Landscape model.

The initial dispersal of crops is affected by several geographical factors such as the location of water bodies, environmental barriers, the suitability of environments to grow the crop, and prehistoric human population density. To model the relative influence of different factors, we use conductance matrices derived from gridded geographic data. In the conductance matrix, each grid cell is represented by a row with values indicating the conductance or relative ease of crop dispersal and gene flow to other cells on the grid. A grid with n cells produces an n×n cells conductance matrix. Generally, we connect pairs of spatially adjacent cells, which receive non-zero values in the conductance matrix, while unconnected pairs of cells receive a zero. Spatially non-adjacent cells could be connected in the conductance matrix to represent long-distance ‘leap-frog’ movements. Here, to keep the model simple and following a number of existing models in archaeology [18]–[20] and spatial genetics [30]–[31], we connect spatially adjacent cells only.

What we call a “conductance matrix” is called a “weighted adjacency matrix” in graph theory. In this context, however, we prefer the term “conductance matrix” to avoid confusion, as adjacency between nodes in the graph derived from the grid does not necessarily imply spatial adjacency between the cells represented by the nodes. A further reason for this terminology is that the ease of transition can be seen as equivalent to conductance in electrical terms [30], [35]–[36]. Grid cells represent the nodes of a mesh, each connected to its neighbors by resistors. Conductance is the reciprocal of resistance (conductance = 1/resistance), which in turn is equivalent to friction or cost, which are terms more commonly used in geospatial analysis.

Using conductance matrices has a number of advantages. In geospatial analysis, least-cost distances are generally calculated from a cost or friction grid [37]. However, a conductance matrix is more versatile as it can represent connections between non-adjacent cells and anisotropy (e.g., the friction from cell i to j being unequal to the friction from j to i). Also, a conductance matrix can be used directly to derive distance metrics based on random walks (see below). Conductance matrices generally contain a large number of zeros and few non-zero values. Hence, conductance matrices can be handled as sparse matrices because most values are zero. Sparse matrices only store (indexed) non-zero values, which is very efficient memory-wise. Also, fast computational methods are available for sparse matrices.

Conductance values are determined from the values of the two grid cells that are connected, using different functions. Simple functions, such as the average, or functions that require parameters can be used. The conductance values need to be corrected for (1) differences between diagonal and non-diagonal connections between cells if cells are connected in more than four directions and (2) distance distortions, specifically the decreasing W-E distance between cell centres on a longitude-latitude grid when moving from the equator towards the poles. Both issues are addressed by dividing conductance values by the distances between the cell centres.

We refer to the final conductance matrix used to calculate the distance metrics as the landscape model (note that the term ‘landscape’ does not imply a certain range of geographical extent or resolution; our landscape model can cover an entire continent). The landscape model is constructed by combining various conductance matrices for the selected variables that might influence crop dispersal. Conductance matrices based on different variables are combined into a single conductance matrix (giving each variable a certain weight), which forms our landscape model. The complexity of the landscape model can vary, depending on the number of conductance matrices (i.e., weight parameters) and the number of parameters required by the function(s) used to determine the values in each of the conductance matrices.

Given a certain geographical origin of crop dispersal and a landscape model, we can predict the movement of crops in geographical space and, consequently, the age of archaeobotanical crop remains, heterozygosity, and genetic distances between crop populations. The following sections discuss the construction of predictor variables from the landscape model and the model-fitting procedure. The modelling approach requires measures of goodness-of-fit between the landscape model on the one hand and the archaeobotanical and genetic data on the other. To derive these measures, we use and adapt elements of existing modelling approaches in both archaeology and geographical genetics.

Modelling crop remain radiocarbon ages.

We use a variant of existing spatial diffusion models in archaeology for the post-domestication diffusion of crops [16]–[20]. Given a landscape conductance matrix and the location of the putative cradle area of the crop, we calculate the least-cost distance to all archaeological site locations for which we have dated prehistoric crop remains, following [18]. These distances are then used as a predictor of the arrival date of the crop at those sites. The earliest dates for each area are the most relevant, as these are indicative of the introduction of the crop to that area, while later dates correspond to local expansion of the crop within the area or to a failure to detect earlier crop remains. To focus our analysis on the sites with the oldest crop remains rather than the average age for a given area and to avoid defining discrete areas, we use quantile regression [38]. Quantile regression is used for regression on a quantile (τ), like the median (τ = 0.5). By setting τ to a relatively extreme value (τ>0.75), we increase the influence of the oldest sites in the analysis. Quantile regression is robust in dealing with skewed distributions and outliers, which makes it especially suited to our approach, which precludes checking error distributions and removing outliers. We use the pseudo-R² of quantile regression, R¹, as the goodness-of-fit [39].

Modelling heterozygosity.

Dispersal is expected to leave a mark on the diversity within and between populations. During the expansion of humans out of Africa and spread across the world, each time generally small groups split off to occupy new areas, taking with them only a portion of the alleles from their original population. As a result, human populations show a regular decline in heterozygosity from Africa to the southern tip of South America [32]. For crops, a similar effect is to be expected. In grain crops, mostly whole infructescences are selected for seed. This reduces the number of maternal parent plants and hence the effective population size. If seed lots are relatively small, genetic bottlenecks occur. In established traditional farming systems, pollination between fields and seed mixing tend to counteract the resulting loss of alleles and maintain diversity levels [40]–[41]. During range expansion, however, seed lots are taken into new territory, beyond the reach of these diversity-restoring processes. Also, seed quantities during crop expansion may have been limited by several causes. If crop expansion was due to human migration, such migration movements may have been motivated by push factors like marginalization or persecution. If so, migrants may have arrived with few resources, including seeds. If crop expansion was a process of cultural exchange, hunter-gatherers with no agricultural experience or farmers adopting a new crop were the ones who took the crop further into new territory. Therefore, cultivation in these new locales must often have been precarious and experimental in nature, possibly leading to small seed lots and low crop plant survival rates. Therefore, we think it is reasonable to expect a loss of diversity during crop dispersal, resulting in a declining gradient of crop diversity from the origin.

Like for crop remain ages, the least-cost distance from the origin of the crop should therefore be a good predictor for heterozygosity levels. Here, for simplicity, we determine the cost distance to predict heterozygosity from the same landscape model as we use for crop remain ages. This assumes that the loss of diversity due to genetic bottlenecks through each area is proportional to the time it took to cross these areas. This may not be realistic. The intensity of genetic drift is related to population size, which may change over time and among agricultural systems. Also, the number of bottlenecks over a given distance may differ. See below, under Discussion–Next modelling iterations, for a refinement of this aspect of the approach.

Selection, introgression from wild populations, as well as recent founder effects and subsequent hybridization may all confound the spatial pattern of heterozygosity. However, as long as the pattern is mainly due to the initial wave of dispersal and not to subsequent long-distance gene flow events or introgression from wild relatives, the net effect of these subsequent demographic events will be to decrease heterozygosity locally. If this is the case, the upper limit of heterozygosity will be largely determined by the least-cost distance from the crop origin. Heterozygosity levels that fall short of this maximum will have undergone more recent drift or selection. Only the upper limit of heterozygosity contains information about prehistoric crop dispersal. This problem is similar to the one encountered with the crop remain age data, where we are also primarily interested in the highest values of each area. Therefore, the quantile regression approach introduced above can be used for heterozygosity as well.

Modelling genetic distances.

During dispersal, the genetic divergence between populations is due to the progressive isolation of populations as their trajectories split. The earlier trajectories split, the more genetic divergence is to be expected. On the other hand, populations that share a large part of their trajectory will undergo a common loss of alleles (alleles which may continue on pathways in other directions from the origin) and have a higher degree of common ‘surfing’ alleles, which have emerged at intermediate locations [42]. Both effects will lead to a higher genetic similarity between populations that share a longer trajectory from the origin.

Ramachandran et al. predicted genetic distances between human populations with distances along dispersal routes out of Africa through waypoints [32]. This gave better results than predictions with direct geographic distances between the sampled populations (as in an isolation-by-distance model). The distance via migration waypoints corresponds to the divergent part of the prehistoric migration trajectories of each pair of populations. Hence, genetic distances between human populations reflect their migration history and arguably the same is the case for crops. We extend this approach in two ways: (1) using a grid-based landscape model, as described above, thus obviating the need for discrete waypoints, and (2) taking into account not only the divergent part of trajectories, but also the length of the shared part. Again, we assume that genetic drift during crop dispersal was constant in time.

During crop dispersal, the first varieties to reach a new place will have more probability to be taken further than varieties that arrive there later. Hence, the dispersal route of individual alleles will be close to the shortest (least-cost) path from the origin location to the location of the sampled population. However, there will also be movements of sideward gene flow along the expansion front that will canalize genes towards parallel paths, bringing in a random element. Random walks can be modelled with analytical methods, using the analogy with electrical current, to avoid repeated simulations to determine probabilities [35]–[36]. Random walk distances are useful to model genetic distances in heterogeneous landscapes for gene flow in an equilibrium situation [30]. However, gene flow during range expansion is intermediate between a least-cost path and a random walk. This could be represented with randomized shortest paths, which allow varying the degree of constraint to the least-cost path, changing the value of θ [43]. Parameter θ can be varied between 0 (random walk) and ∞ (shortest path). A randomized shortest path is also somewhat longer than the shortest path, but in this context, the correlation between the two types of distances is high and tends to 1 when θ approaches ∞.

For a given origin, destination, conductance matrix, and a value for θ, we calculate the net number of times of transition over each cell connection, i.e., the number of transitions not reciprocated by transitions in the opposite direction. We take this to correspond to the probability of passage (P) of the forward movement of an allele during the wave of expansion. With a given origin, we calculate the matrix P for each sample location. Each transition probability matrix P_a represents the stochastic trajectory from the origin to point a. The probability that two different trajectories (P_a and P_b) coincide in connections between cells can be calculated by multiplying the two matrices:(1)P_joint is a matrix with the probabilities of joint passage for each cell connection. Likewise, to determine to what extent connections between cells are part of the divergent part of the trajectories, we calculate the probability that the most probable trajectory crosses a cell connection and the least probable trajectory fails to do so. If this probability exceeds the probability that the least probable trajectory crosses the cell, there is enough asymmetry between the trajectories to consider the cell connection as part of the divergent part of the trajectory:(2)Figure 1 illustrates these calculations by showing the transition probabilities by cell. We multiply the obtained matrices P_joint and P_disjunct with the resistance matrix, R. (Here, we determine R as the reciprocal of the conductance matrix of the landscape model, but see below under Discussion–Next modelling iterations for an extension to this.) We then sum the values of the whole matrix.(3)(4)We repeat this procedure for all pairs of sample locations. The two obtained variables, path overlap and path divergence, can be compared with the pairwise genetic distances using regression methods.

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Figure 1. Example of trajectory overlap and divergence calculation for two populations.

Both populations are from South America (point locations A and B). The origin of the crop is in Mexico. A. Probability of passage from origin to location a. B. Probability of passage from origin to location b. C. Overlap of the trajectories. D. Divergence of the trajectories.

https://doi.org/10.1371/journal.pone.0012060.g001

Fitting the model.

Using the computational strategies outlined above, we can derive from the landscape model different distance measures that relate to crop remain age, heterozygosity, and genetic distances. We use multiple-criteria optimization to evaluate how well our model can explain the archaeobotanical and genetic observations. Multiple-criteria optimization is an underutilized technique with interesting applications to detect conflicts between model structure and patterns in the data [44].

Multiple goodness-of-fit values are determined by regression of the predictors against the archaeobotanical and genetic data. A genetic algorithm optimizes two or more goodness-of-fit measures through an iterative search of the best parameter values and origin coordinates. The outcome of this optimization is a Pareto front of solutions. Pareto solutions are those for which improvement in one goodness-of-fit dimension can only occur with the worsening of at least one other goodness-of-fit dimension. The shape of the Pareto front gives a good indication of the degree of convergence or conflict between the two datasets, given the model structure. A pointed, convex front (seen from the cloud of possible solutions) is evidence for convergence around the same solution. Inspecting the parameters and origin coordinates of the different solutions can provide insights into the source of the conflict and thus help in improving the model structure.

Computational implementation.

The data analysis was done in R, a free and open-source computer program and language for data analysis [45]. The examples can be replicated with the script and data provided as Supporting Information. Basic geographic grid manipulation and calculations were done with functions from R package (plug-in) raster [46]. The creation of conductance matrices from grid data, their manipulation, and the calculation of geographic distance measures were supported by functions in gdistance [47]. Package gdistance makes use of sparse matrices [48]. Methods to analyze distance matrices and calculate genetic distances are implemented in gdistanalyst [49]. We used the R package mco for the multi-criteria optimization [50], which implements the algorithm NSGA-II [51]. An R script to replicate the procedure is available as Supplementary Information (file S1).

Landscape model.

We modelled maize dispersal and diversity with a simple landscape model. We used a grid of 0.5 by 0.5 degree resolution, covering the study area. Grid cells were connected in eight directions (queen's case) to form conductance matrices. The area of origin was modelled as a single cell, which could be anywhere on land. The landscape model includes only information about the shape of the landmass to keep our example as simple as possible, but additional variables could be added (see Discussion). The landmass grid is included as Supplementary Information (file S2).

The conductance of between-cell connections on land was set to 1. The conductance of major water bodies was modelled with a decay function and a weight relative to the conductance of the landmass (p₁), following [20]. Conductance decays with the distance away from the coast (, the average distance from the coast of cell i and j) with a constant decay rate (p₂, the conductance half-value distance). Conductance over water bodies was calculated as(5)We symmetrically normalized the water body conductance matrix [52]. All conductance values (land and water) were divided by cell-to-cell distances (i.e., distances between cell centres) in order to correct for the differences between diagonal and straight distances between cells and to correct for variations in spherical distances between cells.

Archaeobotanical data.

We limited the analysis to macrobotanical remains of maize, which at the moment is the only type of archaeological remains of this crop for which there is a somewhat complete coverage of the Americas. Radiocarbon dates were derived from [53] and supplemented with a number of additional dates from other publications (Supplementary Information, file S3). We calibrated the raw dates with OxCal [54], using IntCal04 [54] for the Northern Hemisphere and ShCal04 [56] for the Southern Hemisphere. In the tropics, we took a weighted average of the two median values, the weights depending on the latitude of the sample location. Outside the tropics, we used the corresponding median values. In a small number of cases, uncalibrated dates were not available and published calibrated dates were used directly.

Genetic data.

We used genetic data from [22] on 193 maize landrace accessions from across the Americas, based on one plant per sample and 99 SSR markers. We calculated the heterozygosity for all samples and used the logarithm of the shared proportion of alleles as the genetic distance between the samples. As we reorganized the data, in order to replicate the procedure, we supply it here as Supplementary Information (files S4 and S5).

Model fitting and evaluation.

We first optimized the landscape model with (1) the age of the archaeobotanical crop remains and (2) the heterozygosity of contemporary maize samples. For a number of the Pareto solutions obtained, we then evaluated the goodness-of-fit with (3) the genetic distances. We choose this setup in two rounds to reduce computation time and to test the performance of our new path overlap and divergence metrics independently. The path overlap and divergence metrics should predict the genetic distances well if the modelling approach is coherent.

In the first round, the goodness-of-fit was determined with quantile regression, setting τ to 0.8 for both radiocarbon age and heterozygosity. Since the archaeobotanical data were highly unequally spread with an especially high density of observations in Colorado, New Mexico, and Arizona, we weighted each observation by 1/number of observations within a radius of 100 km from that observation (including the observation itself). Hence, an observation with one neighbour within 100 km distance received the weight 0.5, while an isolated observation was weighted as 1.

We optimized with a population of 200 during 60 generations. Further improvements after 200 generations were minimal and solutions showed a regular pattern. We selected for further analysis a subset of nine representative Pareto solutions. We calculated path overlap and divergence based on these solutions for various values of θ. We evaluated the correspondence between path overlap/divergence and genetic distances with linear permutational regression with 999 permutations [57].