Constructing and Using Irradiance Response Curves

For each wavelength, plot the IRC as log photon flux (x-axis) vs response magnitude (y-axis) as shown in Fig. 5A. The EC50 is then calculated from this

a LOG photons

478 nm template

350 400 450 500 550 600

B Wavelength (nm)

Fig. 5. Principles of constructing action spectra. (A) Irradiance response curves are first constructed, plotting the logarithm of light dose against biological response. A sigmoid model is then fitted to the data. (B) Relative sensitivity is then calculated from the photon dose required to elicit a 50% response at each wavelength (EC50). This is plotted on a logarithmic scale and fitted against a visual pigment template (see Subheading 5.5. for details).

IRC by fitting a sigmoid function to the IRC. A sigmoid curve may be defined by four parameters (Eq. 3):

where "Top" and "Bottom" correspond to the maximum saturating response and baseline response, respectively; the EC50 is the photon flux necessary to elicit a 50% maximum response (around which the curve rotates); and k is the Hill slope of the curve. Ideally, one should obtain a saturating response for every wavelength, and use a four-parameter model for every IRC. However, in practice this is often not possible, as it may not be feasible to produce a sufficient photon flux to produce a maximal response from the available light source. If the saturating response is known, then this value may be used as the maximum for all IRCs. In many biological systems no detectable response is obtained with a low photon flux, and thus the baseline can be set to 0. This leaves just two remaining parameters to fit, in what is often termed a "global" model (ref. 34; and see Note 5). Unlike linear regression, the best fit of a nonlinear model such as a sigmoid curve cannot be solved analytically, and numerical methods are therefore required. This requires an iterative approach to minimize the sum of squares of the model (SSmodel), which is equal to the sum of the difference between the data and the sigmoid model, squared (so as to become sign-independent) at each point on the IRC, i.e., (data - model)2 (Fig. 6A). This is compared with the total sum of squares (SStotal), which assumes a straight line through the mean (Fig. 6B). The measure of fit, R2, is a value between 0 and 1, and may be thought of as the fraction of the variance explained by the model, and is derived from 1 - (SSmode/SStotal). When R2 = 0, then the model is no better than a straight line through the data, and when R2 = 1, the model describes the data perfectly (R2 is normally used to denote the fit of nonlinear regression, whereas r2 is used to denote linear regression).

Most statistical packages (e.g., GraphPad) will perform these dose-response curves to give the best possible fit. However, the above is provided to enable researchers to have a better understanding of the calculations these packages are actually performing; they may be emulated simply within Excel using the Solver Add-in (Microsoft) for those without access to such statistical software (see Note 6). In addition, the same protocol is required for fitting the resulting action spectra with a photopigment template.

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