Trade-offs are a fundamental concept in evolutionary biology because they are thought to explain much of nature’s biological diversity, from variation in life-histories to differences in metabolism. Despite the predicted importance of trade-offs, they are notoriously difficult to detect. Here we contribute to the existing rich theoretical literature on trade-offs by examining how the shape of the distribution of resources or metabolites acquired in an allocation pathway influences the strength of trade-offs between traits. We further explore how variation in resource distribution interacts with two aspects of pathway complexity (i.e., the number of branches and hierarchical structure) affects tradeoffs. We simulate variation in the shape of the distribution of a resource by sampling 10⁶ individuals from a beta distribution with varying parameters to alter the resource shape. In a simple “Y-model” allocation of resources to two traits, any variation in a resource leads to slopes less than -1, with left skewed and symmetrical distributions leading to negative relationships between traits, and highly right skewed distributions associated with positive relationships between traits. Adding more branches further weakens negative and positive relationships between traits, and the hierarchical structure of pathways typically weakens relationships between traits, although in some contexts hierarchical complexity can strengthen positive relationships between traits. Our results further illuminate how variation in the acquisition and allocation of resources, and particularly the shape of a resource distribution and how it interacts with pathway complexity, makes it challenging to detect trade-offs. We offer several practical suggestions on how to detect trade-offs given these challenges.

Overview of Flux Simulations

To study the strength and direction of trade-offs within a population, we developed a simulation of flux in a simple metabolic pathway, where a precursor metabolite emerging from node A may either be converted to metabolic products B₁ or B₂ (Fig. 1). This conception of a pathway is similar to De Jong and Van Noordwijk’s Y-model (Van Noordwijk & De Jong, 1986; De Jong & Van Noordwijk, 1992), but we used simulation instead of analytical statistical models to allow us to consider greater complexity in the distribution of variables and pathways. For a simple pathway (Fig. 1), the total flux J_total (i.e., the flux at node A, denoted as J_A) for each individual (N = 10⁶) was first sampled from a predetermined beta distribution as described below. The flux at node B₁ (J_B1) was then randomly sampled from this distribution with max = J_total = J_A and min = 0. The flux at the remaining node, B₂, was then simply the remaining flux (J_B2 = J_A - J_B1). Simulations of more complex pathways followed the same basic approach as described above, with increased numbers of branches and hierarchical levels added to the pathway as described below under Question 2. The metabolic pathways were simulated using Python (v. 3.8.2) (Van Rossum & Drake Jr., 2009) where we could control the underlying distribution of metabolite allocation. The output flux at nodes B₁ and B₂ was plotted using R (v. 4.2.1) (Team, 2022) with the resulting trade-off visualized as a linear regression using the ggplot2 R package (v. 3.4.2) (Wickham, 2016). While we have conceptualized the pathway as the flux of metabolites, it could be thought of as any resource being allocated to different traits.

Question 1: How does variation in resource distribution within a population affect the strength and direction of trade-offs?

We first simulated the simplest scenario where all individuals had the same total flux J_total = 1, in which case the phenotypic trade-off is expected to be most easily detected. We then modified this initial scenario to explore how variation in the distribution of resource acquisition (J_total) affected the strength and direction of trade-offs.  Specifically, the resource distribution was systematically varied by sampling n = 10³ total flux levels from a beta distribution, which has two parameters alpha and beta that control the size and shape of the distribution (Miller & Miller, 1999). When alpha is large and beta is small, the distribution is left skewed, whereas for small alpha and large beta, the distribution is right skewed.  Likewise, for alpha = beta, the curve is symmetrical and approximately normal when the parameters are sufficiently large (>2). We can thus systematically vary the underlying resource distribution of a population by iterating through values of alpha and beta from 0.5 to 5 (in increments of 0.5), which was done using the NumPy Python package (v. 1.19.1) (Harris et al., 2020). The resulting slope of each linear regression of the flux at B₁ and B₂ (i.e., the two branching nodes) was then calculated using the lm function in R and plotted as a contour map using the latticeExtra Rpackage (v. 0.6-30) (Sarkar, 2008).

Question 2: How does the complexity of the pathway used to produce traits affect the strength and direction of trade-offs?

Metabolic pathways are typically more complex than what is described above. Most pathways consist of multiple branch points and multiple hierarchical levels. To understand how complexity affects the ability to detect trade-offs when combined with variation in the distribution of total flux we systematically manipulated the number of branch points and hierarchical levels within pathways (Fig. 1). We first explored the effect of adding branches to the pathway from the same node, such that instead of only branching off to nodes B₁ and B₂, the pathway branched to nodes B₁ through to B_n (Fig. 1B), where n is the total number of branches (maximum n = 10 branches). Flux at a node was calculated as previously described, and the remaining flux was evenly distributed amongst the remaining nodes (i.e., nodes B₂ through to B_nwould each receive J_2-n = (J_total - J_B1)/(n - 1) flux). For each pathway, we simulated flux using a beta distribution of J_totalwith alpha = 5, beta = 0.5 to simulate a left skewed distribution, alpha = beta = 5 to simulate a normal distribution, and with alpha = 0.5, beta = 5 to simulate a right skewed distribution, as well as the simplest case where all individuals have total flux J_total = 1.

We next considered how adding hierarchical levels to a metabolic pathway affected trade-offs. We modified our initial pathway with node A branching to nodes B₁ and B₂, and then node B₂ further branched to nodes C₁ and C₂ (Fig. 1C). To compute the flux at the two new nodes C₁ and C₂, we simply repeated the same calculation as before, but using the flux at node B₂, J_B2, as the total flux. That is, the flux at node C₁ was obtained by randomly sampling from the distribution at B₂ with max = J_B and min = 0, and the flux at node C₂ is the remaining flux (J_C = J_B2 - J_C1). Much like in the previous scenario with multiple branch points, we used three beta distributions (with the same parameters as before) to represent left, normal, and right skewed resource distributions, as well as the simplest case where J_total = 1 for all individuals.

Quantile Regressions

We performed quantile regression to understand whether this approach could help to detect trade-offs. Quantile regression is a form of statistical analysis that fits a curve through upper or lower quantiles of the data to assess whether an independent variable potentially sets a lower or upper limit to a response variable (Cade et al., 1999). This type of analysis is particularly useful when it is thought that an independent variable places a constraint on a response variable, yet variation in the response variable is influenced by many additional factors that add “noise” to the data, making a simple bivariate relationship difficult to detect (Thomson et al., 1996). Quantile regression is an extension of ordinary least squares regression, which regresses the best fitting line through the 50^th percentile of the data. In addition to performing ordinary least squares regression for each pairwise comparison between the four nodes (B₁, B₂, C₁, C₂), we performed a series of quantile regressions using the ggplot2 R package (v. 3.4.2), where only the q_th quantile was used for the regression (q = 0.99 and 0.95 to 0.5 in increments of 0.05, see Fig. S1) (Cade et al., 1999).