Predict Selection Response

This procedure predicts the short-term response to selection on shape, using a user-specified selection vector and a genetic and phenotypic covariance matrix. The selection vector can either be a selection differential or a shape variable that is a scaled version of a selection gradient.

Background

These analyses are based on the multivariate breeders' equation: Δμ = GP-1s = , where Δμ is the response to selection, G is the additive genetic covariance matrix, P is the phenotypic covariance matrix, s is the selection differential and β is the selection gradient (Lande 1979).

The selection response Δμ can be thought of as the difference between the mean shapes of the parental and offspring generations. The selection differential s can be thought of as the difference between the shape means in the parental generation before and after selection (under truncation selection), or as the vector of covariances between the shape variables and relative fitness. The selection gradient β can be interpreted as the vector of regression coefficients from a regression of relative fitness on shape.

Together with estimates of G and P, either the selection differential or the selection gradient are sufficient to predict the response to selection.

Selection differential versus selection gradient

The selection differential can be interpreted as a vector in shape (tangent) space. Therefore it can be directly displayed as a shape change.

The difficulty with the selection differential is that it does not support an interpretation about the features of shape that are targeted by selection, because phenotypic covariances can 'distribute' selection to other shape features. Even if selection is acting just on one aspect of shape, the averages of selected and unselected individuals will also differ for other aspects if the respective shape variables are correlated phenotypically.

In contrast, the selection gradient specifically 'holds constant' the effects of those shape features that are not under selection, but are phenotypically correlated. Therefore, the selection gradient is usually favored for studying directional selection (Lande 1979; Lande and Arnold 1983).

However, the selection gradient is not a vector in shape space (e.g. it is in units of change in relative fitness per unit of shape change, not in units of shape change; Klingenberg and Leamy 2001; Klingenberg and Monteiro 2005). It is not entirely impossible to display β as a shape change, but it takes an algebraic 'trick': introducing an auxiliary shape variable a = β(βTβ)-0.5, which has a direction corresponding to β, and can be related to it by a proportionality constant c, so that β = ca (Klingenberg et al. 2010; Martínez-Abadías et al. 2012).

MorphoJ offers both possiblities for specifying directional selection. Users should use either one with a considerable amount of caution.

Projection to the tangent space

Because shape variation is limited to the shape tangent space in the analyses implemented in MorphoJ, selection gradients and selection differentials are automatically projected to the tangent space before computations of the multivariate breeders' equation.

This projection step brings the selection differential or gradient into the same space as the variation characterized by the G and P matrices (assuming that they have been computed from coordinates generated in MorphoJ). There is therefore no selection for components such as variation in centroid size or overall translations or rotations, which will be unattainable because they have been excluded from the variation in the data by the Procrustes fit.

This step has been added to MorphoJ and has not been part of the procedure as initially described (Klingenberg and Leamy 2001). The modification has been introduced in more recent applications of the approach (Klingenberg et al. 2010; Martínez-Abadías et al. 2012).

Total, direct and correlated response to selection

The predicted selection response Δμ usually is in a direction that is different from the direction of selection that was entered into the analysis. Therefore, MorphoJ uses a decomposition of the total response predicted by the breeders' equation into a component of direct response in the direction of the selection gradient or differential and a component of correlated response perpendicular to it (Klingenberg and Leamy 2001).

The user can choose at the outset of the analysis whether the direction of the selection gradient or of the selection differential is to be used for defining the direct and correlated response.

Requesting the analysis

To obtain a prediction of selection response, the user needs to provide estimates of the additive genetic and phenotypic covariance matrices and of selection.

The estimated G and P matrices need to be obtained outside of MorphoJ, for instance, using the 'animal model' based on Procrustes coordinates or principal component scores previously exported from MorphoJ. Such estimated covariance matrices can be imported (Import Covariance Matrix in the File menu) and, if needed, converted from the coordinate system of PC scores back to the original landmark coordinates (Convert Covariance Matrix to Landmark Coordinates in the Variation menu).

For specifying the selection vector, MorphoJ provides a graphical user interface. Users can use estimated selection gradients or differentials from empirical data, or they can simulate selection regimes defined according to anatomical or developmental criteria (e.g. Klingenberg and Leamy 2001; Klingenberg et al. 2010; Martínez-Abadías et al. 2012).

Once the genetic and phenotypic covariance matrices are available in the MorphoJ project, the user can request the analysis by selecting Predict Selection Response from the Genetics menu. A dialog box like the following will appear:

At the top of dialog box, there is a text field for entering a name for the analysis, which will be shown in the Project Tree.

The two buttons below that are for selecting whether a selection gradient or selection differential is to be defined. To determine whether the direction for assessing the direct response to selection.

Moreover, there are two drop-down menus for selecting the genetic and phenotypic covariance matrices.

To go on to define a selection gradient or differential, click Continue. Alternatively, the procedure can be stopped by clicking Cancel.

Specifying a selection gradient

If the user selected to define a selection gradient in the preceding dialog box, a user interface like the following will appear:

The largest part of the user interface is taken up by a graphics window that is similar to other graphs used in MorphoJ to show shape changes, including similar functionality of the popup menu. For instance, it is possible to change the type of graph, adjust the scale factor, or to select a different pair of axes to be displayed (for 3D data).

In this graph, however, it is possible to drag the landmark points to specify a shape change. A dragging movement in the graph can be undone by selecting Undo Last Move from the popup menu.

Alternatively, the elements of the selection gradient can be entered into the table to the right of the user interface. It is also possible to paste values from another source (e.g. an Excel spreadsheet) into this table. The graph to the left will adjust accordingly.

Clicking the Clear button will reset all elements of the selection gradient to zero.

The button To tangent space will perform the projection of the selection gradient to the shape tangent space.

The magnitude of the selection gradient is displayed in a text field as the standardized selection gradient (amount of change in relative fitness per standard deviation of change in the auxiliary shape variable a = β(βTβ)-0.5). The value in this text field is updated after every change of the selection gradient in either the graph window or the table.

The user can also type a new value for the standardized selection gradient into the text field and then click the button labeled Rescale selection gradient. If this is done, the selection gradient is scaled to have the standardized selection differential indicated in the text field.

Clicking the Accept button will start the calculations of the predicted selection response.

If the user interface was accessed from the dialog box for requesting a new analysis, clicking the Cancel button will abort the analysis. Otherwise, if the interface was accessed from the graphical output of an existing analysis, clicking Cancel will leave the selection gradient of that analysis unchanged.

Specifying a selection differential

If the user selected to specify a selection differential, the following user interface will appear:

This interface is the same as the one for specifying a selection gradient, except for the options for scaling its magnitude.

There are two text fields for specifying the magnitude of the selection differential either in units of Procrustes distance (as an absolute shape change) or as a standardized selection differential, in standard deviations of the shape variable corresponding to s.

If a new value is entered into one of these text fields and the button Rescale selection differential is clicked, the selection differential is recalculated so that it has the magnitude indicated in the text field that was edited last.

Graphical output

The graphical output of the procedure consists of five graphs showing different shape changes:

The popup menus for the panels for the selection differential and gradient each contain an additional item, Change Selection Differential and Change Selection Gradient. When one of these is selected, the interface for specifying the respective selection vector is activated.

Text output

The output in the Results tab contains the vectors of the selection differential, selection gradient, total response, direct response and correlated response, as well as the different measures of the magnitudes of these.

References

Klingenberg, C. P., and L. J. Leamy. 2001. Quantitative genetics of geometric shape in the mouse mandible. Evolution 55:2342–2352.

Klingenberg, C. P., and L. R. Monteiro. 2005. Distances and directions in multidimensional shape spaces: implications for morphometric applications. Syst. Biol. 54:678–688.

Klingenberg, C. P., V. Debat, and D. A. Roff. 2010. Quantitative genetics of shape in cricket wings: developmental integration in a functional structure. Evolution 64:2935–2951.

Lande, R. 1979. Quantitative genetic analysis of multivariate evolution, applied to brain:body size allometry. Evolution 33:402–416.

Lande, R., and S. J. Arnold. 1983. The measurement of selection on correlated characters. Evolution 37:1210–1226.

Martínez-Abadías, N., M. Esparza, T. Sjøvold, R. González-José, M. Santos, M. Hernández, and C. P. Klingenberg. 2012. Pervasive genetic integration directs the evolution of human skull shape. Evolution 66:1010–1023.