Mathematical description of the “Stress Addition Model” (SAM)

The 3 principal assumptions for the combined impact of independent stressors, in this case, toxicants and additional environmental stressors can be described as follows:

The first assumption of SAM is that each individual has a certain capacity to tolerate all types of stress, its general stress capacity. We assume this individual stress capacity to be symmetrically distributed over the finite interval [0, 1]. We argue that the stress-dependent population sensitivity follows the same distribution. Individuals with a stress capacity below a given stress level S will die, whereas individuals with a stress capacity above a given stress level will survive. Hence, the stress-dependent population sensitivity is parameterized by the following beta distribution

where p(S) represents the probability density of individuals to tolerate a general stress S, p and q as the non-negative shape parameters of the distribution and B(p, q) as the beta function which is a normalization constant to ensure that the total probability integrates to 1. We postulated symmetry of individual stress capacity (p = q). The parameters were set to p = q = 3.2 which resulted in the best fit between observed and predicted LC10 and LC50 shifts of the 23 experimental study-pairs (see Figure S2). Best model fit was determined by linear least-squares fitting which was applied on studies with an environmental stress mortality of greater than 0%. Goodness of fit was quantified as R² that gives the proportion of variance explained by the model (shift of LC10: R² = 0.49; shift of LC50: R² = 0.38; see Figure S2).

The integral of the density function gives the population size N under non-stress conditions

The stress-dependent survival is calculated as

where N(S) = 1 (100% survival) for the general stress S = 0 and N(S) = 0 (0% survival) for the general stress S ≥ 1.

(2) The second assumption of the SAM is that every specific unit of a given stressor can be transferred into a general stress level. This conversion uses the stress-related mortality as a linking factor. For instance, if a temperature stress or toxicant stress causes a mortality of 10%, the general stress level is given by the 10% quantile of the beta distribution in Equation 2.

(3) The third assumption of SAM is that the general stress levels of independent stressors are additive, with the sum determining the total general stress exerted on a population. The total general stress S is given as the sum of general stress levels S_i of all independently acting stressors.

All studies of the meta-analysis combine a fixed environmental stress level S_ENV with an increasing toxicant stress level S_TOX. Accordingly, the general stress S, calculated for an environmental stressor and a toxicant stressor, is thus expressed as follows:

The resulting survival of the population exposed to the general stress S can be determined applying equation 4. Due to the low number of suitable studies (n = 23) we used the information of all experiments to parameterize the SAM. A validation of the approach requires additional experiments. The practical use of the SAM is facilitated by an Excel spreadsheet available as download within the SI (SAM_Calculator.xlsx).

The SAM combines empirical knowledge with mechanistic concepts in order to predict the combined effects of stressors. The empirical “share” comprises the parameterization of the beta distribution and also the parameterization of the average concentration-response relationship over all studies of the meta-analysis (Fig. 3). The mechanistic “share” of the SAM is based on the concept of subtracting general stress levels, the common currency of various stressors, from the individuals stress capacity as illustrated in Fig. 2.