Optogenetically mediated large volume suppression and synchronized excitation of human ventricular cardiomyocytes

Abstract

A major challenge in cardiac optogenetics is to have minimally invasive large volume excitation and suppression for effective cardioversion and treatment of tachycardia. It is important to study the effect of light attenuation on the electrical activity of cells in in vivo cardiac optogenetic experiments. In this computational study, we present a detailed analysis of the effect of light attenuation in different channelrhodopsins (ChRs)-expressing human ventricular cardiomyocytes. The study shows that sustained illumination from the myocardium surface used for suppression, simultaneously results in spurious excitation in deeper tissue regions. Tissue depths of suppressed and excited regions have been determined for different opsin expression levels. It is shown that increasing the expression level by 5-fold enhances the depth of suppressed tissue from 2.24 to 3.73 mm with ChR2(H134R) (ChR2 with a single point mutation at position H134), 3.78 to 5.12 mm with GtACR1 (anion-conducting ChR from cryptophyte algae Guillardia theta) and 6.63 to 9.31 mm with ChRmine (a marine opsin gene from Tiarina fusus). Light attenuation also results in desynchrony in action potentials in different tissue regions under pulsed illumination. It is further shown that gradient-opsin expression not only enables suppression up to the same level of tissue depth but also enables synchronized excitation under pulsed illumination. The study is important for the effective treatment of tachycardia and cardiac pacing and for extending the scale of cardiac optogenetics.

Appendix

Appendix A Four-state model of opsin photocurrent

On illumination, the opsin molecule switches from the closed ground state-C₁ to open state-O₁. From O₁, the molecule either decays back to C₁ or transits to the second open state-O₂, which is less conductive in comparison to O₁ but has a longer lifetime. The reversible transition between O₁ and O₂ can be both light and thermal induced. The molecule in O₂ state can also transit to second closed-state C₂. From C₂, the molecule thermally relaxes to the ground state C₁ or can be photo-excited back to O₂. The transition from C₂ to C₁, also called the recovery process of ChRs, is the slowest process in the photocycle [38,39,40,41, 50]. Recent experimental results have shown that ChRmine exhibits the fastest recovery kinetics [21].

If C₁, O₁, O₂, and C₂ denote the instantaneous fraction of opsin molecules in each of the four states and follow the constraint C₁ + O₁ + O₂ + C₂ = 1, the time-dependent transitions among these states can be defined by the following set of rate equations,

$${\dot{C}}_1={G}_{d1}{O}_1-{G}_{a1}\left(\phi \right){C}_1+{G}_r{C}_2$$

(5)

$${\dot{O}}_1={G}_{a1}\left(\phi \right){C}_1-\left({G}_{d1}+{G}_f\left(\phi \right)\right){O}_1+{G}_b\left(\phi \right){O}_2$$

(6)

$${\dot{O}}_2={G}_{a2}\left(\phi \right){C}_2-\left({G}_{d2}+{G}_b\left(\phi \right)\right){O}_2+{G}_f\left(\phi \right){O}_1$$

(7)

$${\dot{C}}_2={G}_{d2}{O}_2-\left({G}_r+{G}_{a2}\left(\phi \right)\right){C}_2$$

(8)

The transitions C₁ → O₁, C₂ → O₂, O₁ → C₁, O₂ → C₂, O₁ → O₂, O₂ → O₁, and C₂ → C₁,respectively, are governed by the rate constants G_a1, G_a2, G_d1, G_d2, G_f, G_b, and G_r, respectively. Among these rate constants, the light-dependent rate constants vary as ${G}_{a1}\left(\phi \right)=\upvarepsilon {k}_1{\phi}^p/\left({\phi}^p+{\phi}_m^p\right)$, G_a2(ϕ) $=\upvarepsilon {k}_2{\phi}^p/\left({\phi}^p+{\upphi}_m^p\right)$, G_f(ϕ) $={G}_{f0}+\upvarepsilon {k}_f{\phi}^q/\left({\phi}^q+{\phi}_m^q\right)$ and G_b(ϕ) $={G}_{b0}+{\upvarepsilon k}_b{\phi}^q/\left({\phi}^q+{\phi}_m^q\right)$, where ɛ is a wavelength-dependent parameter. Since there are two open states, f_ϕ(ϕ, t) = O₁ + γO₂ where γ = g_O2/g_O1. g_O1 and g_O2 are the conductances of states O₁ and O₂, respectively.

Appendix B Ionic currents through natural ion channels and pumps in human ventricular cardiomyocytes

The ionic currents namely, I_Na, I_to, I_Kr, I_CaL, and I_Ks can be expressed as ${I}_f=\kern0.5em {g}_f{m}_1^p{m}_2^q{m}_3^r{m}_4^s\left(V-{E}_f\right)$, where g_f and E_f denote the maximal conductance and reversal potential of each ion channel, and m₁, m₂, m₃, and m₄ are different gating variables (with exponent p, q, r, and s, respectively) (Tables 2 and 3). Other ionic currents are expressed as follows:

$${I}_{K1}={G}_{K1}\sqrt{\frac{K_0}{5.4}}\ {x}_{K1\infty }\ \left(V-{E}_K\right)$$

(9)

$${\alpha}_{K1}=\frac{0.1}{1+{e}^{0.06\left(V-{E}_K-200\right)}}$$

(10)

$${\beta}_{K1}=\frac{3{e}^{0.0002\left(V-{E}_K+100\right)}}{1+{e}^{-0.5\left(V-{E}_K\right)}}+\frac{e^{0.1\left(V-{E}_K-10\right)}}{1+{e}^{-0.5\left(V-{E}_K\right)}}$$

(11)

$${x}_{K1\infty }=\frac{\alpha_{K1}}{\alpha_{K1}+{\beta}_{K1}}$$

(12)

$${I}_{Na Ca}={k}_{Na Ca}\frac{e^{\gamma VF/ RT}{Na}_i^3{Ca}_0-{e}^{\left(\gamma -1\right) VF/ RT}{Na}_0^3{Ca}_i\alpha }{\left({K}_{mNai}^3+{Na}_0^3\right)\left({K}_{mCa}+{Ca}_0\right)\left(1+{k}_{sat}{e}^{\left(\gamma -1\right) VF/ RT}\right)}$$

(13)

$${I}_{Na K}={P}_{Na K}\frac{K_0{Na}_i}{\left({K}_0+{K}_{mK}\right)\left({Na}_i+{K}_{mNa}\right)\left(1+0.01245{e}^{-0.1 VF/ RT}+0.0353{e}^{- VF/ RT}\right)}$$

(14)

$${I}_{pCa}={G}_{pCa}\frac{Ca_i}{K_{pCa}+{Ca}_i}$$

(15)

$${I}_{bCa}={G}_{bCa}\left(V-{E}_{Ca}\right)$$

(16)

$${I}_{bNa}={G}_{bNa}\left(V-{E}_{Na}\right)$$

(17)

$${I}_{pK}={G}_{pK}\frac{V-{E}_K}{1+{e}^{\left(25-V\right)/5.98}}$$

(18)

$${I}_{leak}={V}_{leak}\left({Ca}_{sr}-{Ca}_i\right)$$

(19)

$${I}_{up}=\frac{V_{maxup}}{1+{K}_{up}^2/{Ca}_i^2}$$

(20)

$${I}_{rel}={V}_{rel}O\left({Ca}_{SR}-{Ca}_{SS}\right)$$

(21)

$$O=\frac{k_1{Ca}_{SS}^2\overline{R}}{k_3+{k}_1{Ca}_{ss}^2}$$

(22)

$$\frac{d\overline{R}}{dt}=-{k}_2{Ca}_{SS}\overline{R}+{k}_4\left(1-\overline{R}\right)$$

(23)

$${k}_1=\frac{k_{1^{\prime }}}{k_{casr}},{k}_2={k}_{2^{\prime }}{k}_{casr}\ \textrm{and}\ {k}_{casr}={\mathit{\max}}_{sr}-\frac{{\mathit{\max}}_{sr}-{\mathit{\min}}_{sr}}{1+{\left( EC/{Ca}_{SR}\right)}^2}$$

(24)

$${I}_{xfer}={V}_{xfer}\left({Ca}_{SS}-{Ca}_i\right)$$

(25)

where the Nernst equation has been used to model the variable reversal potential for each ion channel that includes E_Na, E_K, E_Ca, and E_Ks as,

$${E}_X=\frac{RT}{zF}\ \mathit{\ln}\left[\frac{{\left[X\right]}_0}{{\left[X\right]}_i}\right],\kern0.5em X={Na}^{+},{Ca}^{2+},{K}^{+}$$

(26)

$${E}_{Ks}=\frac{RT}{F}\ \mathit{\ln}\frac{K_0+{p}_{KNa}{Na}_0}{K_i+{p}_{KNa}{Na}_i}$$

(27)

where [Na⁺]_i, [Ca²⁺]_i, and [K⁺]_i are intracellular ion-concentrations, which have been initially considered [Na⁺]_i = 7.67 mM, [Ca²⁺]_i = 0.00007 mM and [K⁺]_i = 138.3 mM and z = 1 for Na⁺and K⁺, z = 2 for Ca²⁺ [51, 52].

The rate of change of in [Na⁺]_i, [Ca²⁺]_i and [K⁺]_i can be expressed as,

$$\frac{d{Na}_i}{dt}=-\frac{I_{Na}+{I}_{bNa}+3{I}_{Na K}+3{I}_{Na Ca}}{V_cF}$$

(28)

$$\frac{d{K}_i}{dt}=-\frac{I_{K1}+{I}_{to}+{I}_{Kr}+{I}_{Ks}-2{I}_{NaK}+{I}_{pK}+{I}_{stim}}{V_cF}$$

(29)

$$\frac{d{Ca}_{itotal}}{dt}=-\frac{I_{bCa}+{I}_{pCa}-2{I}_{NaCa}}{2{V}_cF}+\frac{V_{sr}}{V_c}\left({I}_{leak}-{I}_{up}\right)+{I}_{xfer}$$

(30)

$$\frac{d{Ca}_{SRtotal}}{dt}=\left({I}_{up}-{I}_{leak}-{I}_{rel}\right)$$

(31)

$$\frac{d{Ca}_{SS total}}{dt}=-\frac{I_{Ca L}}{2{V}_{SS}F}+\frac{V_{sr}}{V_{ss}}{I}_{rel}-\frac{V_c}{V_{ss}}{I}_{xfer}$$

(32)

$${Ca}_{ibufc}=\frac{Ca_i\times {Buf}_c}{Ca_i+{K}_{bufc}}$$

(33)

$${Ca}_{sr bufsr}=\frac{Ca_{sr}\times {Buf}_{sr}}{Ca_{sr}+{K}_{bufsr}}$$

(34)

$${Ca}_{ss bufss}=\frac{Ca_{ss}\times {Buf}_{ss}}{Ca_{ss}+{K}_{bufss}}$$

(35)

$${b}_{Ca}={Buf}_c-{Ca}_{ibufc}-{Ca}_{itotal}-{\left[{Ca}^{2+}\right]}_i+{K}_{bufc}$$

(36)

$${c}_{Ca}={K}_{bufc}\left({Ca}_{ibufc}+{Ca}_{itotal}+{\left[{Ca}^{2+}\right]}_i\right)$$

(37)

$$new\ {\left[{Ca}^{2+}\right]}_i=\left(\sqrt{{b_{Ca}}^2+4{c}_{Ca}}-{b}_{Ca}\right)/2$$

(38)

$$\frac{d{\left[{Ca}^{2+}\right]}_i}{dt}= new{\left[{Ca}^{2+}\right]}_i-{\left[{Ca}^{2+}\right]}_i$$

(39)

where Ca_itotal is total cytoplasmic Ca²⁺ concentration, Ca_SRtotal is total sarcoplasmic reticulum Ca²⁺ concentration, Ca_SStotal is total diadic subspace Ca²⁺ concentration, Ca_i is free cytoplasmic Ca²⁺ concentration, Ca_SR is free sarcoplasmic reticulum Ca²⁺ concentration, Ca_ss is free diadic subspace Ca²⁺ concentration, I_rel is calcium-induced calcium release current, I_up is sarcoplasmic reticulum Ca²⁺ pump current, I_leak is sarcoplasmic reticulum Ca²⁺ leak current, I_xfer is diffusive Ca²⁺ current between diadic Ca²⁺ subspace and bulk cytoplasm, and O and $\overline{R}$ are proportion of open and closed I_rel channels, respectively [51, 52]. Each gating function x (m₁, m₂, m₃, and m₄) obeys the first-order kinetics as $\dot{x}=\left({x}_{\infty }-x\right)/{\tau}_x$. The voltage-dependent functions (x_∞and τ_x) and values of parameters are given in Tables 2 and 3, respectively [51, 52].

Appendix C Geometrical spreading, Gaussian distribution, absorption, and scattering of light under fiber optic illumination

Light emitted from the optical fiber spreads as a cone of light with a divergence half angle (θ_div) which depends on the NA and the refractive index of the tissue (n_tissue) as follows,

$${\theta}_{div}={\mathit{\sin}}^{-1}\left( NA/{n}_{tissue}\right)$$

(40)

The radius of the light cone (R) at tissue depth z from the optical fiber with radius R₀ spreads as,

$$R(z)={R}_0+z\ \mathit{\tan}\left({\theta}_{div}\right)$$

(41)

For myocardium, n_tisssue = 1.4. Here, R₀ is considered to be 0.2 mm, and NA is considered within the range 0.1–0.4, as used in most of the optogenetics experiments [53, 54].

If I is the irradiance at a distance z from the optical fiber, the radiant power (P) at each point in space, when independently considering geometrical effects, is defined as,

$$P=I(z)\pi\ R{(z)}^2={I}_0\pi {R}_0^2$$

(42)

Therefore, the transmittance due to geometrical spreading is defined as follows,

$$C(z)={\left[\frac{R_0}{R(z)}\right]}^2$$

(43)

The Gaussian distribution of light G(r, z) emitted by an optical fiber can be approximated as,

$$G\left(r,z\right)=\frac{1}{\sqrt{2\pi }}\mathit{\exp}\left\{-2{\left[\frac{R_0}{R(z)}\right]}^2\right\}$$

(44)

The Kubelka-Munk general theory of light propagation in diffuse scattering media has been used to capture the effect of scattering and absorption [55, 56]. The transmittance of light through the myocardium tissue is given by,

$$M\left(r,z\right)=b/\left[a\;sinh\;\left(bS\sqrt{r^2+z^2}\right)+b\;cosh\;\left(bS\sqrt{r^2+z^2}\right)\right]$$

(45)

where a = 1 + K/S, $b=\sqrt{a^2}-1$. This study assumes that the myocardium tissue is optically homogeneous and illuminated normally from its surface with monochromatic light. The absorption and scattering coefficients are wavelength dependent.

Table 2 Gating function parameters of ion channels in ten Tusscher cardiac cell circuit model [10, 35, 51, 52]

Table 3 Model parameters of human ventricular cardiomyocytes [10, 35, 51, 52]

Table 4 Effect of surface irradiance on the activity of different opsin-expressing human ventricular cardiomyocytes (HVCMs) under uniform illumination with sustained light

Table 5 Effect of surface irradiance on the activity of different opsin-expressing human ventricular cardiomyocytes (HVCMs) under fiber optic illumination (NA = 0.1) with sustained light

Table 6 Effect of numerical aperture (NA) on the activity of different opsin-expressing human ventricular cardiomyocytes (HVCMs) in axial and transverse directions under fiber optic illumination with sustained light at 10 mW/mm²

Table 7 Effect of opsin expression density on the activity of different opsin-expressing human ventricular cardiomyocytes (HVCMs) under uniform illumination with sustained light

Table 8 Effect of opsin expression density on the activity of different opsin-expressing human ventricular cardiomyocytes (HVCMs) under fiber optic illumination (NA = 0.1) with sustained light at 10 mW/mm²

Fig. 8

Variation of membrane potential with time in ChR2(H134R)-expressing human ventricular cardiomyocytes (HVCMs) on illuminating with 6-s light pulse at indicated irradiances at 470 nm

Fig. 9

Variation of membrane potential with time in GtACR1-expressing human ventricular cardiomyocytes (HVCMs) on illuminating with 6-s light pulse at indicated irradiances at 515 nm

Fig. 10

Variation of membrane potential with time in ChRmine-expressing human ventricular cardiomyocytes (HVCMs) on illuminating with 6-s light pulse at indicated irradiances at 585 nm

Fig. 11

Variation of membrane potential with time in ChRmine-expressing human ventricular cardiomyocytes (HVCMs) on illuminating with 6-s light pulse at indicated irradiances at 650 nm