Overview and underlying assumptions
The eDITH model (Carraro et al. 2018; Carraro, Mächler, et al. 2020; Carraro, Stauffer, and Altermatt 2021; Carraro, Blackman, and Altermatt 2023) makes use of spatially replicated eDNA measurements within a river network to infer the spatial distribution of any taxon (or OTU, or ASV; hereafter simply “taxon”) of interest across that river network. The key underlying concept is that eDNA particles are advected downstream by streamflow, and hence an eDNA sample is not only representative of the location where it is taken, but it provides information about taxon abundance (in the case of qPCR single-species data) or biodiversity (for metabarcoding data) for a certain area upstream of the sampling location. By exploiting information from multiple sampling sites distributed in space, and embedding a model for transport and concurrent decay of DNA, the eDITH model is able to disentangle the various sources of DNA shedding (and hence, of the target taxon’s abundance). As a result, eDITH complements pointwise eDNA measurements by projecting taxon distributions (and thus biodiversity information) into space-filling catchment maps.
DNA production and decay rates
The DNA production (i.e., shedding) rate of a taxon in stream water is assumed to be proportional to its density (Lodge et al. 2012; Apothéloz-Perret-Gentil et al. 2017). The validity of this assumption is restricted to the time point when eDNA is sampled. Thus, it is irrelevant if a taxon varies in its release of DNA across the season, as long as DNA released is proportionate to taxon density at the time of sampling. For the same reason, hydrological variables (such as discharge and velocity) are assumed to be constant in time (not in space!) across the duration of a sampling campaign. For instance, if multiple days are required to perform eDNA sampling across all sampling sites, these days would be assumed to constitute a unique time point, during which hydrological conditions can be adequately represented by time-averaged values of discharge and velocity.
The decay of DNA is assumed to be expressed by first order kinetics (i.e., the rate of change of DNA concentration decreases linearly with time). Similarly to the production rate, it is assumed that a single value of decay time is representative for a given taxon at a given time point.
Sampling design
The spatial extent of the eDNA sampling must adequately cover the entire river network, in order to allow the model to make robust predictions. As a rule of thumb, there should be one sampling site per each 10–20 km2 of drained area.
Moreover, all main tributaries of a river network should be sampled in a spatially hierarchical design. Ideally, sampling sites should be located just upstream of a confluence, so that the independent signals from the joining tributaries can be gauged. See (Carraro, Stauffer, and Altermatt 2021) for further information on optimal eDNA sampling strategies.
The mean distance (along the river network) between eDNA samples should not be too large, otherwise the eDNA found at an upstream site would be totally depleted before reaching the next downstream site. In this case, the eDITH model could not make inferences based on these multiple sites. To make an example, if we consider first order kinetics, a water velocity of 1 ms-1 and a decay time of 4 h (i.e., a half life of 2.77 h), we would be able to measure only about 50% of the concentration that we would measure 10 km upstream, and 3% of the concentration that we would measure 50 km upstream. As a rule of thumb, sites that are more than 50 km apart (along the river network) should be considered as independent samples. See also (Deiner and Altermatt 2014; Pont et al. 2018) with respect to transport distances of eDNA.
Finally, the model assumes that eDNA is well mixed in the water column, and/or that samples at a site adequately span the river cross section (e.g., sampling at both banks and at the centre, for large rivers, while in small rivers sampling at a single bank might be sufficient, by assuming that eDNA is sufficiently well mixed). Analogous care should be taken in the choice of a representative sampling volume to be filtered (Altermatt et al. 2023).
Measurement errors
Variability in DNA detection through sampling (e.g., filtration) or laboratory procedures (e.g. DNA extraction, Polymerase Chain Reaction (PCR), or sequencing) do not introduce a systematic bias in DNA concentrations across samples.
If eDITH is used to model metabarcoding data, the expected read count at a given site for a given taxon is assumed to be proportional to the underlying DNA concentration.
Replicated eDNA measures at the same site and time point are treated as independent measures. No distinctions is made on whether these are physical (i.e., different water samples) or laboratory (different PCR runs) replicates.
What taxa can be modelled?
In principle, the eDITH model is suitable for aquatic macroorganisms that shed their DNA into stream water. Nonetheless, the model can also be extended for microorganisms that are sampled in their entirety (e.g., free floating bacteria), in which case their upstream sources can be considered as biofilm colonies, and the decay time does not refer to DNA molecules but rather to the bacteria lifetime.
The eDITH model does not make a distinction on whether the eDNA data is referred to an assigned taxon (say, at a species, genus or family level) or an unassigned cluster such as OTUs or ASVs. What matters is that the assumptions on production and decay rates (see DNA production and decay rates) must hold.
In principle, it is also possible to estimate the spatial distribution of terrestrial, rather than aquatic, taxa, provided that the assumption of proportionality between DNA production rate and taxon density holds. In this case, the spatial unit on which predictions are performed is the subcatchment, i.e., the portion of land that directly drains towards the associated reach.
River network model
The domain on which the eDITH model is applied is a river network discretized into reaches, i.e. segments of river not interrupted by confluences and treated as smallest spatial units. Each reach is considered as a node of the network, and the ensemble of the reaches covers the entire river network. It is assumed that reaches have no internal variability (e.g., the exact coordinates of a sampling site do not matter, provided that the sampling site is associated to the same reach). Note that the along-stream distance between two consecutive confluences can be covered by a sequence of different reaches, to allow for a finer discretization of the river network. Indeed, number and maximum length of reaches can be tuned by using function aggregate_river of rivnet. See also (Carraro, Bertuzzo, et al. 2020) and the documentation of the OCNet package for details on aggregation of a river network into reaches.
Lakes and braided channels
The eDITH model is designed to work for river networks not containing lakes or reservoirs. Dynamics of eDNA in lakes are still largely unclear, but the very large residence times of water particles in lakes compared to those of a river reach imply that all dissolved DNA entering a lake either degrades or is retained by substrate particles. Particle-bound DNA can eventually be resuspended (Shogren et al. 2017), but this likely occurs at long time scales (say, monthly, yearly or even longer), hence the assumption of first-order decay kinetics with a single decay time value across the river network would be violated.
Similarly, the river networks produced by rivnet do not admit bifurcations in the downstream direction (e.g., braided or artificial channels creating loops in the river network). Hence, the different braids or channels cannot be treated as independent reaches in the eDITH model. In such a case, the eDITH model could still be applied by considering a single channel as a conceptual equivalent of the multiple real braids or channels.
The governing equations
The main equation of the eDITH model results from a mass balance of eDNA across a cross-section of a river, and it reads:
\[\begin{equation}
C_j = \frac{1}{Q_j}\sum_{i \in \gamma(j)} p_i A_{S,i} \exp\left(-\frac{L_{ij}}{\overline{v_{ij}} \tau}\right)
\end{equation}\]
where:
- \(C_j\) is the DNA concentration at a sampling site \(j\) (strictly speaking, \(j\) is the reach where a sampling site is located);
- \(Q_j\) is the water discharge in \(j\);
- \(\gamma(j)\) identifies the set of reaches upstream of \(j\);
- \(p_i\) is the DNA production rate at an upstream reach \(i\);
- \(A_{S,i}\) is the source area in \(i\) (i.e., extent of the node; considering a river reach and an aquatic taxon, this could e.g. be equal to the product of its length and width);
- \(L_{ij}\) is the length of the along-stream path joining \(i\) to \(j\);
- \(\overline{v_{ij}}\) is the average water velocity along \(L_{ij}\);
- \(\tau\) is a characteristic decay time for DNA in stream water.
Assuming that the morphology and hydrology of the river network are known (and thus lengths, areas and discharges), the above equation links DNA concentrations to the unknown parameters \(\tau\) and \(\mathbf{p}=\left( p_1, \dots, p_N\right)\) (where \(N\) is the total number of reaches). While the former is linked to the behaviour of DNA in stream water (and could in principle be measured, or at least its value be constrained), the estimation of the latter is the actual goal of the eDITH model. Contrasting observed and modelled DNA concentrations thus enables the estimation of maps of \(\mathbf{p}\) across all \(N\) reaches constituting the river network, and hence of relative taxon density, given the initial assumption.
It is often convenient (both from a modelling and interpretation viewpoint) to express the DNA production rate \(p_i\) as a function of environmental covariates, possibly related to the spatial patterns of the investigated taxon:
\[\begin{equation}
p_i = p_0 \exp\left( \boldsymbol\beta^T \mathbf{X}(i) \right)
\end{equation}\]
where \(\mathbf{X}(i)\) is a vector of covariates evaluated at reach \(i\); \(\boldsymbol\beta\) a vector of covariate effect sizes; and \(p_0\) a baseline production rate, i.e., the DNA production rate at a site where all covariates are at a null level. This reduces the number of unknowns from \(N\) (size of \(\mathbf{p}\)) to the number of selected covariates (length of \(\boldsymbol\beta\)) plus one (\(p_0\)).
If the eDNA data are in the form of read counts, the above equations remain applicable. Indeed, \(C_j\) would play the role of the expected read number at site \(j\) (proportional to the underlying DNA concentration, as per the initial hypothesis). Consequently, \(\mathbf{p}\) would in this case represent the DNA production rates multiplied by such constant of proportionality.
Estimating detection probability
Once model parameters are estimated (either via a Bayesian method or likelihood maximization), it is possible to transform DNA production rates \(\mathbf{p}\) into corresponding detection probabilities. This is done by exploiting the assumption on the probability distribution used to model measurement errors (and hence formulate the likelihood, see Likelihood function and model parameters).
In particular, this is done by calculating, for each reach \(j\), the expected eDNA value \(\widetilde{C_j}\) (concentration or read count) if that reach were detached from the river network (that is, in the absence of upstream inputs, with the water discharge in the reach being equal to the locally produced discharge). This is then transformed into a detection probability value, calculated as the probability of observing a non-null eDNA value under the assumed error probability distribution and the expected value \(\widetilde{C_j}\).
The so-obtained detection probability maps can be further transformed into presence-absence maps and used to assess biodiversity patterns (e.g., Carraro, Blackman, and Altermatt 2023).
