1 Appendix on the 3 3 -FRET method. Supplemental Data for: Erickson et al., Neuron 31, pp

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1 upplemental Data for: Erickson et al., Neuron 31, pp upplemental ppendices on the Three-Cube Method and Extensions of the Method to Characterize Properties of Binding Between Donor and cceptor Molecules 1 ppendix on the method 1.1 epresentation of microscope optical properties and fluorescence To aid understanding of calculated values that are based upon multiple measurements with different fluorescence cubes, we have found it convenient to model the properties of the epifluorescence microscope and fluorophore by the following formalism. In our system, there are two types of fluorophores, donor D or and acceptor or, each of which can exist in the ground D, or excited states D*, *. In the field of view, there are N and N D donor and acceptor molecules, respectively. D b is the fraction of donor molecules bound by an acceptor, and b is the fraction of acceptor molecules bound by a donor. It is assumed that no occurs between unassociated donor and acceptor molecules. To quantitate fluorescence behavior of donor and acceptor molecules there are then three subsystems to consider, which we describe in turn Fig.. The first concerns the excitation of fluorophore. The excitation rate in units of transitions per second of a single ground-state fluorophore is given by I o x y,λ ex,x, where I o is the overall intensity of the xenon lamp over all wavelengths, x specifies which of three optical cubes is being used,, or, y specifies whether we are concerned with a donor or acceptor molecule D or, and λ ex,x is the predominant wavelength of excitation light determined mainly by the excitation filter of cube x. x y,λ ex,x is thus a constant that incorporates spectral properties of the lamp, optical properties of the excitation filter and dichroic mirror of cube x, and wavelength-dependent absorption properties of the fluorophore in question as given by a molar extinction coefficient ε y λ. The second subsystem concerns the rate-constant model Fig. describing the probabilities of occupying, *, D, and D* states, and the probability flux of transitions among the various states. For purposes of calculating fluorescence emission, we need only consider the steady-state behavior of the system because measurement times are far longer than the characteristic relaxation times. standard assumption for derivation of equations is that the system is in the low-excitation limit, where excitation power is low enough that the steadystate probabilities of being in D or P D or P are essentially unity. We verified experimentally that we are orders of magnitude from exceeding the low-excitation limit. With this simplification in mind, it is straightforward to calculate that the steady-state probability of occupying D* is: P D* 1-D b I o x D,λ ex,x / k D + D b I o x D,λ ex,x / k T +k D [] i

2 where k T is the rate constant for between donor and acceptor molecules all rate constants in units of s -1, and k D is the rate constant for non- relaxation from D* to D. The rate constants k T and k D are independent of wavelength, but k T is a function of donor-acceptor distance according to the Förster equation. Likewise, under the low-excitation limit, the steadystate probability of occupying * is given by P * I o x,λ ex,x / k + b [ I o x D,λ ex,x / k T +k D ] k T / k [2] where k is the wavelength-independent rate constant for non- relaxation from * to. The first term relates to excitation of the acceptor by the xenon lamp which occurs regardless of whether a donor is bound, and the second term concerns excitation of the acceptor which only can occur if donor is bound. Figure Fluorescence behavior of donor and acceptor molecules in a microscope field of view, represented quantitatively as three sequential subsystems. The excitation subsystem incorporates properties of excitation light source, excitation filter, dichroic mirror, and flurophore excitation spectrum. The fluorophore-rate-constant subsystem models behavior of fluorophores as a state diagram with interstate transitions governed by various rate constants. The rate constants relating to emission of fluorescent photons k D and k are functions of intrinsic properties of donor and acceptor molecules, and are independent of the wavelengths used for excitation or detection of emission. The rate constant pertaining to resonance energy transfer k T is a function of many factors including distance and orientation between bound donor and acceptor molecules, as well as overlap between emission and excitation spectra of donor and acceptor molecules, respectively. However, k T is also independent of the wavelengths used for excitation or detection of emission. The emission-detection subsystem incorporates properties of the emission filter, dichroic mirror, photomultiplier electronics, as well as fluorophore emission spectrum and quantum yield. The three output signals on the right are those that comprise aggregate fluorescence output obtained with any of the filter cubes. ii

3 The third subsystem relates to the efficiency by which we can detect excited donor and acceptor relaxations by fluorescence measurements. In relation to excited donor relaxations possibly giving rise to fluorescence emission, the rate of such relaxations is k D P D*. The fluorescence output from the photomultiplier tube PMT mv output per second arising from excited donor relaxations is given by N D k D P D* F x D,λ em,x where N D is the number of donor molecules in the field of view, λ em,x is the predominant wavelength or wavelength range of the output segment of cube x, and F x D,λ em,x is an output transfer function with units of mv per non- donor relaxation. F x D,λ em,x is a constant that incorporates the emission spectrum and quantum yield of the donor, the dichroic mirror and emission filter optical properties of cube x, and frequency-dependent sensitivity of the PMT detector. Inserting Eq. into the above expression yields the full equation for fluorescence output resulting from donor fluorescence, as measured with cube x: x λ ex,x,λ em,x, N D k D [1-D b / k D + D b / k T +k D ] I o x D,λ ex,x F x D,λ em,x [3] Likewise, analogous reasoning and Eq. 2 provide the full equation for fluorescence output resulting from acceptor fluorescence, as measured with cube x. This entity is given by two terms relating to and excitation: x λ ex,x,λ em,x, N k [ I o x,λ ex,x /k ] F x,λ em,x x λ ex,x,λ em,x, N b [ I o x D,λ ex,x / k T +k D ] k T F x,λ em,x [4] [5] These descriptors include any idiosyncrasies of the actual optical elements employed, and provide a convenient means of tracking effects of such peculiarities in extended calculations of the ratio F. In fact, by this means, we will be able to prove that the method self normalizes for the inevitable peculiarities of physical optical elements. 1.2 Nomenclature for various fluorescence measurements In the Methods section of the paper, the actual fluorescence signal output obtained from a given sample with a certain optical cube is denoted by the descriptor x PECIMEN, where x is the name of the cube,, and PECIMEN is either the donor D, acceptor, or both. In this ppendix section, we expand this formalism to include the predominant wavelengths in nm for excitation of the specimen λ ex,x, and for emission λ em,x as detected by the photodetection device. This yields the term x PECIMEN,λ ex,x,λ em,x. For example, the signal output obtained from with the cube is D,440,480. iii

4 1.3 Measurement ratios for transformation of optically isolated signals from donor or acceptor s described in the text of the paper, certain fluorescence measurements obtained from a mixture of both donor and acceptor molecules can be attributed primarily to donor or acceptor only. To transform such optically isolated fluorescence signals into those that would be in effect using excitation and emission wavelengths where both donor and acceptor fluorescence would be appreciable such as near 535 nm where we focus our analysis of sensitized acceptor fluorescence, we make use of three predetermined ratios of measurements taken from cells expressing only acceptor, or only donor. We restate these ratios here, and go on to specify their relation to the core formalism of section 1.1 which is accomplished via Eqs. 3 5.,440,535,500,530LP,440 F,500 F, 535, 530LP [6] D1 D,440,535 D,440,480 D,440 F D,440 F D, 535 D, 480 [7] D,500,530LP D,500 F, 530LP [8] D,440,480 D,440 F D, 480 Note that these ratios are independent of the excitation intensity I o and the number of donor or acceptor molecules in the field of view N D or N. Thus they can be determined in cells in which acceptor or donor alone are expressed, and then applied to transformation of signals obtained from different cells in which mixtures of acceptor and donor are expressed. The utility of these ratios can be illustrated by a simple case example. uppose we used the cube to obtain a fluorescence measurement from a cell expressing both donor and acceptor. Because does not detectably emit photons in the 480 nm range not shown, but rigorously tested, the cube measurement,440,480 is equivalent to an entity related to fluorescence alone, or 440,480, from Eq. 3. How can we convert this into the fluorescence that would be present using the cube, which is equivalent to from Eq. 3? Inspection of Eqs. 3 and 7 shows that the desired transformation can be determined by the relation D1,440,480 [9] Thus, the ratio D1 allows us to transform the optically isolated signal,,440,480, into the contribution to fluorescence at 535 nm, where both and fluorescence are appreciable. remarkable feature of the transformation is that it is rather exact, regardless of the concentrations of donor and acceptor in the field of view, the possibility of binding and between donor and acceptor molecules, the excitation power, and the inevitable idiosyncrasies of the optical cubes involved. These factors have all been incorporated into Eqs. 3 and 7, which were used to solve for Eq. 9. The features of exactness and generality pertain to all subsequent calculations as well. iv

5 1.4 Determination of ratio F by the method The ratio F, which we wish to determine experimentally by the method, is defined as F + [0] where the terms refer to fluorescence due to and excitation, as defined in Eqs. 4 and 5. The numerator of the expression is easily determined from the experimental measurement,440,535 by considering its constituent components Fig. and Eqs. 3-5:,440, [1] We have an experimentally determined measure of the third term in Eq. 9. Thus, the numerator of the F expression in Eq. 0 is experimentally determined as +,440,535 D1,440,480 [2] To solve for, the denominator of the F expression in Eq. 0, we first express the cube measurement,500,530lp in terms of its three constituent components Fig.. With reference to Eqs. 3-5, we can then write an expression strictly analogous to Eq. 1,500,530LP 500,530LP, + 500,530LP, + [3] The second term dominates the expression, consistent with the near selective excitation of with the cube. The third term is considerably smaller, and it will turn out that the first term is even smaller and negligible from a practical standpoint. The equations used in the Methods therefore ignore the first term, but here we will treat all terms rigorously so as to prove that first term can be practically ignored. By analogy to the derivation for Eq. 9, Eqs. 3 and 8 can be combined to specify the third term as a function of experimentally determined measures, according to: 500,530LP,,440,480 [4] olving Eq. 3 for 500,530LP, and substituting from Eq. 4 yields: 500,530LP,,500,530LP,440, ,530LP, 500,530LP, [5] v

6 With the aid of Eqs. 4 and 6, the product 500,530LP, can be shown to be exactly equal to. Hence, multiplying Eq. 5 by yields [,500,530LP,440,480] 500,530LP, [6] Eq. 5 allows us to relate 500,530LP, to by the relation 500,530LP, D,500,440 D,440, Y [7] ubstituting Eq. 7 into Eq. 6 yields [8] [,500,530LP,440,480] Y Finally, Eqs. 2 and 8 can be solved simultaneously to give the denominator term for F in Eq. 0, in terms of experimentally measurable entities, as given by 1 1- Y [,500,530LP Y [ 1 Y, 440,535,440,480] D1,440,480] [9] ubstituting Eqs. 2 and 9 into the F expression in Eq. 0 provides us with the full specification of ratio, all in terms of experimental measures. The complete relation is given by, F [20] [1 Y] [,440,535 D1,440,480] [,500,530LP,440,480] Y [, 440,535,440,480] D1 The magnitude of Y turns out to be exceedingly small, and can be estimated from the ratio of molar extinction coefficients for and ε λ or ε λ, as given by Y D,500 D,440,440,500 ε ε 500 ε 440 ε [21] vi

7 We have determined the ratios of molar extinction in brackets, using excitation spectra for and. These ratios indicate that Y < We have calculated all Fs in the experiments using Eq. 20 and Y The F values were in all instances less than 0.1% different than when we set Y 0. Hence, for all practical purposes with the - pair, Y can be set to zero, yielding the F equation reported in the Methods and restated here. ll values reported in the text and figures of the paper have therefore been calculated by Eq. 22. F [ [,440,535 D1,500,530LP,440,480],440,480] [22] It is worth emphasizing again that this determination of F holds true, regardless the concentrations of donor and acceptor in the field of view, the possibility of binding and between donor and acceptor molecules, the excitation power, and the inevitable idiosyncrasies of the optical cubes involved. These factors have all been incorporated into all of the equations from which Eq. 22 is derived, and they cancel out in the final analysis. 2 ppendix on the extension of the method to characterize properties of binding between donor and acceptor molecules 2.1 Effect of incomplete labeling of acceptor molecules on calculated ratio To avoid the possibility of trivial resulting from high bulk concentrations of acceptor and donor, we intentionally limited the supply of donor-tagged CaM by using an expression plasmid pv2 with a weak promoter and no V40 ori to avoid plasmid replication by T antigen. Channel subunits, being generally harder to express than CaM, were kept in an expression plasmid pcdn3 with a strong CMV promoter and intact V40 ori. The potential consequence of using lower expression levels of donor-tagged CaM was that not every acceptortagged channel subunit would be guaranteed to have associated with it a tagged CaM. What is the impact of this possibility on the experimentally determined ratios F? The answer comes with considering the definition of F in Eq. 0, and transforming it by the mechanistic expressions for the various forms of fluorescence Eqs This provides an equation that specifies how F would change with different b, the fraction of acceptor molecules with an associated donor. The resulting equation is F 1+ D,440 E, F max b [23] where E k T / k T + k D is defined as the efficiency of a donor-acceptor pair, and the ratio of D,440/,440 is essentially equal to the ratio of molar extinction coefficients ε 440 / ε 440. The equation highlights three important features of incomplete labeling of acceptor molecules. First, the measured F varies linearly with an increasing fraction of acceptor bound to donor, according to slope F max. econd, the equation also indicates that the efficiencies calculated in Figs. 3-4 are actually effective efficiencies E eff E b. Finally, to vii

8 calculate the true efficiency E, we would need to estimate F max from some type of regression analysis based upon measured F as a function of b. The genuine value of E would be required to constrain actual distances between donor and acceptor moieties according to the Förster equation. The last point underscores the need for an experimental estimate of b, a challenge that we engage in the next section. 2.2 Using measurements to estimate the fraction of acceptor molecules bound to donor To estimate b from measurements on a single cell, we begin by considering the binding equation between donor and acceptor molecules. We assume that acceptor molecules are membrane associated like tagged channels, free donor molecules are soluble cytoplasmic moieties like tagged CaM, and the stoichiometry of donor-acceptor binding is 1:1. b is then given by the classic binding equation b 1 / K d / [D free ] [24] where K d is the dissociation constant in M units, [D free ] is the concentration of free unbound donor molecules in M units, and the factor of 2 relates to the fact that donor molecules can only bind to acceptor from the cytoplasmic side of the membrane. This can be restated in terms of the total number of donor and acceptor molecules in a cell which is always within a field of view as b 1 / [ K d V N avogadro / N D - b N ] [25] where N avogadro is vogadro s number, N D and N are number of donor and acceptor molecules in the cell, and V is the volume of the cell in litres. We can then solve this equation for b, yielding N b D + N + 2 N avogadro K d V - N + N + 2 N K V D 2 N - 4 N [26] Now we are in a position to consider optical means of estimating N D and N. From Eqs. 9 and 8, we have optical means of calculating and. From Eqs. 4 and 5, these are related to N D and N by the equations avogadro d 2 D N N D k D [1-D b / k D + D b / k T +k D ] I o D,440 F D,535 N I o,440 F,535 [27] [28] iven that efficiencies E k T / k T + k D are less than 10-20% in Figs. 3-4, k T must be less than 11-25% the value of k D in Eq. 27. Hence, to a good degree of approximation, k T can be dropped from Eq. 27 yielding viii

9 N D I o D,440 F D,535 [29] The and F terms in Eqs. 28 and 29 can be estimated by,440 F,535 C [ε λ] λ nm [ f λ] λ nm D,440 F D,535 C [ε D λ] λ nm [ f D λ] λ nm [30] [31] where C is a constant, [ε λ] λ nm is the average molar extinction coefficient of over the bandwidth of the cube excitation filter nm; [ε D λ] λ nm is the average molar extinction coefficient of over the same bandwidth; [ f λ] λ nm is the average value of the emission spectrum over the bandwidth of the cube emission filter nm; and [ f D λ] λ nm is the average value of the emission spectrum over the same emission filter bandwidth. Prior to averaging f and f D, each function is scaled such that the total area under each spectrum is equal to the quantum yield of or, respectively. The approximation relies on the fact that the optical transfer functions for the excitation and emission paths of the microscope are nearly constant over their respective bandwidths. We have calculated these averages from experimentally determined excitation and emission spectra, and find that M [ε λ] λ nm [ f λ] λ nm 0.036, and M D [ε D λ] λ nm [ f D λ] λ nm ubstituting Eqs. 30 and 31 into Eqs. 28 and 29 then yields the following expressions for N and N D N I o C M N D I o C M D [32] [33] ubstituting Eqs. 32 and 33 into Eq. 26 yields an experimentally-based estimate of b, according to b ET + ET + K d,eff + + K ET 2 ET ET d,eff 2-4 ET ET [34] ET M D [35] ET M [36] K d, EFF 2 K d V N avogadro I o C [37] ix

10 Now we are in a position to use regression analysis to estimate b in individual cells. given cell provides the experimentally determined ratio F exp and three measurements. If we pick the parameters F max and K d,eff, then Eq. 34 will translate the measurements into a prediction of b, and Eq. 23 will in turn translate the predicted b into a predicted F F predicted. We can adjust the parameters F max and K d,eff until the squared error F exp - F predicted 2 is minimized. In itself, this would not be a very stringent constraint on the parameters. However, the same F max should apply to different cells expressing variable numbers of donor and acceptor molecules. In addition, if the volume of cells V is roughly comparable, then the same K d,eff should apply to different cells. We could then apply a single pair of F max and K d,eff values to all cells, and calculate an aggregate squared error F exp - F predicted 2 summed from all cells. F max and K d,eff could then be adjusted to minimize the aggregate error, thus providing a much more stringent constraint on these parameters. This is what we have done in Fig. 5, using a numerical solver. The analysis immediately provides several dividends. First, the overall linearity of the F exp versus b plot, based upon an optimal pair of F max and K d,eff values, provides some evidence that donor and acceptor molecules interact via 1:1 saturable binding, although the scatter in the data precludes a definitive interpretation. econd, the estimated F max value permits us to estimate the true efficiency Eq. 23, which is required to engage in calculation of donor-acceptor distance. Finally, the estimated K d,eff values determined for different channel-cam interactions provides an indication of relative affinity, even without explicitly determining the relationship to actual K d values which, in principle, could be done according to Eq. 37. x