images were acquired on PIXEL (http://pixel.univ-rennes1.fr/) (facility of GIS EUROPIA, University of Rennes 1, France). The SHG imaging system consists of a confocal model No. FV1000 scanning head (Olympus, Tokyo, Japan) mounted on an upright microscope (model No. BX61WI; Olympus) and equipped with a MaiTai femtosecond laser (Spectra Physics, Santa Clara, CA). High NA water immersion objective (model No. LUMFL 60 W × 1.1 NA; Olympus) was used for applying 10–20 mW of 940 nm excitation at the sample. The SHG signal was collected in forward direction using a water immersion condenser (model No. IX2-DICD 0.9 NA; Olympus). A BG39 bandpass filter (Schott, Duryea, PA) and a 470-nm IR filter (10 nm, full width at half-maximum (FWHM)) were placed before the photomultiplier tube. All specimens were positioned on the fixed x,y stage of the microscope with light propagating in the z direction. The input polarization was chosen along the y axis and perpendicular to the main x axis of the myofibrils. Images, either from SHG or EM microscopies, were analyzed with open source ImageJ software (National Institutes of Health; http://rsb.info.nih.gov/ij/).

Results

SHG images of fresh nonproteolyzed Xenopus tadpoles' tails (see Materials and Methods) are shown in Fig. 2 A. They are characterized by a regular bright single-band (SB) pattern in agreement with our previous study (20,22). Moreover, an increase on intensity is clearly visible at the center of each sarcomere, as previously reported (21). However, the SHG signal of 24 h proteolyzed muscle tissue is characterized (see Fig. 2 B) by a decrease on intensity at the middle of the sarcomere, resulting in the predominance of double-band (DB) sarcomeric SHG intensity pattern as previously observed (20). To understand this drastic modification of sarcomeric SHG intensity patterns, we have undertaken calculation of the far-field SHG radiation pattern, taking into account both the particular organization of myosin thick filaments within the sarcomere and the geometry of the incident laser beam as illustrated in Fig. 3.

Figure 2.

Typical optical sections of SHG images of Xenopus tadpole tail muscles (A and B). (A) Fresh muscles with characteristic SB pattern. (B) Proteolyzed muscles with most SB converted to DB pattern. (Inset) Experimental values (dots) of the SHG intensity (arbitrary units) obtained along the dotted line with the theoretical intensity I^T_2ω (solid line) obtained for M′ = 110 nm (A) and M′_a = 310 nm (B). L = 2 μm, A = 1.6 μm, n_ω = n_2ω = 1.33, w_xy = 0.48 μm, w_z = 2.4 μm, and λ_ω = 940 nm were used in all simulations. Bars are 2 μm.

Figure 3.

Schematic view of a sarcomere in presence of the laser beam. Nonlinear sources are drawn with opposite arrows to take into account the inversion of polarity +− at the center of the sarcomere. L, A, and M′ are, respectively, sizes of sarcomere, myosin thick filaments, and of antiparallel overlapping of myosin tails. The Gaussian laser beam (shaded) with lateral beam waist w_xy is located at Δ from the center of the sarcomere. The radiated pattern of SHG is measured for each observation point r with spherical coordinates r, θ, and φ. The x,z plane of incidence is normal to the x,y plane of observation.

There are two main approaches to treat SHG theoretically. The first one developed by Armstrong et al. (25) and Bloembergen (26) involves direct solution of Maxwell's equations, which is very useful for nonlinear homogeneous crystals. The second one is based on the summation of the second harmonic fields radiated by the nonlinear polarization induced by the incident laser electric field. The advantage of this later method is to enable us to easily take into account the spatial organization of the nonlinear sources in the interacting region. It had been used in the 1970s for inhomogeneous ferroelectric crystals with polar domains (27–29) and developed later by Freund et al. (1) for biological tissue presenting polar ordering. Our starting point is the inhomogeneous vector wave equation giving the second-harmonic electric field E_2ω emitted at 2ω in the presence of a nonlinear polarization P_2ω (30)

$$\nabla \times \nabla \times {\mathit{E}}_{2\omega }-k_{2\omega }^2{\mathit{E}}_{2\omega }=\frac{4\pi {\omega }^2}{c^2}{\mathit{P}}_{2\omega },$$ (1)

where k_2ω is the wave-vector of the harmonic wave and c is the velocity of light in vacuum. The electric field E_2ω, which is the solution of Eq. 1, can be expressed in terms of Green's function. Its expression at any position r (r, θ, φ) in the observation plane perpendicular to z (see Fig. 3), is obtained by coherent summation over all the nonlinear sources located at r₀ in the medium (31,32) as

$${\mathit{E}}_{2\omega }\left(\mathit{r}\right)=\frac{-4\pi {\omega }^2}{c^2}\int G\left(\mathit{r}-{\mathit{r}}_0\right)\times \left(I-\mathit{ss}\right)·{\mathit{P}}_{2\omega }\left({\mathit{r}}_0\right)d^3{r}_0.$$ (2)

Here, $\mathit{G}\left(\mathit{r}-{\mathit{r}}_0\right)=e^{i{k}_{2\omega }\left|\mathit{r}-{\mathit{r}}_0\right|}/4\pi \left|\mathit{r}-{\mathit{r}}_0\right|$ is the scalar Green's function, and I − ss is a projection operator formed by the dyadic product ss and used to select the transverse part of P_2ω(r₀). I is the second-order unity tensor and s the unitary vector along k_2ω. Note that the discrete summation over the nonlinear sources has been replaced by a continuous integration because the distances between individual emitters are much lower than the optical wavelength (11,24,33). For far-field radiation r >> r₀, |r – r₀| is approximated to |r| – r · r₀/|r| and E_2ω can be recast as

$${\mathit{E}}_{2\omega }\left(\mathit{r}\right)=\frac{-{\omega }^2}{c^2}\frac{e^{i{k}_{2\omega }\left|\mathit{r}\right|}}{\left|\mathit{r}\right|}\int e^{-i{\mathit{k}}_{2\omega }\left(\mathit{r}\right)\cdot {\mathit{r}}_0}\times \left(I-\mathit{ss}\right)·{\mathit{P}}_{2\omega }\left({\mathit{r}}_0\right)d^3{\mathit{r}}_0.$$ (3)

We assume that the laser beam propagates in the z direction with wave vector k_ω and with input polarization taken along y. Following Mertz (34–36), we assume the excitation field to be three-dimensional Gaussian at the vicinity of the focus

$${\mathit{E}}_{\omega }\left({\mathit{r}}_0\right)=\hat{y}{E}_{\omega }^0{e}^{\frac{x_0^2+y_0^2}{w_{xy}^2}-\frac{z_0^2}{w_z^2}+i\left(k_{\omega }{z}_0-\varphi \left(z_0\right)\right)}.$$ (4)

The values w_xy and w_z are, respectively, the lateral and axial beam waists, $\varphi \left(z_0\right)=arc\tan \left(z_0/z_r\right)$ is the Gouy phase mismatch due to focusing (37) and $z_r=\pi n_{\omega }{w}_{xy}^2/{\lambda }_{\omega }$ is the Rayleigh range. Near the focus, the assumption of a linear dependence of the Gouy phase variation with z is also made ϕ(z₀) ≈ z₀/z_r (34–36). To take into account the polarity inversion of the nonlinear dipoles along direction x, we introduce a L-periodic modulation function g₀(x₀) defined as follows within a sarcomere with the origin chosen at its center (see Fig. 3)

$$g_0\left(x_0\right)=\{\begin{array}{ll}+1\hfill & \text{for}{M}^{\prime }/2<x_0\le A/2\hfill \\ -1\hfill & \text{for}-A/2\le x_0<-M^{\prime }/2\hfill \\ 0\hfill & \text{elsewhere}.\hfill \end{array}$$

To take into account the laser scan, we define the translated modulation function

$$g_{\Delta }\left(x_0\right)=g_0\left(x_0-\Delta \right),$$

with Δ the laser beam position along the x axis (see Fig. 3). Under these conditions, the nonlinear polarization ${\mathit{P}}_{2\omega }\left({\mathit{r}}_0\right)={\chi }^{\left(2\right)}\left({\mathit{r}}_0\right){\mathit{E}}_{\omega }^2\left({\mathit{r}}_0\right)$ is given by

$${\mathit{P}}_{2\omega }\left({\mathit{r}}_0\right)=\hat{\mathit{x}}{\chi }_{15}{g}_{\Delta }\left(x_0\right){\mathit{E}}_{\omega }^2\left({\mathit{r}}_0\right)$$ (5)

with ${\chi }_{15}=\left(1/2\right)N\beta \cos \alpha {\sin }^2\alpha$ (8). The values N, β, and α are, respectively, the number density of active harmonophores, the hyperpolarizability coefficient, and the α-helix angle. We have previously shown that α is unchanged between SB and DB sarcomeric SHG patterns (22). Because thick filaments have a constant size with no strain during contraction (38,39) and are not degraded during the proteolysis process (see EM images in Fig. 6 below), we assume that N, β, and χ₁₅ remain constant. Therefore, orientation of dipoles, their density and the distance between myosin thick filaments could not be crucial parameters modulating the sarcomeric SHG intensity profile. The SHG intensity $I_{2\omega }\left(\mathit{r}\right)={\left|{\mathit{E}}_{2\omega }\left(\mathit{r}\right)\right|}^2$ radiated at r is derived from Eq. 3 using Eqs. 4 and 5

$$I_{2\omega }\left(\mathit{r}\right)=\frac{{\left({\chi }_{15}{E}_{\omega }^0{E}_{\omega }^0\right)}^2}{r^2}\times I^D\times I^X\times I^Y\times I^Z,$$ (6)

with

$$|\begin{array}{l}\begin{array}{l}{I}^D=1-{\sin }^2\theta {\cos }^2\phi \\ I^X={\left|F_{x_0}\left[g_{\Delta }\left(x_0\right)e^{-2x_0^2/w_{xy}^2}\right]\left({\tilde{k}}_x\right)\right|}^2\end{array}\hfill \\ I^Y={\left|F_{y_0}\left[e^{-2y_0^2/w_{xy}^2}\right]\left({\tilde{k}}_y\right)\right|}^2\hfill \\ I^Z={\left|F_{z_0}\left[e^{-2z_0^2/w_z^2}\right]\left({\tilde{k}}_z\right)\right|}^2.\hfill \end{array}$$

$F_{{\eta }_0}\left[f\left({\eta }_0\right)\right]\left(\tilde{k}\right)=\int f\left({\eta }_0\right)e^{-2i\pi \tilde{k}{\eta }_0}d{\eta }_0$ denotes the Fourier transform (40) of function f, respectively to η₀ = x₀, y₀, z₀ and $\tilde{k}={\tilde{k}}_x$, ${\tilde{k}}_y$, ${\tilde{k}}_z$ with ${\tilde{k}}_x=k_{\omega }/\pi \sin \theta \cos \phi ,$${\tilde{k}}_y=k_{\omega }/\pi \sin \theta \sin \phi$, and ${\tilde{k}}_z=k_{\omega }/\pi \left(\cos \theta -1+2k_{\omega }^{-2}{w}_{xy}^{-2}\right)$. I^D is the result of the projection operator and corresponds to the angular dependence of the radiation of the dipole. I^Y and I^Z are directly derived as

$$I^Y=\frac{\pi }{2}{w}_{xy}^2{e}^{-k_{\omega }^2{w}_{xy}^2{\sin }^2\theta {\sin }^2\phi }$$

and

$$I^Z=\frac{\pi }{2}{w}_z^2{e}^{-k_{\omega }^2{w}_z^2{\left(1-\cos \theta -2k_{\omega }^{-2}{w}_{xy}^{-2}\right)}^2}.$$

I^Z accounts for the phase-matching (PM) condition and the Gouy effect. Note that we have made the assumption that fundamental and harmonic waves travel in phase (n_ω = n_2ω) because the coherence length of the medium L_c = 1/2(n_2ω = n_ω)/λ_ω (∼12.5 μm considering dispersion of water (41)) is >w_z (= 2.4 μm, see later), which is the useful distance to produce the SHG signal. In low focusing condition (NA < 0.4), the Gouy effect is small, at k_ω⁻² w_xy⁻² < 1, and the PM condition is such that fundamental and harmonic waves are mainly emitted in the same direction z. In the case of high focusing condition (NA > 0.7) that we are considering here, the effect of the Gouy phase shift becomes important and the PM condition is such that most of the light is emitted at an angle θ_PM = cos⁻¹(1 – 2 k_ω⁻² w_xy⁻²) ∼ 27° (λ_ω = 940 nm, w_xy = 0.48 μm, and n_ω = 1.33). This off-axis emission is similar to that obtained in a biological membrane labeled with nonlinear amphiphilic chromophores (35,36,42) where SHG light is emitted along two lobes. In our case, the emission pattern due to the Gouy effect has a circular symmetry because diffraction takes place from a volume instead of a bilayer surface.

Figure 6.

EM images of fresh (left column) and one-day (right column) proteolyzed Xenopus tadpole tail muscles. (A) Representative EM image of healthy tissue showing tightly joined myofibrils and well-aligned sarcomeres. (B) Magnified EM image of delimited area of (A) (rectangle with dotted line) showing two canonical sarcomeres within a myofibril. (C) Representative EM image of proteolyzed tissue. Note that myofibrils are disjoined and sarcomeres are disorganized. (D) Magnified EM image of delimited area (C, rectangle with dotted line) showing two disorganized sarcomeres with disjoint myofibrils. (Arrowheads) Z-lines. (Asterisks) Gaps between myofibrils. (Brackets) Delimited M-bands. (Arrows) Z-lines, which are deviated from vertical. Bars are 2 μm.

The periodical function g_Δ(x₀) can be expanded in a Fourier series (40) as

$$g_{\Delta }\left(x_0\right)=\sum _{-\mathrm{\infty }}^{+\mathrm{\infty }}{a}_{\Delta ,n}{e}^{2i\pi n{x}_0/L},$$

with Fourier coefficients

$$a_{\Delta ,n}=\frac{1}{L}{\int }_{-L/2}^{-L/2}{g}_{\Delta }\left(x\right)e^{-2\pi inx/L}dx=\frac{-2i}{\pi n}\sin \left[\pi n\left(A+M^{\prime }\right)/2L\right]\times \sin \left[\pi n\left(A-M^{\prime }\right)/2L\right]\times \exp \left(-2i\pi n\Delta /L\right)$$

for n ≠ 0 and a_Δ,0 = 0 for n = 0, such that I^X is derived as

$$I^X=\frac{\pi }{2}{w}_{xy}^2{\left|\sum _{-\mathrm{\infty }}^{+\mathrm{\infty }}{a}_{\Delta ,n}{e}^{-\frac{1}{2}{k}_{\omega }^2{w}_0^2{\left(\sin \theta \cos \phi -\frac{n\pi }{L{k}_{\omega }}\right)}^2}\right|}^2.$$

The radiated SHG intensity I_2ω(r) given by Eq. 6 with the above expressions of I^D, I^X, I^Y, and I^Z is now explicitly expressed in terms of laser beam waists w_xy, w_z, sarcomere parameters L, A, M′, and laser position Δ. MATLAB (The MathWorks, Natick, MA) was used for the simulation. Beam waists w_xy, w_z were estimated from the two-photon excitation point spread function obtained from 0.17 μm diameter fluorescent micro beads (PS-Speck Microscope Point Source Kit No. P7220; Molecular Probes, Eugene, OR). Lateral and axial FWHM were found to be FWHM_xy = 0.40 μm and FWHM_z = 2.0 μm at 940 nm using open source MATLAB code PSFAnalyzer (http://pixel.univ-rennes1.fr/PSF Analyzer/PSFAnalyzer.m). w_xy = 0.48 μm and w_z = 2.4 μm were estimated from these values, using the relation $w_{xy,z}=\mathrm{FWH}{\mathrm{M}}_{xy,z}/\sqrt{\ln 2}$ (43).

Sarcomere parameters have been estimated to be ∼L = 2 μm and A = 1.6 μm (23,24). M′ was varied to fit the experimental SHG intensity profile. Because the summation in I^X is converging rapidly, it was truncated at |n| = 10. I^D, which corresponds to the radiation of the dipole, is plotted in Fig. 4 A. As expected, I^D is maximal (I^D = 1) in the plane perpendicular to the dipole (φ = 90°) and is minimal (I^D ∼ 1/2) in the plane of the dipole (φ = 0°) for the maximum collection angle of the water immersion condenser (∼42.6°). As previously said, I^Z which is plotted in Fig. 4 B exhibits a circular symmetry around z with maximum signal around θ_PM. To gain insight into variation of I^X when the laser beam is scanned along a sarcomere, two particular positions of the laser, either at the center of an hemi-filament or at the center of a sarcomere, are considered.

Figure 4.

Theoretical SHG radiation pattern (A–F) and schematic view of the anisotropic emission of the A-band (G). (A) Radiation pattern I^D of a dipole aligned along x. (B) Diffraction pattern I^Z due to Gouy effect. (C) Diffraction pattern $I^Y{I}_{++}^X$ for dipoles of identical polarity ++. (D) Radiated SHG intensity $I_{2\omega }^{++}\propto I^D\times I^Z\times I^Y\times I_{++}^X$ for dipoles of identical polarity ++. (E) Diffraction pattern $I^Y{I}_{+-}^X$ for dipoles of inverse polarity +−. (F) Radiated SHG intensity $I_{2\omega }^{+-}\propto I^D\times I^Z\times I^Y\times I_{+-}^X$ for dipoles of inverse polarity +−. All constant factors in I^X, I^Y, and I^Z are taken to unity. L = 2 μm, A = 1.6 μm, n_ω = n_2ω = 1.33, w_xy = 0.48 μm, w_z = 2.4 μm, λ_ω = 940 nm, and M′ = 0 were used in the simulation. (G) Schematic view of the anisotropic emission of the A-band in the direction of θ_PM. The Gouy effect results in cone-like emission of harmonic waves (shown in green) with angle θ_PM (see also panel B showing the transversal section of the cone). In direction θ_PM, interference between harmonic waves is either destructive or constructive when waves are generated from dipoles located in the same hemi-filament (blue) or in adjacent (blue and red) hemi-filaments.

If the laser is focused at the center of an hemi-filament, all dipoles involved in the interaction have the same polarity ++ (neglecting edge effect, g = 1) and I^X reduces to

$$I_{++}^X=\frac{\pi }{2}{w}_{xy}^2{e}^{-k_{\omega }^2{w}_{xy}^2{\sin }^2\theta {\cos }^2\phi }.$$

The product $I^Y{I}_{++}^X$, which is plotted in Fig. 4 C, represents the diffraction of a Gaussian circular aperture with a spread-out driven by the transverse size of the focus beam. The SHG intensity $I_{2\omega }^{++}$ radiated by dipoles of identical polarity ++ is thus given by Eq. 6 using $I^D,I_{++}^X,I^Y,I^Z$ and is plotted in Fig. 4 D.

If the laser is focused at the center of the sarcomere (Δ = 0) and for M′ = 0, all dipoles of each side of the y,z plane have opposite polarity +−. $I^X$, calculated for Δ = 0, is denoted $I_{+-}^X\equiv {I^X|}_{\Delta =0}$. The product $I^Y{I}_{+-}^X$, which is plotted in Fig. 4 E, has a different angular behavior than that of $I^Y{I}_{++}^X$ because two lobes appear with zero signal at θ = 0°. Indeed, harmonic waves emitted from each side of the sarcomere add destructively along the z axis. By analogy with the double-slit diffraction of Young's formula, the condition of constructive interference is such that the angle of emission of the harmonic waves generated by the two illuminated zones of each side of the y,z plane, and separated by a distance w_xy, should correspond to an optical path difference of δ = w_xy n_2ω sin θ – λ_2ω /2. This condition leads to θ = 21.6°, which is close to the simulated value θ = 18° (see Fig. 4 E). Such interference effect is similar to the one which is observed when two GUV membranes are brought together (44). Indeed, when the distance of the two membranes is such that the direction of constructive interference between harmonic waves is in the direction of PM, the SHG intensity is maximum. Here, the SHG intensity $I_{2\omega }^{+-}$ radiated by dipoles of opposite polarity +− and for a laser beam focused at the center of the sarcomere is given by Eq. 6, using $I^D,I_{+-}^X,I^Y,\text{and}{I}^Z$, and is plotted in Fig. 4 F. Note that the maximum value of $I_{2\omega }^{+-}$ is much greater than that of $I_{2\omega }^{++}$ due to the better overlap between I^Z with $I^Y{I}_{+-}^X$ than that of I^Z with $I^Y{I}_{++}^X$ (see Fig. 4, B, C, and E).

The total SHG intensity I^T_2ω collected by the condenser is obtained by angular integration of Eq. 6 over the condenser aperture

$$I_{2\omega }^T=\int I_{2\omega }\left(\mathit{r}\right)d\Omega ,$$ (7)

with dΩ = sin θ d θ d φ as the differential solid angle in spherical coordinates r, θ, and φ. $I_{2\omega }^T$ is computed numerically using MATLAB trapz function and is plotted in Fig. 5 A as a function of the laser beam position Δ for different values of M′. We note that an increase on intensity at the center of the sarcomere is observed for values of M′ < 200 nm. This result invalidates previous suggestions that a DB sarcomeric SHG intensity pattern could result from intrathick filaments centrosymmetry due to tail-tail interactions of myosin molecules (10,13,14,18). Moreover, such increase on intensity is also observed in single isolated myofibrils (21) refuting also that optical axial resolution could explain the change of the sarcomeric SHG intensity profile (14). Because the width M′ of the intrathick filaments centrosymmetry (resulting from the tail-tail interactions of myosin molecules) has been estimated in the literature to be ∼130–160 nm (45–47), a sarcomeric SHG intensity pattern is expected to be SB in physiological condition according to our model as it is observed in Fig. 2 A.

Figure 5.

Sarcomeric SHG intensity pattern (A) and effect of the condenser NA on the sarcomeric SHG intensity pattern (B and C) of healthy muscle. (A) Theoretical SHG intensity I^T_2ω emitted along one sarcomere of width 2 μm as a function of the indicated size of antiparallel myosin tails overlapping M′. (B and C) The same SHG images of a thin cryostat-sliced section of healthy muscle for two values of the condenser aperture diaphragm. (B) Condenser aperture diaphragm fully open (NA = 0.9). (C) Condenser aperture diaphragm partially open (NA = 0.3). The image in panel C has been acquired with a larger photomultiplier tube gain compared to panel B to obtain approximately the same gray intensity level between the two images. (Inset) Experimental values (dots) of the SHG intensity profiles (arbitrary units) obtained (dotted line) with the theoretical intensity I^T_2ω (solid line) obtained for M′ = 90 nm, L = 2 μm, A = 1.6 μm, n_ω = n_2ω= 1.33, w_xy = 0.48 μm, w_z = 2.4 μm, and λ_ω = 940 nm. Bars are 2 μm.

Such an increase on intensity at the center of the sarcomere, even in the presence of a region with no SHG signal emitted, is caused by the combination of Gouy effect I^Z and of light diffraction I^Y I^X, which is different along the sarcomere as illustrated in Fig. 4 G. In direction θ_PM, interference between harmonic waves generated from dipoles separated by ≈w_xy and located in the same hemi-filament (blue color) is destructive because their optical path difference is δ ≈ λ_2ω/2. In the same direction, interference between harmonic waves generated from antipolar dipoles located in adjacent hemi-filaments (blue and red color) is constructive due to the extra π-phase shift associated with polarity inversion. This explains, even in presence of a central region of width M′ with no SHG emission, the observed increase on intensity at the center of the sarcomere and the SB sarcomeric SHG intensity pattern. We have experimentally observed dispersion of sarcomere sizes ranging from 1.8 to 2.6 μm in fresh and proteolyzed muscles. In this limited range, varying the value of L in the model does not change the SHG pattern. However, the SHG intensity at the Z-line decreases with the increase of L because the density of harmonophores N in the focus spot is reduced.

To highlight the high degree of angular anisotropy of the SHG radiation pattern, image of a thin cryostat-sliced section of healthy muscle (see Material and Methods) is shown in Fig. 5, B and C, for two positions of the condenser aperture diaphragm corresponding to, respectively, NA = 0.9 (Fig. 5 B) and NA = 0.3 (Fig. 5 C). The bright strip at the M-band in Fig. 5 B is replaced by a dark strip in Fig. 5 C. In this later case, the maximum size of the aperture (∼13°) is such that the two lobes emitted from the central part of the sarcomere are blocked by the aperture diaphragm (see Fig. 4 F). The SHG intensity detected from the central part of each hemi-filament is thus greater than the one detected from the center of the sarcomere (see Fig. 4, D and F).

We have also experimentally checked that the SHG signal was unchanged when closing the aperture diaphragm until NA ∼ 0.75 (∼ 34°), confirming that the second harmonic signal is emitted with an angular aperture which is lower (see Fig. 4, D and F) than that of the maximum collection angle of the condenser (∼ 42.6°); this is in agreement with a previous report (36). Therefore, under our experimental conditions with an excitation objective of NA = 1.1, the use of a condenser with NA = 0.9 is sufficient to detect the entire SHG signal. We have seen that the amplitude of the increase on intensity at the M-band is very sensitive to M′ (see Fig. 5 A). The best fit of the experimental sarcomeric SHG intensity profiles of healthy muscles along the dotted lines of Fig. 2 A and Fig. 5, B and C, using Eq. 7, is obtained for M′ = 110 nm and M′ = 90 nm, respectively, and is shown in the inset of each figure.

For M′ values >200 nm, the model predicts a DB sarcomeric SHG intensity pattern, which cannot be achieved by antiparallel tail packing at the M-band (45–47). To gain insight into the ultrastructural arrangement of thick filaments responsible for the appearance of the predominant feature of DB pattern in proteolyzed muscles (20), we undertook EM study of fresh and proteolyzed tissues (see Fig. 6). Fresh muscle presents tightly closed myofibrils, which are well registered at Z-lines (Fig. 6 A). At higher magnification, thick filaments are individualized and well organized (Fig. 6 B). In proteolyzed tissue (Fig. 6 C), myofibrils are disjointed and clefts appear between them. Z-lines are also deviated in an oblique manner. At higher magnification, thick filaments are laterally disorganized (Fig. 6 D). We can also note that Z-lines are no longer continuous but are changed into dark aggregates in the center of I-bands (arrowheads in Fig. 6, B and D).

The electron-dense M-line protein scaffold (brackets in Fig. 6 B) has disappeared in proteolyzed muscle (brackets in Fig. 6 D). Such disorganization can be easily explained, because at this early stage of muscle proteolysis, titin and M-line proteins, which respectively align and center myosin thick filaments, are the first to be broken (48–51). M″ value of sarcomeric axial thick-filaments' displacement can be determined from EM images using M″ (μm) = 1.6 × σ/A, where σ and A are, respectively, displacement and size of thick filaments measured on EM images assuming a constant size (1.6 μm) of thick filaments (38,39). The average value was found to be M″ = 200 nm ± 100 nm from 50 randomly chosen proteolyzed sarcomeres. A large number of thick filaments is involved in the nonlinear interaction, considering that the approximate number of thick filaments per myofibril is ∼1000 and that 1–3 myofibrils are within the excitation volume. As a consequence, an antiparallel interthick filaments centrosymmetry is generated, resulting in an apparent antipolar arrangement M′_a = M′+ M″ at the center of the sarcomere.

The measured thick filaments displacement M″ obtained from EM images was therefore translated in our model as an extension of the thick filaments centrosymmetry (M′ → M′_a) at the center of the sarcomere because we have assumed for simplicity that thick filaments are constrained to be perfectly aligned. Therefore, intrathick filaments centrosymmetry of width M′ = 110 nm (determined from the fit of Fig. 2 A for fresh muscle) and interthick filaments centrosymmetry of width M″ = 200 nm (determined from EM measurements) lead, for proteolyzed muscle, to a global apparent intra, and inter, centrosymmetry of M′_a = 310 nm.

As expected, such a value yields to a DB sarcomeric SHG intensity pattern (see Fig. 5 A) in agreement with the observed SHG intensity pattern (see Fig. 2 B). The fit of the experimental SHG intensity profile along the dotted line of Fig. 2 B using this value is shown in the inset. Finally, mean values of M′ and M′_a, obtained from the fit of the sarcomeric SHG intensity profile of 20 randomly chosen sarcomeres, are M′ = 115 ± 25 nm and M′_a = 320 ± 25 nm for, respectively, fresh and proteolyzed tissues. As mentioned in the Introduction, nonuniform distribution of myosin heavy chain MHC A and B isoforms with different hyperpolarizability coefficients along thick filaments in Caenorhabditis elegans has been proposed to explain the DB sarcomeric SHG pattern (2). This cannot be taken into account here in the case of muscle proteolysis where the hyperpolarizability coefficient β remains constant because thick filaments remain intact (See Fig. 6) and myosin molecules are not degraded.

Discussion

The above results report theoretical calculation of the SHG intensity emitted by Xenopus skeletal muscles. Theoretical analysis highlights the high degree of angular anisotropy of the SHG emission due to the concomitant effect of 1), Gouy phase shift, 2), diffraction of the harmonic waves, and 3), antipolar distribution of myosin filaments within the sarcomere.We show that diffraction is highly dependent on the organization of the nonlinear dipoles within the focusing region. Our model simulates sarcomeric SHG intensity pattern and predicts that, unexpectedly, maximum SHG intensity is produced at the M-band even with the presence of antiparallel overlapping of myosin tails. Comparison with experimental results allows us to estimate the width M′ of this overlapping to be ∼115 ± 25 nm. Despite the high variability of thick filament length from 1.6 μm in vertebrates up to 10 μm in C. elegans, there is a consensus concerning the width of the M-band of both vertebrates and invertebrates deduced from EM, low angle x-ray diffraction studies, and analysis of myosin genes' amino-acid sequences.

From EM studies, the mean M-band width was found to be 149 nm in frog whole thigh muscles (52), 154 nm in myosin filaments from goldfish skeletal muscle under relaxing conditions (46), and 154 nm in adult chicken pectoralis major muscle (53). From cryo-EM studies, the mean M-band width was found to be 148 nm in fish plaice fin muscle in the relaxed state and 177 nm in the rigor state (54). From ionic charge interaction analysis of the myosin II rod sequence, the mean M-band width was found to be 164 nm in human tibialis anterior muscle (47). From genetic studies and myosin-rod sequence analysis of the soil nematode C. elegans, an M-band width of 160 nm was suggested (45). Considering the length of the myosin tail sequence and the ionic amino-acid interactions, the antiparallel packing width M′ (inside the M-band) has been estimated from above measurements to be of ∼130 nm in nematode (45), 154 nm in fish (46), and 158 nm in human (47).

Our model gives M′ = 115 ± 25 nm, which is close, but lower than these values. M′ is obtained from the fit of the experimental sarcomeric SHG intensity pattern by the theoretical model using sarcomere (L, A) and Gaussian beam (w_xy, w_z) parameters. Sarcomere parameters have been estimated to be ∼ L = 2 μm, A = 1.6 μm (23,24) and beam waists w_xy = 0.48 μm, w_z = 2.4 μm are deduced from two-photon excitation point-spread function measurements. These later values can be compared with the theoretical ones, w_xy = 0.40 μm and w_z = 1.2 μm, calculated from Zipfel et al. (43) for our experimental conditions.

The slight discrepancy between theoretical and experimental values, especially in direction z, indicates the presence of spherical aberrations which were not taken into account in our model and which could influence the result of the theoretical simulation. Indeed, θ_PM is driven not only by w_xy but also by the angular width of the laser excitation cone. In our model, we only took into account w_xy but not objective NA. Presence of spherical aberrations leads us to underestimate the width of the excitation cone and of NA because the focusing spot is not diffraction-limited. In this respect, we may consider that θ_PM as well as M′ are also underestimated. Whereas $I_{2\omega }^{+-}$ is not very sensitive to small change of θ_PM due to the good overlap between I^Z and $I^Y{I}_{+-}^X$ (see Fig. 4, B and E), $I_{2\omega }^{++}$ would change significantly for slight variation of θ_PM due to the poor overlap between I^Z and $I^Y{I}_{++}^X$ (see Fig. 4, B and C).

As a result, positive variation of θ_PM would increase the ratio $I_{2\omega }^{+-}/I_{2\omega }^{++}$, the increase on intensity at the center of the sarcomere and, in consequence, the value of the fitting parameter M′. We checked that a small positive variation of θ_PM by 1 ∼ 2° is sufficient to obtain increase of M′ by a few tens of nanometers. Our model needs further refinements to better estimate M′, taking into account spherical aberrations, but it is sufficient to give the main feature of the sarcomeric SHG intensity pattern and to explain the increase on intensity observed at the center of the sarcomere for healthy muscles.

Concerning proteolyzed muscles, experimental sarcomeric SHG intensity patterns are well fitted by our model assuming apparent antipolar arrangement M′_a = 320 ± 25 nm due to intrathick filaments' centrosymmetry M′ and interthic filaments centrosymmetry M″ estimated from thick filaments displacements obtained from EM images. Nevertheless, the model can be used to estimate the amount of thick filaments disorder that could occur during muscular dysfunction and pathology. Interestingly it has been shown recently that Duchenne muscular dystrophy of mdx mouse is characterized by an increase in Y-shape SHG intensity pattern called “verniers” (55). Most of these patterns are reminiscent of DB patterns, which could therefore be used as markers of myopathies. Moreover, thick filaments misalignment has also been observed in different extraphysiological and biochemical experimental conditions, e.g., rat adductor longus muscle lesion after eccentric challenge (56), mice with calpain 3 null mutation (57), zebrafish runzel muscular dystrophy linked to titin gene (58), and rabbit soleus and psoas muscles during prolonged passive-force generation from relaxed fibers (59).

Conclusion

We have successfully explained the experimental sarcomeric SHG intensity patterns of fresh and proteolyzed muscles which are due to modification of the supramolecular centrosymmetry at the M-band with a theoretical model based on the summation of the second harmonic fields, taking into account both Gouy effect and light diffraction. Our results highlight the sensitivity of SHG microscopy to reveal supramolecular antipolar organization at the nanometric scale and the potential usefulness of this technique to diagnose proteolysis-generating muscular disorders.