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Selective activation of photoactivatable fluorescent protein based on binary holography

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Abstract

The ability to deliver laser doses to different target locations with high spatial and temporal resolution has been a long-sought goal in photo-stimulation and optogenetics research via, for example, photoactivatable proteins. These light-sensitive proteins undergo conformational changes upon photoactivation, serving functions such as triggering fluorescence, modulating ion channel activities, or initiating biochemical reactions within cells. Conventionally, photo-stimulation on light-sensitive proteins is performed by serially scanning a laser focus or via 2D projection, which is limited by relatively low spatiotemporal resolution. In this work, we present a programmable two-photon stimulation method based on a digital micromirror device (DMD) and binary holography to perform the activation of photoactivatable green fluorescent protein (PAGFP) in live cells. This method achieved grayscale and 3D selective PAGFP activation with subcellular resolution. In the experiments, we demonstrated the 3D activation capability and investigated the diffusion dynamics of activated PAGFP on the cell membrane. A regional difference in cell membrane diffusivity was observed, indicating the great potential of our approach in interrogating the spatiotemporal dynamics of cellular processes inside living cells.

© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

The growing toolbox of photoactivatable proteins, such as photoactivatable fluorescent proteins and light-sensitive proteins for optogenetic applications, has provided unprecedented capability to spatially and temporally label or regulate specific structures and biological activities against a non-photo-stimulated “dark” background, addressing many challenges that cannot be resolved by conventional methods [1,2]. Optogenetics is an emerging biotechnique that uses light to control cells genetically modified to express light-sensitive proteins. It has revolutionized neuroscience research through the highly selective control of neurons and greatly advanced our understanding of many other biological processes, including gene expression [3], organelle transport [4,5], and signal transduction [6,7], to name a few. On the other hand, photoactivatable fluorescent proteins are featured by a marked increase in the fluorescent signal upon photo-stimulation of a specific wavelength and intensity. This property has not only led to novel imaging techniques but also facilitated new biological discoveries through the precise optical labeling and tracking of proteins, organelles, and cells within living systems [8]. The effective and precise activation of photoactivatable proteins, which is central to the success of optogenetic or photoactivatable fluorescent protein applications, hinges on the accurate control over the timing, location, and intensity of photo-stimulation.

Conventionally, in most photoactivation systems, the spatial, temporal, and dosage resolution of photo-stimulation is very limited due to the lack of efficient and versatile light modulation methods. Specifically, in terms of spatial resolution, most conventional activation methods operate via continuous wave lasers. These methods, however, tend to lack precision in the axial direction even with the help of a confocal pinhole [9]. In contrast, two-photon activation ranging from 720 to 840 nm provides better penetration depths and higher spatial resolution in all three dimensions [10]. Yet, the limitation of the low temporal resolution is still a problem in these systems as most two-photon activation methods rely heavily on galvanometer (galvo) mirrors, which has a limited frame rate of ∼30 Hz [11,12]. The inertia scanning mechanism also prohibits advanced features such as 3D programmable activation and precise grayscale control, which hinders high-precision applications of light-sensitive proteins and new discoveries.

To address these challenges, we present a digital micromirror device (DMD)-based 3D random-access scanner and demonstrate high spatiotemporal resolution of photo-stimulation using photoactivatable green fluorescent protein (PAGFP). PAGFP is one of the most widely used photoactivatable fluorescent proteins. It undergoes photoconversion upon 400-nm laser exposure, leading to an approximately 100-fold fluorescence enhancement at 488-nm excitation. Unlike conventional DMD projecting systems [13], the DMD in our setup works as a programmable binary mask, which rapidly generates user-defined laser foci based on binary holography at 22.7 kHz [14]. This random-access method can induce photo-stimulation with laser foci individually controlled in terms of 3D position, intensity, and exposure duration. Based on the system, 3D grayscale PAGFP activation experiments were designed and performed on the cell membrane of COS-7 cells. To demonstrate the capability of our method, two kinds of diffusion behaviors were triggered and observed under custom-designed activation patterns. An image-based approximation algorithm was developed to interpret the diffusion dynamics and identify the corresponding diffusion coefficients. The results show that the diffusion coefficients vary significantly in different regions of the cell membrane. These results demonstrate the spatial and temporal resolution of our method in delivering photo-stimulation and activating photoactivatable proteins, indicating its great potential for decoding the spatiotemporal dynamics of various biological processes.

2. Methods

2.1 Optical setup

Figure 1 presents the two-photon PAGFP activation and imaging system. A Ti:sapphire femtosecond laser (Chameleon Vision S, Coherent) provided an activation wavelength at 800 nm and an imaging wavelength at 900 nm. (800 nm and 900 nm were selected as they offer less switching time between the imaging and the activation modules, while keeping a high activation and imaging efficiency on PAGFP [10].) The group velocity dispersion (GVD) was compensated by the built-in dispersion control module of the laser system. The input laser power was electrically controlled by a polarizing beam splitter (PBS, PBS122, Thorlabs) and a broadband half-wave plate (HWP, AHWP25-SNIR-A-M, LBTEK) via a micro servo (DS215MG, KST). For the 3D random-access activation of PAGFP, a DMD scanner (activation module) was installed to work with the 800-nm femtosecond laser. A high-efficiency transmission grating (T-1400-800-2415-94, LightSmyth) was placed before the DMD (V-7001, ViALUX) to pre-compensate the angular dispersion induced by the DMD. A 1:1 telescope (L3 and L4, f = 200 mm) with a spatial filter was installed to spatially select the +1th (or -1th) order diffraction beam. A galvo scanner (Sutter Instrument) and a photo-multiplier tube (PMT, R10699, Hamamatsu) were installed to provide video rate (∼30 Hz) two-photon images under 900-nm illumination (imaging module). A second PBS (PBS2, PBS252, Thorlabs) was installed to recombine the activation and imaging beams. Both beams were relayed to the objective lens (OL, Plan Fluor 40×/1.30, Nikon) by a 4-f system, which included a scan lens (SL, LSM54-850, Thorlabs) and a tube lens (L5, f = 200 mm).

 figure: Fig. 1.

Fig. 1. Optical configuration for the PAGFP activation and imaging system. HWP: broadband half-wave plate; PBS1-PBS2: polarizing beam splitters; L1-L5: lenses; M1-M4: high reflectivity mirrors; G1: grating; BE: beam expander; SL: scan lens; DM: dichroic mirror; OL: objective lens; pink beam: 800-nm femtosecond laser for PAGFP activation; red beam: 900-nm femtosecond laser for PAGFP imaging; green beam: fluorescent signals emitted by post-activation PAGFP; dash line inset: sample cell with a user-defined PAGFP activation pattern; solid line inset: illustration of DMD-based random-access scanning for PAGFP activation (left) versus galvo-based raster scanning for post-activation PAGFP imaging (right).

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Instead of using the reflection beam in projection systems, the DMD in the activation module uses the diffracted beam to realize single- or multi-focus 3D random-access scanning [15,16], and precise power control [17]. Before the experiments, parameters such as number and location of laser foci as well as the intensity in each focus were defined and substituted into Eq. (1) to generate binary holograms:

$$ h(i, j)=\left\{\begin{array}{l} 1,-\frac{q}{2} \leq \frac{\varphi(x, y, z)}{2 \pi} \leq \frac{q}{2} \\ 0, \text { otherwise } \end{array}\right. $$
where $h({i,\; j} )$ represents a binary hologram; $i,\; j$ are the pixel column and row index, respectively; $0,\; 1$ refer to the on/off states of a DMD pixel; $\varphi ({x,y,z} )$ is the phase distribution on the DMD chip, which can be calculated by backpropagating the laser foci through the optical system; q is a binarization constant that controls the power of the corresponding foci. When the designed binary holograms were sequentially displayed on DMD, the corresponding foci would scan along the designated trajectory in a discrete manner. In our setup, the DMD could display holograms at a rate of 22.7 kHz. Accordingly, the temporal resolution of PAGFP activation was greatly enhanced at the diffraction limited spatial resolution. (The axial resolution was enhanced by the two-photon effect.)

To realize post-activation PAGFP imaging using the same laser source, an automatic beam switching and isolating mechanism was developed to switch between the imaging and activation module while preventing potential interference. Specifically, when the activation module was working, the laser output an 800-nm beam to the DMD with a GVD compensation of 21000 fs2. During this time, the galvo scanners were set to a rest position so that no light would pass through the imaging module. When the imaging module was in operation, the laser output a 900-nm beam with a GVD compensation of 11600 fs2. Notably, during this time, the transmission grating in the activation module would direct the 900-nm incident beam off the DMD (by 7.7° in our setup) according to the grating equation (Eq. (2)) to prevent it from entering the objective lens. (Note that the GVD compensation values are different for the imaging and activation modules owing to the different optical path lengths. After compensation, both modules output a pulse width of ∼80 fs.)

$$d({sin{\theta_i} - sin{\theta_m}} )= n\lambda $$

In Eq. (2), d is the groove spacing length of the grating; ${\theta _i}$ and ${\theta _m}$ are the incident and diffraction angle of the laser beam, respectively; n is the target diffraction order; and $\lambda $ denotes the laser wavelength.

2.2 Plasmid construction

The plasmid was cloned in the mammalian expression vector pEGFPN1. Plasmid PAGFP-CAAX was made by inserting the CAAX sequence at the C terminus of PAGFP by the ligation method. The ability of CAAX to target the cell membrane has been established in previous reports [21].

2.3 Cell culture and cell transfection

COS-7 cells, from African green monkey kidney, were cultured in Dulbecco’s modified eagle medium (DMEM) supplemented with 10% fetal bovine serum (FBS) and plated on 35-mm poly-l-lysine (PLL) coated confocal dishes with a confocal region of either 1.33 cm2 or 3.14 cm2 (SPL Life Sciences). After 1-2 days of growth, the cells were transfected with the desired plasmid using Lipofectamine 3000 (Thermo Fisher Scientific) following the manufacturer's protocol. Transfected cells were allowed to recover and express the desired protein overnight in a complete medium. Fluorescence imaging and photo-activation test of the transfected cells were performed one day after transfection.

3. System characterization

In this section, we demonstrate and characterize the 3D selective activation capability, spatial resolution, dosage control, and repeatability of our system. Figure 2(a) and 2(b) present the 2D and 3D selective activation results on the COS-7 cell membrane. In Fig. 2(a), a 2D “CUHK” pattern was activated on the cell membrane, showing the random-access activation capability of our system. To distinguish our method from other DMD projection methods, we conducted a 3D activation experiment. Specifically, in Fig. 2(b), the laser focus was scanned along a designed sine trajectory on the y-z plane (color-labeled in the x-y plane). Accordingly, the intersection between the laser trajectory and the cell membrane generated three activated spots shown in Fig. 2(b) inset, which demonstrates the 3D selective activation capability of our system. (Note that Fig. 2(b) is a time-lapse two-photon image of multiple activation processes.)

 figure: Fig. 2.

Fig. 2. Selective PAGFP activation on COS-7 cells expressing plasma membrane-localized PAGFP-CAAX. (a) Activation of a 2D “CUHK” pattern on the membranes of four COS-7 cells. (b) Selective 3D activation of three spots (inset) on a COS-7 cell membrane by a designed sinusoidal scanning trajectory. Color bar: depth of laser scanning trajectory. Scale bar: 30 µm.

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Figure 3 presents lateral resolution characterization results. Specifically, we activated a series of hollow squares with decreasing sizes until the hollow part could not be resolved under the two-photon imaging module. The intensity profile along the solid gray line of the smallest square is presented on the right, which suggests a minimum distinguishable activation size of 4.21 ± 0.32 µm. Note that the minimal activation size is much larger than the laser spot size [15] owing to the diffusion effect of PAGFP on the cell membrane [1]. On the other hand, it took five seconds for the system to switch from the activation module to the imaging module, which resulted in a more pronounced diffusion effect. As the diffusion effect is isotropic in the medium, the minimum activation size in z-direction, is presumably the same as the lateral resolution when activating in a 3D medium with the same diffusivity.

 figure: Fig. 3.

Fig. 3. Characterization of minimal distinguishable distance for membrane-localized PAGFP-CAAX activation on COS-7 cell membrane. Scale bar: 30 µm.

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Precise laser dose control can be achieved by adjusting q values in the binary holograms, as expressed in Eq. (1), which simultaneously controls the position of the laser focus [15]. In the experiment, we projected four holograms with increasing q values from 0.4 to 1.0, while keeping the input laser power and total exposure time (five seconds) constant. Figure 4 presents the experimental results, where Fig. 4(b) and 4(c), respectively, show measured fluorescent intensity and calculated imaging contrast (F/F0). Figure 4(a) is a control image before laser activation.

 figure: Fig. 4.

Fig. 4. Precise laser dose control for membrane-localized PAGFP-CAAX activation via binary holograms. COS-7 cells expressing plasma membrane-localized PAGFP were used. (a) Fluorescent image before PAGFP activation. (b) Post-activation image of the same COS-7 cell with four different q values. Upper left: q = 1.0, upper right: q = 0.8, lower left: q = 0.6, lower right: q = 0.4. (c) Heat map of F/F0 calculated via (b) and (a). Color bar: F/F0 value for the heatmap in (c). Scale bar: 30 µm.

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To demonstrate the temporal control capability of our system, we designed a repeating PAGFP photo-activation experiment. Figure 5(a)–5(e) present the results, where PAGFP activation was repeated on the same cell for five consecutive cycles. Specifically, the activation module generated four laser foci in the boxed region in Fig. 5(a) with equal power and illuminated the specimen for five seconds in each activation cycle. The system then switched to the imaging module immediately after the activation. A 6-minute rest was arranged between each activation process, allowing previously activated PAGFP to diffuse across the entire cell membrane to gain a low background for the next activation cycle. As shown in Fig. 5(f), the gradual decrease in the contrast of mean fluorescence intensity before and after each round of activation indicates the accumulation of activated PAGFP on the cell membrane.

 figure: Fig. 5.

Fig. 5. Repeated activation of membrane-localized PAGFP-CAAX on the same cell. COS-7 cells expressing plasma membrane-localized PAGFP were used. (a)-(e) Fluorescent images after each activation. (f) Mean contrast (±SEM, n = 4 for each experiment) of the activation sites before and after each activation. Scale bar: 30 µm.

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4. System application

To demonstrate the potential applications of our system, we measured the diffusion coefficient of COS-7 cell membrane by activating pre-defined PAGFP patterns within the selected regions of interest (ROIs). Comparing with previous methods for measuring diffusion coefficients, such as fluorescence recovery after photobleaching (FRAP) or inverse FRAP (iFRAP), our method presents two unique advantages. Firstly, it uses significantly lower exposure dosage that prevents possible cell damage and allows for long-term study. Secondly, it enables photo-activation of customized 2D and 3D patterns.

In the experiment, we activated two types of PAGFP patterns, each exhibiting a unique diffusion behavior within the selected ROI (Fig. 6). The first type of PAGFP pattern emulated FRAP pattern with a hollow rectangular shape. In the experiment shown in Fig. 6(a), the mean intensity within the selected ROI first increased, which resembled the diffusion behavior commonly seen in FRAP experiments. However, because only a limited region was activated in our setup, the mean intensity within ROI gradually decreased due to the diffusion of activated PAGFP throughout the cell. To show the versatility of our method, an iFRAP pattern with a solid square was also activated. As shown in Fig. 6(b), the activated PAGFP exhibits a diffusion behavior identical to previous studies [18,19], which further validates the compatibility of our method.

 figure: Fig. 6.

Fig. 6. Different diffusion behaviors under the hollow/solid membrane-localized PAGFP-CAAX patterns on COS-7 cells. (a) Diffusion process and mean ROI intensity change for a hollow rectangular PAGFP pattern. (b) Diffusion process and mean ROI intensity change for a solid rectangular PAGFP pattern. Red rectangle: selected ROI for mean intensity measurement. Scale bar: 30 µm.

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The selective photo-activation in our method has offered greater flexibility to related studies but lacked straightforward mathematical solutions to decode the diffusion dynamics into the corresponding diffusion coefficients. To address this issue, we used a finite difference approximation method to simulate the diffusion partial differential equation (PDE, shown in Eq. (3)) with fluorescent images as the initial condition (IC) and cell boundaries as the boundary condition (BC), as expressed in Eq. (3):

$$ \left\{\begin{array}{l} P D E: \frac{\partial C(x, y, t)}{\partial t}=D \nabla^2 C(x, y, t) \\ I C:\left.\quad C(x, y, t)\right|_{t=0}=C_0 \\ B C:\left.\quad \nabla C(x, y, t)\right|_{(x, y) \in \text { cell boundary }}=0 \end{array}\right. $$
where $C({x,y,t} )$ represents the concentration/fluorescent intensity of the activated PAGFP on the cell membrane; D is the diffusion coefficient (constant during each PDE simulation); ${C_0}$ is the fluorescent image (typically the first post-activation image) given to the PDE simulator; and cell boundary represents the manually/automatically labelled cell boundary in the fluorescent image. After all the conditions were set, the diffusion coefficient D would be approximated by a simplified hill-climbing algorithm, as illustrated in Fig. 7(d), to minimize the mean square error (MSE) of the ROI intensity between the PDE simulation and actual data.

 figure: Fig. 7.

Fig. 7. Approximation method for complex post-activation PAGFP patterns. (a) First post-activation image that serves as the IC. (b) Manually labelled cell boundary (red contour) that serves as the BC. (c) Selected ROI (yellow box) for MSE calculation between the simulation and actual mean ROI intensity. (d) Flow diagram for diffusion coefficient approximation. Scale bar: 30 µm.

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Different ICs and BCs were applied to test the efficacy of the proposed algorithm. Specifically, three different adiabatic boundaries, including an image-sized square, a cell-sized oval, and a cell-shaped boundary were used in the algorithm, respectively. The approximation successfully converged within 50 iterations for all three conditions. The results are presented in Fig. 8(a), where all cases share the rising and falling trend. Notably, the cell-shaped BC shows the best fitting result, which indicates a more accurate prediction of the diffusion coefficient (D). Figure 8(b) presents experimental results under three different BCs while using 0-, 1-, or 2-second fluorescent image as IC, respectively. The approximated diffusion coefficient shows good data convergence within each group. Notably, the cell-shaped BC shows the lowest MSE between the simulation and the actual mean ROI intensity.

 figure: Fig. 8.

Fig. 8. Efficacy test for the approximation algorithm. (a) PDE simulation results under different BCs using the approximated diffusion coefficients. (b) Experimental result (±SEM, n = 3 for each experiment) for the diffusion coefficient approximation under different BCs. Red contours: adiabatic boundary for different BCs. Scale bar: 30 µm.

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Lastly, we measured the difference of diffusion coefficients within the same cell. As shown in Fig. 9, three different ROIs were activated using a solid square pattern, where a 6-minute rest was arranged between each activation. Each ROI was subjected to a 5-second exposure for PAGFP activation, followed by two-photon imaging (∼200 seconds) to record the post-activation PAGFP dynamics. The diffusion coefficients were approximated using the 0-, 1-, or 2-second post-activation images as the ICs, respectively, and the cell boundary as the BC. The approximated diffusion coefficients were 0.3590 ± 0.0036 µm2/s, 0.3614 ± 0.0053 µm2/s, and 0.3251 ± 0.0005 µm2/s for ROI 1, ROI 2, and ROI 3, respectively. Comparing with the previous FRAP results on COS-7 cells [20], the diffusion coefficients fall within the range of previous studies with substantially improved precision. Notably, the diffusion coefficient in ROI 3 was significantly lower than that in ROI 1 and ROI 2, which suggests a regional diffusivity difference on the cell membrane. This result may be attributed to the inherent heterogeneity in the lipid-protein matrix of the cell membrane. For instance, areas with a higher concentration of cholesterol or sphingolipids may exhibit less fluidity compared to those with more phospholipids. On the other hand, the presence of membrane domains, changes in cell physiology, and specific protein-lipid interactions may also lead to the observed differences in diffusion coefficients. Future investigations are needed to elucidate the mechanisms underlying such phenomena.

 figure: Fig. 9.

Fig. 9. Measurement of diffusion coefficient within a single cell. (a) Selected ROI for measurement. (b) Experimental results (±SEM, n = 3 for each experiment) of the diffusion coefficient. Yellow box: ROI 1; red box: ROI 2; blue box: ROI 3. Scale bar: 30 µm. * p = 0.00154; ** p = 0.00521.

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5. Conclusion

We have designed and characterized a two-photon activation and imaging system for PAGFP and applied it to examine the regional diffusivity on the cell membrane. 3D selective PAGFP activation was realized by a DMD random-access scanner working under 800-nm femtosecond laser. A separate galvo-scanner was integrated in the system for video-rate two-photon imaging at a wavelength of 900 nm to study the post-activation PAGFP dynamics. Compared with other photoactivation systems, our setup achieves a minimum distinguishable photo-activation size of ~4 µm and an ultrahigh temporal resolution of 22.7 kHz. 2D and 3D selective activation on COS-7 cell membranes were performed to demonstrate the capability of our system in activating complex spatial patterns at the subcellular resolution. The dosage of PAGFP activation can be directly controlled by the designed scanning hologram with high precision. Lastly, we used our system to selectively activate PAGFP patterns on the cell membrane and record the subsequent diffusion processes. An approximation algorithm was developed to estimate the diffusion coefficient based on the post-activation dynamics.

Funding

Research Grants Council, Collaborative Research Fund (C5031-22GF); Centre for Perceptual and Interactive Intelligence (A-CUHK-16-5-14); National Natural Science Foundation of China, Young Scientists Fund (32201208); Research Grants Council, Young Collaborative Research Grant (C4001-22Y); National Natural Science Foundation of China/Research Grants Council Joint Research Scheme (N_CUHK489/22); Basic and Applied Basic Research Foundation of Guangdong Province (2023A1515011865).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

References

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Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (9)

Fig. 1.
Fig. 1. Optical configuration for the PAGFP activation and imaging system. HWP: broadband half-wave plate; PBS1-PBS2: polarizing beam splitters; L1-L5: lenses; M1-M4: high reflectivity mirrors; G1: grating; BE: beam expander; SL: scan lens; DM: dichroic mirror; OL: objective lens; pink beam: 800-nm femtosecond laser for PAGFP activation; red beam: 900-nm femtosecond laser for PAGFP imaging; green beam: fluorescent signals emitted by post-activation PAGFP; dash line inset: sample cell with a user-defined PAGFP activation pattern; solid line inset: illustration of DMD-based random-access scanning for PAGFP activation (left) versus galvo-based raster scanning for post-activation PAGFP imaging (right).
Fig. 2.
Fig. 2. Selective PAGFP activation on COS-7 cells expressing plasma membrane-localized PAGFP-CAAX. (a) Activation of a 2D “CUHK” pattern on the membranes of four COS-7 cells. (b) Selective 3D activation of three spots (inset) on a COS-7 cell membrane by a designed sinusoidal scanning trajectory. Color bar: depth of laser scanning trajectory. Scale bar: 30 µm.
Fig. 3.
Fig. 3. Characterization of minimal distinguishable distance for membrane-localized PAGFP-CAAX activation on COS-7 cell membrane. Scale bar: 30 µm.
Fig. 4.
Fig. 4. Precise laser dose control for membrane-localized PAGFP-CAAX activation via binary holograms. COS-7 cells expressing plasma membrane-localized PAGFP were used. (a) Fluorescent image before PAGFP activation. (b) Post-activation image of the same COS-7 cell with four different q values. Upper left: q = 1.0, upper right: q = 0.8, lower left: q = 0.6, lower right: q = 0.4. (c) Heat map of F/F0 calculated via (b) and (a). Color bar: F/F0 value for the heatmap in (c). Scale bar: 30 µm.
Fig. 5.
Fig. 5. Repeated activation of membrane-localized PAGFP-CAAX on the same cell. COS-7 cells expressing plasma membrane-localized PAGFP were used. (a)-(e) Fluorescent images after each activation. (f) Mean contrast (±SEM, n = 4 for each experiment) of the activation sites before and after each activation. Scale bar: 30 µm.
Fig. 6.
Fig. 6. Different diffusion behaviors under the hollow/solid membrane-localized PAGFP-CAAX patterns on COS-7 cells. (a) Diffusion process and mean ROI intensity change for a hollow rectangular PAGFP pattern. (b) Diffusion process and mean ROI intensity change for a solid rectangular PAGFP pattern. Red rectangle: selected ROI for mean intensity measurement. Scale bar: 30 µm.
Fig. 7.
Fig. 7. Approximation method for complex post-activation PAGFP patterns. (a) First post-activation image that serves as the IC. (b) Manually labelled cell boundary (red contour) that serves as the BC. (c) Selected ROI (yellow box) for MSE calculation between the simulation and actual mean ROI intensity. (d) Flow diagram for diffusion coefficient approximation. Scale bar: 30 µm.
Fig. 8.
Fig. 8. Efficacy test for the approximation algorithm. (a) PDE simulation results under different BCs using the approximated diffusion coefficients. (b) Experimental result (±SEM, n = 3 for each experiment) for the diffusion coefficient approximation under different BCs. Red contours: adiabatic boundary for different BCs. Scale bar: 30 µm.
Fig. 9.
Fig. 9. Measurement of diffusion coefficient within a single cell. (a) Selected ROI for measurement. (b) Experimental results (±SEM, n = 3 for each experiment) of the diffusion coefficient. Yellow box: ROI 1; red box: ROI 2; blue box: ROI 3. Scale bar: 30 µm. * p = 0.00154; ** p = 0.00521.

Equations (3)

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h ( i , j ) = { 1 , q 2 φ ( x , y , z ) 2 π q 2 0 ,  otherwise 
d ( s i n θ i s i n θ m ) = n λ
{ P D E : C ( x , y , t ) t = D 2 C ( x , y , t ) I C : C ( x , y , t ) | t = 0 = C 0 B C : C ( x , y , t ) | ( x , y )  cell boundary  = 0
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