Evolution, at all biological organizational levels (from DNA to complex morphologies or behavior), is contingent upon genetic variability. When dealing with the evolution of complex systems, like the mammalian skull, not only should genetic variation of each individual dimension be considered, but perhaps more importantly, the amount and pattern of covariation between traits (**G**-matrix) also needs to be taken into account. Genetic and phenotypic patterns of variation and covariation among traits are critical for understanding the evolution of complex systems, whatever the evolutionary processes (genetic drift or natural selection) impacting the population. The reason is straightforward: genetic covariance is produced when traits share some of their genetic basis and this covariance implies that traits cannot evolve independently (Lande 1979; Lande and Arnold 1983; Arnold et al. 2001). Therefore, **G**-matrices are the quantitative expression of constraints (Cheverud 1984; Arnold 1992), interacting with genetic drift and selection to produce evolutionary change and at same time also being shaped by those same process along with mutation (Wagner et al. 2007).

In a pathbreaking paper, Schluter (1996) examined the relationship between the **G**-matrix and evolutionary divergence. In particular, the focus was on what he called the “genetic line of least resistance” or **g _{max}** for simplicity. This line of least resistance is a single simple factor: the multivariate direction of greatest genetic (or its phenotypic surrogate) variation (

**g**or

_{max}**p**). This factor is a linear combination of a suite of morphological traits that displays the maximum within-population variance (first principal component [PC1]). It defines a line of least resistance to evolutionary change by either natural selection or genetic drift (Arnold et al. 2001) simply because it is the axis with the largest genetic variance. Schluter (1996) showed that this line of least resistance can affect both the direction and the magnitude of evolutionary change. It is also possible to extend the same rationale to other eigenvectors holding sequentially smaller portions of the total variation embodied in the

_{max}**G**-matrix (Blows and Higgie 2003; Blows et al. 2004; McGuigan et al. 2005).

One way to appreciate the effects of genetic covariance upon the magnitude and direction of evolution is portrayed in Figure 1 (modified from Arnold et al. 2001). In this hypothetical adaptive landscape, three populations differ in their current position in relation to the adaptive peak. All the three populations share the same basic genetic covariance pattern. Selection will push all the three populations to the nearest adaptive peak, but the orientation of the selection gradient will differ due to their differences in current position on the landscape. Although selection in all the three populations specifies the shortest linear path to the peak, the realized evolutionary trajectory (**Δz**) from generation to generation may be quite different. In fact, if the two axes of major genetic variance are not aligned (population C) with the direction of selection, the evolutionary response to selection will be curvilinear (Fig. 1). Furthermore, this curvilinear trajectory would be biased by the line of least resistance embodied in the **G**-matrix (gray ellipses in the figure). Because in this simple example (with only two dimensions) the first line of least resistance holds almost twice as much variation as the second one, the initial response in population C would be strongly biased in the direction of the largest genetic variance (**g _{max}**). It is also important to note how the line of least resistance influences not only the direction but also the magnitude of the evolutionary response along the path of selection. This point is made it clear when comparing populations A and B in Figure 1. The magnitude of the response in population A is much larger than in population B. This reflects the fact that in population A the first line of least resistance is aligned with the selected dimension whereas in population B the second line of least resistance is the one aligned with the direction of selection (

**β**). The difference in response magnitude between A and B due to the variance differences even overcomes differences in strength of selection due to the fact that the path between A and the peak has a shallower slope than the path between B and the peak that is steeper and therefore reflects stronger selection.

Another important question, and one that it is still underexplored in the evolutionary literature, is the relationship between the adaptive landscape and constraints embodied in the **G**-matrix (Cheverud 1984; Arnold et al. 2001; Arnold 2005). Although our comparisons of **P**- and **G**-matrices in mammals (Cheverud 1995, 1996; Marroig and Cheverud 2001; Oliveira et al. 2009; Porto et al. 2009) with an a priori hypotheses based on functional and developmental factors imply that stabilizing selection should be an important force keeping those patterns stable in mammals (at least in the skull), two fundamental processes that potentially have an impact on structuring the **P**- and **G**-matrices are not usually addressed: mutation and directional selection. The difficulties in estimating the **M**-matrix (the variance/covariance matrix of mutational effects) usually prevents it from being measured, even though it has been suggested for some time now (Cheverud 1984) that the interaction between the **M**-matrix and the **W**-matrix (the matrix describing the covariance among traits due to selection) is critical to the problem of stability or plasticity in **G**-matrices. In other words, in an adaptive radiation like the one observed in New World Monkeys (NWMs) where **P**- and **G**-matrices remain relatively stable (Cheverud 1995, 1996; Marroig and Cheverud 2001; see also below) are the long-term patterns of evolutionary change influenced by the constraints (**G**-matrix) irrespective of selection regimes, are they a result of selection and constraints operating together, or finally do both constraint and selection interact to produce evolutionary change whereas the constraints themselves are shaped by selection? If so, should we expect an alignment between them and the covariance in the net-selection gradients through time?

The present paper builds on our previous contributions (see also Material and methods below) that showed that the evolutionary divergence of cranial morphology in NWMs was not compatible with a null hypothesis of genetic drift (Marroig and Cheverud 2004). Additionally, Marroig and Cheverud (2005) showed that the line of least resistance affected both the direction and magnitude of evolutionary change in size during the adaptive radiation of NWMs. However, net-selection gradients (**β**'s) reconstructed by Marroig and Cheverud (2005) were not especially similar to size (**p _{max}**) vectors (fig. 4 in Marroig and Cheverud 2005) suggesting that the bulk of the size evolution observed in NWM crania was due to the influence of the constraints. Yet, the possibility that those net-selection gradients were not representative of the past selection regimes due to either the Turelli's effect (Turelli 1988; see also Jones et al. 2004) or the error involved in matrix estimation, in particular in the inverted matrices (G. Marroig and D. Melo, unpubl. ms.), cannot be disregarded. The so-called Turelli's effect could happen if there is stochastic variation in the

**G**-matrix and a covariance between the

**G**-matrix and the net-selection gradient (Turelli 1988; Jones et al. 2004). However, Jones et al. (2004) present evidence that usually this effect will be small. Furthermore, although the observed macroevolutionary pattern of changes along

**g**could be accounted for by the conservation and influence of the line of least resistance, there was also good evidence that dietary diversification might have been the ecological factor driving such diversification through natural selection on size due to metabolic and foraging restrictions (Marroig and Cheverud 2004, 2005). This correlation between size and diet suggests an alignment of adaptive peaks with size during NWM diversification and raises the question of whether it is possible to test whether or not size evolution is a direct consequence of selection on size (selection + constraints hypothesis) or is an indirect consequence of

_{max}**g**redirecting evolutionary response (constraint hypothesis; Björklund and Merilä 1994). Here an approach based on extensive simulations applying the multivariate response to selection equation (

_{max}**Δz**=

**Gβ**) is used to evaluate these two hypotheses; specifically, looking at the direction and magnitude of evolutionary response under two distinct simulated selection scenarios, nonsize selection and size selection. We also introduce a benchmark to compare the observed amount of evolutionary change in size (or any other multivariate dimension) among groups and infer whether size was under direct selection or not. This benchmark is equal to the correlation of the direction of selection and the line of least resistance when both vectors are normalized to a length of one. Our results show that both constraints alone and selection aligned with constraints can produce evolutionary change aligned with

**g**. However, there is substantial difference in the alignment between evolutionary responses and

_{max}**g**when the two hypotheses are compared in terms of the variance of the two distributions. Also, the magnitude of the evolutionary change along the line of least resistance produced by these two processes is quite different allowing tests of whether selection was operating directly on size or size evolution was an indirect consequence of the attractor effect of the line of least resistance.

_{max}