Live-Cell Transforms between Ca²⁺ Transients and FRET Responses for a Troponin-C-Based Ca²⁺ Sensor

Construction of adenoviral vectors bearing TN-L15

The vector TN-L15/pAdLox was constructed by transferring the coding region for TN-L15 from TN-L15/pcDNA3 (12) into the multiple cloning site of the adenoviral shuttle-vector pAdLox (16–18), via BamHI and EcoRI restriction sites. Recombinant adenovirus, bearing TN-L15, was then generated by Cre-lox recombination, according to previously described methods (18). Briefly, 60-mm diameter dishes containing CRE 8 cells at ∼20% confluence were cotransfected with 3 μg TN-L15/pAdLox and 3 μg ψ5 viral DNA using FuGENE 6 (Roche Diagnostics, Basel, Switzerland). Following observation of cytopathic effects, usually after 7–10 days, cells were harvested, subjected to three freeze-thaw cycles, and centrifuged to remove cellular debris. The resulting supernatant (1 ml) was used to infect a 10-cm dish of 90% confluent CRE 8 cells. Following observation of cytopathic effects after 2–3 days, cell supernatants were used to reinfect a new plate of CRE 8 cells. This procedure was repeated two to three more times, followed by viral expansion and purification as previously described (18). Viral particle numbers, determined by absorbance at 260 nm, were present at ∼10¹² particles ml⁻¹.

Isolation, culture, and infection of rat cardiac myocytes

Adult Sprague-Dawley rats were killed with an overdose of sodium pentobarbital or halothane. Hearts were excised and ventricular myocytes isolated by enzymatic digestion using a Langendorf perfusion apparatus, as previously described (18). Healthy, rod-shaped cardiac myocytes were cultured on 35-mm culture dishes, with laminin-coated No. 0 glass bottoms (MatTek, Ashland, MA). After initial plating, cells were maintained in Medium 199 supplemented with 5 mM carnitine, 5 mM creatine, 5 mM taurine, 1% antibiotic antimycotic, and 5% fetal bovine serum to promote attachment to the coverslips (all from Sigma-Aldrich, St. Louis, MO). After 1 h, the culture medium was switched to serum-free Medium 199, otherwise supplemented as described above. Cultures were equilibrated with 6% CO₂ and 94% air at 37°C in a water-jacketed CO₂ incubator. As needed, cells were infected by adding 2–3 μl of the TN-L15 virus stock directly onto the cells, in a final volume of 200 μl. After 1–2 h, fresh serum-free media was added to a final volume of 2 ml for overnight incubation. For details, see Alseikhan et al. (18).

Indo-1 loading of cardiac myocytes

When required, cells were loaded with Indo-1 by incubating with Indo-1 AM (Teflabs, Suffolk, England) at a bath concentration of 0.5 μM for 10 min at 37°C. Then, the cells were washed thoroughly with HEPES-buffered 2 mM Ca²⁺ Tyrode's solution (pH 7.4) to remove residue Indo-1 AM. Tyrode's solution was comprised of 138 mM NaCl, 4 mM KCl, 1 mM MgCl₂, 10 mM HEPES (pH 7.35 with NaOH), 5 mM glucose (all from Sigma-Aldrich).

Field stimulation of myocytes

Field stimulation (square pulse, 20 V × 4 ms) was produced by a Grass SD9 pulse generator through a pair of platinum electrodes. The electrodes were closely spaced and positioned in the Tyrode's solution directly above the center of the objective lens, so as to achieve local stimulation. To facilitate Ca²⁺ transients in field-stimulated rat cardiac myocytes, cells were initially bathed in a 2 mM Ca²⁺ Tyrode's solution (with 1 μM Forskolin, Sigma-Aldrich) for 15 min, before switching to a 10 mM Ca²⁺ Tyrode's solution (without Forskolin) for fluorescence imaging.

Fluorescence measurements

Cells were imaged with a Nikon TE300 Eclipse microscope (40×, 1.3 n.a. objective) and custom fluorometer system (University of Pennsylvania Biomedical Instrumentation Group, Philadelphia, PA). Excitation light was delivered by a 150-W short-gap Xenon arc-lamp (Optiquip, Highland Mills, NY) with servo control, and gated by a computer-controlled shutter (Vincent Associates, Rochester, NY). Filter cubes (Chroma, Rockingham, VT) employed were (excitation, dichroic, emission): Indo405 (D340×, 380DCLP, D405/30m), Indo485 (D340×, 380DCLP, D485/25m), CFP (D440/20×, 455DCLP, D480/30m) and FRET (D436/20×, 455DCLP, D535/30m). During one cycle of acquisition, the filter cubes were sequentially rotated by a motorized Hex-Filter (Conix Research, Springfield, OR), with a 20-s interval between 2.4-s imaging periods by individual filter cubes. The shutter limited excitation to these imaging periods. Ten cycles of measurements were taken for each myocyte, and the results averaged. Normally, neutral density filters were engaged such that only 1/32 of the light source was transmitted. However, due to the weaker power output of the arc-lamp at ultraviolet wavelengths, it was necessary to boost the excitation signals for Indo-1 by allowing 1/4 of the light source to be transmitted during imaging with Indo-1 cubes. Epifluorescence emission from entire individual cells was selected by an image-plane pinhole, and measured with an ambient temperature photomultiplier tube (PMT) (30 mm EMI 9124B, Electron Tubes, Rockaway, NJ). PMT signals were low-pass filtered at 10 kHz, and then digitized (ITC 18, Instrutech, Port Washington, NY). Shutter control, electrical stimulation, automated switching of filter cubes, dark-current subtraction, and data acquisition were all controlled by custom MATLAB (MathWorks, Natick, MA) software. To correct for autofluorescence and background light scatter, averages from individual uninfected myocytes were subtracted from same-day experimental values for each filter cube.

Numerical processing for forward-backward transforms

Custom scripts in EXCEL 2003 (Microsoft, Seattle, WA) were used to elaborate the Euler integration steps for transforms as described in the Results section.

Statistical analysis

Data were expressed as mean ± SD. Nonlinear curve fitting and least-squares linear regression (Solver add-in tool, EXCEL 2003) were used when appropriate.

Independent detection of Indo-1 and TN-L15 signals within individual cells

To monitor the “genuine” Ca²⁺ signal, we required a customary chemical fluorescent Ca²⁺ dye whose spectral properties would permit optical interrogation, independently from that for the CFP/yellow fluorescent protein (YFP) FRET signals of TN-L15. Given the spectra of CFP and YFP, candidate dyes included those with ultraviolet excitation (e.g., Indo-1 and Fura-2), and others with far red-shifted emission (e.g., Rhod-2). Of these, the ultraviolet candidates were favored because they readily permitted ratiometric determination of Ca²⁺ concentration. One ultraviolet dye, Fura-2, did exhibit detectable (albeit small) cross talk with CFP/YFP FRET (not shown).

We therefore favored the use of the ultraviolet dye Indo-1, which featured no detectable cross talk with CFP/YFP FRET under explicit quantitative testing, as follows. An initial set of experiments was performed in HEK293 cells. When Indo-1 was loaded into HEK293 cells lacking TN-L15 expression, fluorescence output from CFP and FRET cubes (S_CFP and S_FRET) was negligible, despite working levels of Indo-1 as detected by Indo-1 fluorescence (S_Indo405) through the Indo405 cube (Fig. 1 A). In particular, each symbol plots S_CFP vs. S_Indo405 (bottom), or S_FRET vs. S_Indo405 (top), wherein each symbol corresponds to a separate cell. The gray boxes represent the working ranges of TN-L15 (ordinate) and Indo-1 (abscissa) signals in single cardiac myocytes, as spanned by resting and contracting conditions. The negligible ordinate values of symbols, relative to the green regions, explicitly document the absence of Indo-1 to CFP/YFP cross talk. Conversely, for HEK293 cells expressing TN-L15 but lacking Indo-1 loading, fluorescence output from Indo405 and Indo485 cubes (S_Indo405 and S_Indo485) was negligible, despite working levels of TN-L15 (S_CFP). The near-zero ordinate values for plots of S_Indo405 vs. S_CFP (Fig. 1 B, top), or S_Indo485 vs. S_CFP (bottom), demonstrates the absence of appreciable CFP/YFP to Indo-1 cross talk.

Additional experiments within myocytes further substantiated the lack of cross talk between Indo-1 and TN-L15. Myocytes loaded with Indo-1, but lacking TN-L15, were field-stimulated to contract once under each filter cube (at time 0) and imaged. One “cycle” of measurements therefore incorporated data obtained under Indo405, Indo485, CFP, and FRET cubes. As data were highly reproducible over subsequent contractions, we frequently signal averaged data over 10 such cycles. Exemplar traces of this kind are displayed in Fig. 1 C, which illustrates the characteristic rise and fall of respective Indo405 and Indo485 signals (left) during a triggered cytoplasmic Ca²⁺ transient. Importantly, neither S_CFP nor S_FRET signals were detected under these same conditions, demonstrating the absence of cross talk from Indo-1 into TN-L15 optical channels in myocytes. By contrast, for myocytes expressing TN-L15, but lacking Indo-1, the Indo405 and Indo485 channels are devoid of appreciable signal (Fig. 1 D, left), whereas S_CFP and S_FRET channels clearly exhibit TN-L15 responsiveness to a triggered Ca²⁺ transient (right). This outcome directly confirms the absence of cross talk from TN-L15 to Indo-1 optical channels within the myocardial context.

Simultaneous detection of Indo-1 and TN-L15 signals in cardiac myocytes

Having established that Indo-1 and TN-L15 permit independent readouts, we proceeded to simultaneously detect both sensor signals from single myocytes. In these experiments, cardiocytes expressing TN-L15 were also loaded with Indo-1, and imaging was then performed under Indo405, Indo485, CFP, and FRET cubes (Fig. 2 A). For Indo-1 outputs, exemplar traces show the respective rise and fall of Indo405 and Indo485 signals (Fig. 2 B, S_Indo405 and S_Indo485) during a field-stimulated Ca²⁺ transient. Note that the time course of these Indo-1 transients was closely similar to that obtained in the absence of TN-L15 expression (Fig. 1 C), suggesting that our expression of TN-L15 does not appreciably alter native Ca²⁺ transients. The ratio of S_Indo405 and S_Indo485 signals (Fig. 2 B, R_Indo) was then used to calculate the phasic time course of free intracellular Ca²⁺ concentration (Fig. 2 B, [Ca²⁺]), according to the classic equation (5)

(1)

where the apparent dissociation constant K_d,Indo was 800 nM as previously established for myocytes (19); the respective maximal and minimal ratio values, under Ca²⁺-bound and free conditions (R_max,Indo, R_min,Indo), were 2.715 and ∼0.47; the β-parameter (= ratio of S_Indo485 signals for Indo-1 under Ca²⁺-free and bound conditions) was 2.632 (Table 1).

Concerning TN-L15 signals, exemplar traces from the same cell illustrate the respective rise and fall of S_FRET and S_CFP signals (Fig. 2 C), confirming robust sensor function during single Ca²⁺ transients. In principle, the ratio of these two signals (Fig. 2 C, R_TNL = S_FRET / S_CFP, black trace) should then be quantitatively related to the actual [Ca²⁺] waveform (Fig. 2 B, bottom). Indeed, this general interrelation between R_TNL and [Ca²⁺] was qualitatively reproduced across many myocytes. However, R_TNL seemed kinetically slow compared to [Ca²⁺]: R_TNL remained elevated above baseline levels at the 2000 ms, whereas [Ca²⁺] had long reached resting levels by this time point. Such kinetic slowing is common among genetically encoded Ca²⁺ sensors (9,11).

Determination of TN-L15 sensor parameters within live myocytes

Given the apparent reproducibility of the relation between [Ca²⁺] and R_TNL, we wondered whether a quantitatively accurate “transform” between these two entities could be deduced. Such transforms would find broad utility among genetically encoded Ca²⁺ sensors, but the existence of these mapping functions has thus far been little explored (9). Toward this end, a prerequisite was to determine key sensor parameters appropriate for the intracellular milieu of myocytes. The crucial Indo-1 parameters were R_max,Indo, R_min,Indo and β, as defined above. The analogous TN-L15 parameters were R_max,TNL, R_min,TNL, and α (= ratio of S_CFP for TN-L15, under Ca²⁺-free and bound conditions). To determine these parameters, we studied either myocytes that lacked TN-L15 expression while loaded with Indo-1 (for specification of Indo-1 parameters), or myocytes expressing TN-L15 without Indo-1 loading (for TN-L15 parameters). To ensure strong elevation of intracellular Ca²⁺ during determination of R_max, cardiocytes were preexposed for 15 min to a bath solution containing 5 μM 2,3-butanedione monoxime (BDM) to counter hypercontraction during subsequent Ca²⁺ elevation, and 5 μM cyclopiazonic acid (CPA) with 10 μM ryanodine to limit buffering of intracellular Ca²⁺ during ensuing maneuvers that elevate Ca²⁺. The pretreatment was followed by a 15-min exposure to 5 μM ionomycin within a Tyrode's solution containing elevated 10 mM Ca²⁺, before imaging for R_max values. To obtain R_min parameters, myocytes were exposed for 15 min to 5 μM ionomycin in a Ca²⁺-free Tyrode's solution (4 mM EGTA), and then subject to imaging. For the β-parameter, a myocyte was initially pretreated with BDM, CPA, and ryanodine (as above), then exposed for 15 min to 5 μM ionomycin in Ca²⁺-free Tyrode's solution, before measuring the Ca²⁺-free S_Indo485 signal. Subsequently, the same myocyte was exposed to 5 μM ionomycin in a 10 mM Ca²⁺ Tyrode's solution for 15 min, before imaging S_Indo485 in saturating Ca²⁺; β was thus determined as the ratio of these S_Indo485 measurements. An analogous procedure was used to determine the TN-L15 parameter-α. After refinement by a sequential procedure (Supplementary Material, section 1), the average values for sensor parameters are summarized in Table 1.

Mapping from [Ca²⁺] into R_TNL via a three-state mechanism

With sensor parameters in hand, we tested for the existence of a quantitative relationship between dynamic [Ca²⁺] inputs and TN-L15 outputs (R_TNL). Because the steady-state relationship between R_TNL and [Ca²⁺] conforms well to a Hill function with a coefficient of unity (12), we considered approximate mechanistic schemes in which a single Ca²⁺ binding site dominates control of FRET conformational changes. Configurations encompassing only Ca²⁺-unbound (UB) and bound (B) states invariably failed to cohere to the experimental data in Fig. 2, with inability to explain the degree of kinetic slowing in the falling phase of R_TNL (not shown). By contrast, the next simplest model, in which there are two Ca²⁺-bound conformations (Fig. 3 A, B₁ and B₂), proved remarkably successful in mapping Indo-1-derived [Ca²⁺] into experimentally measured R_TNL. This mapping is explicitly illustrated (Fig. 2, B and C) by the conversion of the experimentally determined [Ca²⁺] transient (Fig. 2 B, bottom) into the predicted R_TNL,sim (Fig. 2 C, gray curve in third row), via a “forward transform” algorithm. The specifics of the transformation, with its in-depth validation, are as follows.

To compute R_TNL,sim, we presumed the Ca²⁺-unbound state (UB) to be associated with a FRET ratio of R_min,TNL, whereas both Ca²⁺-bound states (B₁, B₂) would be affiliated with the same maximal FRET ratio of R_max,TNL. Accordingly, R_TNL at any instant of time would be given by

(2)

where UB(t) is the probability of occupying state UB at time t, and B_T(t) is the sum of the probabilities of occupying B₁ and B₂ at time t. All that remained, then, was to evaluate UB(t) and B_T(t) via numerical integration of the equations corresponding to Fig. 3A, utilizing the experimentally determined [Ca²⁺] transient (Fig. 2 B, bottom) as a multiplicative factor specifying the overall rate constant from UB to B₁. For initial conditions at a time point before the onset of contraction (typically t ∼ −400 ms), we set UB(t), B₁(t), and B₂(t) to their predicted steady-state values with respect to the average resting [Ca²⁺] ([Ca²⁺]_rest) determined from TN-L15 measurements, according to the analytical solutions

(3A)

(3B)

(3C)

From this initial time point onward, UB(t), B₁(t) and B₂(t) were calculated by Euler's integration using the iterative equations

(4A)

(4B)

(4C)

Given the strategy embodied in Eqs. 2–4, we asked whether a single set of rate constants (Fig. 3 A, k_on, k_off, k₁₂, and k₂₁) could produce close conformation of simulated R_TNL,sim to experimentally determined R_TNL dynamic waveforms, while simultaneously predicting the steady-state relation between R_TNL and [Ca²⁺]. Accordingly, for the exemplar myocyte in Fig. 2, we numerically minimized the sum of squared errors between R_TNL,sim (Fig. 2 C, gray trace) and the experimental R_TNL waveform (Fig. 2 C, gray trace), yielding the set of rate constant parameters in Table 2. Reassuringly, the steady-state TN-L15 response entailed by these parameters (Fig. 3 B, smooth black curve) closely matched the experimental relation (symbols). The predicted response is computed from the following relation

(5)

To assess the generality of this forward transform, we mapped [Ca²⁺] to R_TNL,sim in other myocytes, using the same set of parameters (Table 2). In fact, the average percentage error between R_TNL,sim and the experimentally observed R_TNL was nearly zero with small standard deviations (Fig. 2 C, bottom), confirming generalization across cells.

As a more stringent test, we considered whether the forward transform would suffice even during closely spaced Ca²⁺ transients, in which TN-L15 responses would become fused. Appropriate prediction of nonlinear summation during these fusion events represents a challenging regime for predictive algorithms (20). Fig. 4 shows the results of experiments in which myocytes were stimulated twice with a coupling interval of 600 ms. The format is closely similar to that for the single-twitch experiment in Fig. 2, and double- and single-twitch data were often obtained in the same myocytes. In fact, the same exemplar myocyte is used for Figs. 2 and 4. The forward transform output (Fig. 4 B, third row, gray curve) indicates excellent prediction of fused events. Moreover, the near-zero average percentage error (Fig. 4 B, bottom) confirms the quality of the predictions for overlapping responses over multiple cells. These results (Figs. 2 and 4) support the broad applicability of the forward transform.

Back calculation of rapid Ca²⁺ transients from TN-L15 FRET responses

Given the existence of a quantitative relation between dynamic Ca²⁺ concentration waveforms and TN-L15 FRET responses, we wondered whether the time course of rapid Ca²⁺ transients could be deduced from TN-L15 signals, based on an inversion of the aforementioned forward transform. Such a capability would enhance the potential of TnC-based Ca²⁺ sensors for quantitative dynamic in vivo measurements. Toward this end, we devised and tested the following “backward transform”.

Because the forward transform resembles low-pass filtering, the reverse operation would approximate a high-pass filtering operation, which would greatly amplify noise. To avoid confounding noise amplification, an initial step is to fit smooth functions to experimentally measured TN-L15 R_TNL outputs, yielding R_TNL,fit waveforms (Fig. 5 A, top row, gray trace). B_T(t) could then be computed as

(6)

With knowledge of B_T(t), B₁(t) and B₂(t) are then deduced in a straightforward manner. Their initial values, at a time point before contraction (e.g., at −400 ms), are specified by considering their steady-state distribution according to Since it follows that

(7)

From this instant of time onwards, we could use numerical integration to calculate B₁ and B₂ using the iterative relations

(8)

(9)

To deduce the back-calculated [Ca²⁺]_back, we recalled that UB(t) = 1 − B_T(t), so that

.

Rearranging yields

We therefore solved for [Ca²⁺]_back as

(10)

Equations 6–10 thereby enabled mapping from R_TNL,fit to [Ca²⁺]_back. Fig. 5 A illustrates this backward transform for an exemplar myocyte (gray waveforms, top and middle row). Visual inspection confirms a close similarity between predicted and measured [Ca²⁺] (middle row, gray and black waveforms). The near-zero average percentage errors for population data establishes the adequacy of the backward transform over multiple cells (Fig. 5 A, bottom row). Fig. 5 B reveals equally impressive performance of the backward transform for the dual-stimulus paradigm. Overall, the existence of a quantitative relation between [Ca²⁺] and TN-L15 FRET responses, together with effective forward and backward transforms, heightens the potential utility of TnC-based GECIs for in vivo imaging.

TnC-based GECIs resist interference by endogenous CaM and contractile elements

Unlike many GECIs that incorporate CaM as the Ca²⁺-motile module (13,21), TnC-based sensors promise less sensitivity to interference by endogenous CaM (11,12,15). However, one remaining concern for these sensors has been the potential for interference by endogenous contractile proteins within striated muscle (13). Thus, whereas these GECIs may perform consistently within neurons (11), they may prove inadequate in contractile tissues. Accordingly, the findings in this study are particularly significant because experimental tests were performed in cardiac myocytes, where both endogenous CaM and contractile proteins are found in abundance (2). The well-behaved profile of TnC-based GECIs within this environment furnishes strong empirical evidence for insensitivity to these forms of interference.

To further exclude the possibility that TnC-based GECIs might be compromised by incorporation into the matrix of muscle contractile proteins, we performed fluorescence recovery after photobleaching (FRAP) experiments on TN-L15 expressed in myocytes. Rapidly motile recovery of photobleached segments (over ∼10 s) directly excludes appreciable incorporation (not shown). Otherwise, darkened regions should persist from appreciably longer durations. This stands in contrast to older generation GECIs based on CaM, wherein a large fraction of such sensors was not freely diffusible as gauged by FRAP (8).

Such resistance to interference from endogenous factors may also be shared by newer generations of CaM-based sensors GCaMP2 reportedly functions well within intact heart of transgenic mice (9). More intriguingly, computationally redesigned CaM/peptide elements within GECIs also hold enormous promise for enhanced resistivity to endogenous interference (13).

Algorithmic correction for slow kinetics of current-generation GECIs

Beyond reproducibility of GECI sensor responses to Ca²⁺ transients, a remaining limitation is the relatively slow kinetic responsiveness of current sensors, compared with the characteristic time constants of native Ca²⁺ transients (10). Here we demonstrate that first-order quantitative maps between Ca²⁺ and sensor outputs may be deduced from simultaneous recordings of a “gold-standard” Ca²⁺ input signal and simultaneous GECI sensor outputs. These maps may then be used to infer rapid Ca²⁺ transient behavior from slowed GECI sensor responses, thereby expanding the capabilities of current-generation sensors.

The approach we have taken for deducing these maps appears convenient and generalizable to other GECIs. The paradigm of obtaining simultaneous recordings of Ca²⁺ concentration “inputs” and FRET sensor “outputs” greatly facilitates deduction of approximate transform mechanisms, with appropriate emphasis on those features with greatest relevance for native Ca²⁺ transient behavior. The adequacy and simplicity of the sensor scheme in Fig. 3 A suggest that a full-bore mechanism may not be required to afford reasonably accurate dynamic maps between Ca²⁺ waveforms and FRET responses. It will be interesting to explore whether a like approach may prove successful for other GECIs like GCaMP2 (9) and novel computationally redesigned CaMs (13).

It is nonetheless worth emphasizing the likely profile of more in-depth formulations of TnC-based sensors. In particular, how can an approximate scheme (Fig. 3 A) emphasize the dominance of a single Ca²⁺ binding event, whereas TnC clearly possesses four Ca²⁺ binding sites? Insight into this question arises from the fact that the two Ca²⁺ binding sites in the C-terminal lobe of TnC may exhibit very high affinity and slow kinetics, such that these sites may already be constitutively occupied if resting Ca²⁺ levels exceed a certain minimum level (22–24). This would leave Ca²⁺ binding to the two sites in the N-terminal lobe as potentially important in gating the FRET response of GECIs. Even here, there is precedent from cardiac TnC that one of these sites may predominate in controlling physiological phasic Ca²⁺ responses (2). Given this context, we hypothesize that the in-depth mechanism for TN-L15 may include ancillary Ca²⁺ binding sites that are constitutively occupied in resting Ca²⁺, leaving a single key Ca²⁺ site to control FRET responses during cardiac twitch Ca²⁺ events (Fig. 6). Ca²⁺ binding to the C-terminal lobe sites of TnC may be necessary, but insufficient to trigger a TnC sensor conformational change that is detectable by an alteration in FRET. Thus, there is no change in the baseline of R_TNL within the Ca²⁺ concentration range from 0 to 100 nM. As Ca²⁺ sweeps higher above the ∼100 nM resting level, the relevant N-terminal lobe site becomes progressively occupied, and this is reflected rather directly by variations in FRET (R_TNL). The Ca²⁺ occupancy of the C-terminal sites may well impact the nature of how Ca²⁺ binding to the critical N-terminal site is transduced to changes in FRET. However, since both C-terminal sites may be saturated at Ca²⁺ concentrations greater than ∼100 nM, R_TNL increases according to Ca²⁺ occupancy of the key N-terminal site over this range of Ca²⁺. Hence, R_TNL would rise with a Hill coefficient of unity, just as observed experimentally for TN-L15 (12).

Limitations to kinetic compensation of GECIs

There are limitations to the back-calculation algorithm, which merit emphasis. First, before back calculation, we found it necessary to smooth the raw TN-L15 outputs (R_TNL waveforms). Specifically, we fit idealized mathematical functions to the actual R_TNL waveforms. This procedure was necessary because back-calculation is rather like taking derivatives of the R_TNL waveforms; unfettered noise would become intolerably amplified in the procedure. Although the smoothing procedure is of little consequence if actual Ca²⁺ transients lack high-frequency components, this procedure would inevitably preclude back calculation of high-frequency components in the Ca²⁺ transients. As a rough estimate, empirical tests indicate that Ca²⁺ transient components exceeding ∼20 Hz cannot be accommodated by the current back-calculation algorithm. Thus, it is unlikely that sub-10-ms Ca²⁺ sparks in heart muscle could be inferred from R_TNL outputs (25). Nonetheless, much useful Ca²⁺ transient information (with components <∼50 Hz) can be recaptured via the back-calculation algorithm.

Overall, as future generations of GECIs are developed, their intrinsic kinetic responsiveness promises to improve. In this evolution, forward-backward “transforms” as devised here will continue to expand the kinetic boundary for inferences regarding the actual underlying Ca²⁺ signaling events.