Dataset: phyto temperature traits
Deployment: Litchman_2012

Text here is modified from the supplementary information of Thomas et al. (2012):

Statistical analysis of thermal tolerance curves

Temperature dependent specific growth rates can be described by the following equation:

f(T) = aebT[1-{(T-z)/(w/2)}2]    - (S.1)

Here specific growth rate f is an explicit function of temperature, T. The shape of the thermal tolerance curve is controlled by two important species traits, z and w. The range of temperatures over which growth rate is positive, or the thermal niche width, is given by w. Species trait z determines the location of the maximum of the quadratic portion of this function. In the case where parameter b = 0, this value is identical to the temperature at which a species achieves its maximum growth rate. However, when b is non-zero, the maximum value of S.1 falls above (or potentially below, b <0) the value of z, and can be found through numerical optimization.

We allowed a and b to be free parameters, fit simultaneously with z and w. To describe the growth data for each isolate, we used a maximum likelihood approach, such that the mean growth rate at a given temperature followed equation (S.1),

µ = f(T) + N(0, σ2)                   -(S.2)

Here observational error was described by a normal distribution with a mean of zero and variance of σ2.

Determining physiological parameter uncertainty

While confidence intervals for the point estimates of the parameters of (S.1) were easy to obtain, it was not straightforward to determine uncertainty for implicit properties such as the temperature at which growth rate is maximized (or, the ‘optimum temperature’). Yet, this was the property that we were mainly interested in, leading us to adopt a parametric bootstrapping approach.

We used a Monte Carlo approach such that for each thermal tolerance curve having n data points, we simulated n new data points, drawn from a normal distribution such that:

1) The mean of the distribution corresponds to the value of (S.1) at each of the original experimental temperatures, given the coefficients previously estimated for the original curve.

2) The standard deviation of the distribution, σ, was obtained by adjusting the original maximum likelihood estimate, to account for uncertainty in its estimation.

Equation (S.1) was then fit to the simulated data using maximum likelihood estimation, and the new parameter values, as well as the numerically estimated optimum temperature, were retained. Repeating this process a total of 10,000 times (for each isolate), yielded bootstrapped distributions of all parameter estimates. From these distributions we calculated the 95% confidence intervals as the range between the 2.5th and 97.5th quantiles. 

Related Reference:

Thomas, M. K., Kremer, C. T., Klausmeier, C. A., & Litchman, E. (2012). A  global pattern of thermal adaptation in marine phytoplankton. Science, 338(6110),  1085-1088.  DOI: 10.1126/science.1224836.

Supplemental materials: http://www.sciencemag.org/content/suppl/2012/10/25/science.1224836.DC1/Thomas.SM.pdf